AP Calculus BC · ⁨AP 미적분학 BC⁩

Tips · ⁨팁⁩

AP Calculus BC contains all of AB and adds parametric, polar and vector-valued functions, advanced integration techniques, improper integrals, logistic growth, and — the largest addition — infinite sequences and series, including Taylor and Maclaurin series with error bounds.

Series is where BC is won or lost. The convergence tests must be chosen, not tried in turn: given a series, you should know within seconds which test the form points to, and be able to name the test and verify its conditions.

BC also reports an AB subscore, so the AB material stays fully examinable and must not be left behind.

The notes cover the AB and BC material in CED order, giving series the space it needs. Released past papers are in the library. Since the AB subscore comes from the same paper, the AB units stay in the notes at full depth rather than being summarised away.

Train all vocabulary · ⁨전체 어휘 연습⁩

Past Papers · ⁨과거 시험지⁩

Limits and Continuity · ⁨한계와 연속성⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

1.1

Introducing Calculus: Can Change Occur at an Instant?

Syllabus

English

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-1 Calculus allows us to generalize knowledge about motion to diverse problems involving change.	CHA-1.A Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.	CHA-1.A.1 Calculus uses limits to understand and model dynamic change. CHA-1.A.2 Because an average rate of change divides the change in one variable by the change in another, the average rate of change is undefined at a point where the change in the independent variable would be zero. CHA-1.A.3 The limit concept allows us to define instantaneous rate of change in terms of average rates of change.

한국어

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-1 미적분을 이용하면 운동에 대한 지식을 변화와 관련된 다양한 문제에 generalize(확장·적용)할 수 있다.	CHA-1.A 특정 instant(순간)에서의 rate of change(변화율)를 해당 instant를 포함하는 intervals(구간)에서의 average rates of change(평균 변화율)로 해석한다.	CHA-1.A.1 미적분은 limits(한계)를 사용하여 dynamic change(동적 변화)를 이해하고 model(모형화)하는 데 사용된다. CHA-1.A.2 평균 변화율은 한 변수의 변화를 다른 변수의 변화로 나누어 정의되므로, 독립 변수의 변화량이 0이 되는 점에서는 평균 변화율이 undefined(정의되지 않음)이다. CHA-1.A.3 limit 개념을 이용하면 instantaneous rate of change(순간 변화율)를 평균 변화율을 통해 정의할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Calculus is the mathematics of change 变化 and of accumulation 累积. It answers two big questions: how fast is something changing right now, and how much has piled up so far? Unit 1 builds the one tool both questions rest on – the limit 极限.

Start with a puzzle. A car's speedometer reads $60$ km/h. What does that mean at a single instant 瞬间? Speed is distance over time. But at one instant no time passes and no distance is covered, so the fraction looks like $\tfrac{0}{0}$ – undefined.

The average rate of change 平均变化率 uses a whole interval 区间: the change in one quantity divided by the change in another. It divides by zero, and so is undefined, when the change in the input would be zero.
The instantaneous rate of change 瞬时变化率 is what we want at a point. It is the value the average rate approaches 趋近 as the interval shrinks toward zero length.

The clever move is not to plug in zero (undefined), but to watch what the average rate approaches as the interval gets smaller and smaller. That approaching value is a limit. So calculus lets us describe change at an instant – as a limit of average rates over ever-shorter intervals. This one idea powers the derivative 导数 (Unit 2) and, run in reverse, the integral 积分 (Unit 6). Everything else in this unit defines limits carefully and computes them reliably.

Explore · ⁨탐색하기⁩

Explore the slope at an instant

y = bx² + d

Slide the point along the curve. The tangent line shows the exact rate of change $\frac{dy}{dx}$ there — the value the average rates approach as the interval shrinks to a single instant. The slope changes with position, so change does have a value at each instant.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
change/tʃeɪndʒ/	변화
accumulation/əˌkjuːmjʊˈleɪʃn/	적분함수(Accumulation function)
limit/ˈlɪmɪt/	한계
at a single instant/ætə ˈsɪŋɡl ˈɪnstənt/	단순 순간에
average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/	평균 변화율
interval/ˈɪntəvl/	구간
instantaneous rate of change/ˌɪnstənˈteɪnɪəs reɪt ɒv tʃeɪndʒ/	순간 변화율
approaches/əˈprəʊtʃɪz/	근사법
derivative/dɪˈrɪvətɪv/	미분Unless derivative.
integral/ˈɪntɪɡrəl/	적분Unless integral.
hole/həʊl/	구멍

1.2

Defining Limits and Using Limit Notation

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.	LIM-1.A Represent limits analytically using correct notation.	LIM-1.A.1 Given a function $f$, the limit of $f(x)$ as $x$ approaches $c$ is a real number $R$ if $f(x)$ can be made arbitrarily close to $R$ by taking $x$ sufficiently close to $c$ (but not equal to $c$). If the limit exists and is a real number, then the common notation is $\lim_{x \to c} f(x) = R$. Exclusion statement: The epsilon-delta definition of a limit is not assessed on the AP Calculus AB or BC Exam. However, teachers may include this topic in the course if time permits.
	LIM-1.B Interpret limits expressed in analytic notation.	LIM-1.B.1 A limit can be expressed in multiple ways, including graphically, numerically, and analytically.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A smooth bridge curve: limits describe the value a graph approaches as we zoom in

Given a function $f$, the limit of $f(x)$ as $x$ approaches $c$ is a real number $R$ if $f(x)$ can be made arbitrarily 任意地 close to $R$ by taking $x$ sufficiently 足够 close to $c$ – but not equal to $c$. We write

$$\lim_{x \to c} f(x) = R$$

and read it: "the limit of $f(x)$, as $x$ approaches $c$, equals $R$."

The last words are the heart of a limit: it describes the behavior 行为 of $f$ near $c$, not the value at $c$. The function may be undefined at $c$, or defined but equal to something else – the limit does not care.

A limit can be shown in three ways: graphically 用图象, numerically 用数值 (a table), and analytically 用解析式 (algebra). Learning to move between these representations is a core skill.

(Note: the epsilon-delta definition of a limit is not tested on the AP Exam, so this handout does not use it.)

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
arbitrarily/ˌɑːbɪˈtrerɪli/	임의로
sufficiently/səˈfɪʃəntli/	충분히
behavior/bɪˈheɪvjə/	동태/성격(behavior)
graphically/ˈɡræfɪkli/	그래프적으로
numerically/njuːˈmerɪkli/	숫자적으로
analytically/ˌænəˈlɪtɪkli/	분석적으로

1.3

Estimating Limit Values from Graphs

Syllabus

English

Enduring Understanding (LIM-1): Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Learning Objective LIM-1.C: Estimate limits of functions.

LIM-1.C.1 The concept of a limit includes one sided limits.
LIM-1.C.2 Graphical information about a function can be used to estimate limits.
LIM-1.C.3 Because of issues of scale, graphical representations of functions may miss important function behavior.
LIM-1.C.4 A limit might not exist for some functions at particular values of $x$. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.
- Illustrative examples for LIM-1.C.4:
  - $\lim_{x \to 0} \dfrac{1}{x^2} = \infty$
  - $\lim_{x \to 0} \dfrac{|x|}{x}$ does not exist.
  - $\lim_{x \to 0} \sin\left(\dfrac{1}{x}\right)$ does not exist.
  - $\lim_{x \to 0} \dfrac{1}{x}$ does not exist.

한국어

지속적 이해(LIM-1): 정의, 정리, 성질을 이용한 추론을 통해 극한에 대한 주장을 정당화할 수 있습니다.

학습 목표 LIM-1.C: 함수의 극한을 추정합니다.

LIM-1.C.1 극한의 개념에는 양측 극한이 포함됩니다.
LIM-1.C.2 함수에 대한 그래프 정보는 극한을 추정하는 데 사용할 수 있습니다.
LIM-1.C.3 스케일 문제 때문에 함수의 그래프 표현은 중요한 함수 행동을 놓칠 수 있습니다.
LIM-1.C.4 특정 $x$ 값에서 일부 함수는 극한이 존재하지 않을 수 있습니다. 극한이 존재하지 않는 경우로는 함수가 무한대인 경우, 이 값 근처에서 진동하는 경우, 좌극한과 우극한이 서로 다른 경우가 포함됩니다.
- LIM-1.C.4의 예시:
  - $\lim_{x \to 0} \dfrac{1}{x^2} = \infty$
  - $\lim_{x \to 0} \dfrac{|x|}{x}$은 존재하지 않습니다.
  - $\lim_{x \to 0} \sin\left(\dfrac{1}{x}\right)$은 존재하지 않습니다.
  - $\lim_{x \to 0} \dfrac{1}{x}$은 존재하지 않습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A graph is often the fastest way to read a limit. To find $\displaystyle \lim_{x \to c} f(x)$, run your finger along the curve toward $x = c$ from each side and ask: what height is the curve heading for?

Trace from the left (inputs smaller than $c$): this gives the left-hand limit 左极限, $\displaystyle \lim_{x \to c^-} f(x)$.
Trace from the right (inputs larger than $c$): this gives the right-hand limit 右极限, $\displaystyle \lim_{x \to c^+} f(x)$.
These are the one-sided limits 单侧极限. If both head to the same height $R$, then the two-sided limit exists and $\displaystyle \lim_{x \to c} f(x) = R$.

Crucially, ignore the point itself. Graphs mark the difference between the limit and the value:

An open circle 空心圆 marks a height the curve approaches but does not reach – a "hole" 空洞.
A closed circle 实心圆 marks the actual value $f(c)$.

So a curve may approach $R = 3$ from both sides (limit is $3$) while a filled dot sits at height $5$ (value $f(c) = 5$). The limit is $3$; the two need not match.

A limit does not exist (often written DNE) when the two sides disagree (a jump 跳跃), when the function is unbounded 无界 (grows without limit), or when it oscillates 振荡 forever near $c$. For example:

$$\lim_{x \to 0} \frac{1}{x^2} = \infty, \qquad \lim_{x \to 0} \frac{|x|}{x}\ \text{DNE}, \qquad \lim_{x \to 0} \sin\!\left(\frac{1}{x}\right)\ \text{DNE}.$$

Watch the scale 比例 of a graph: a zoomed-out picture can hide important behavior near a point, so confirm with algebra when you can.

A limit exists at x=2 even though the function value f(2) is different — The open circle is the height the curve approaches (the limit); the filled dot is the actual value $f(2)$ -- they need not agree.

Explore · ⁨탐색하기⁩

Read a limit off the graph

y = ax² + bx + c

The limit as $x\to c$ is the height the curve heads toward from both sides — it is about where the function is going, not its value at $c$. Follow the curve toward an $x$ and read the $y$ it approaches.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
left-hand limit/left hænd ˈlɪmɪt/	좌측 한계
right-hand limit/raɪt hænd ˈlɪmɪt/	우측 한계
one-sided limits/wʌn ˈsaɪdɪd ˈlɪmɪts/	일변한 극한
open circle/ˈəʊpən ˈsɜːkl/	빈 원
closed circle/kləʊzd ˈsɜːkl/	닫힌 원
jump/dʒʌmp/	점프
unbounded/ʌnˈbaʊndɪd/	무한대
oscillates/ˈɒsɪleɪts/	진동한다
scale/skeɪl/	스케일(scale)

1.4

Estimating Limit Values from Tables

Syllabus

English

Enduring Understanding (LIM-1): Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Learning Objective LIM-1.C: Estimate limits of functions.

LIM-1.C.5 Numerical information can be used to estimate limits.

한국어

지속적 이해(LIM-1): 정의, 정리, 성질을 이용한 추론을 통해 극한에 대한 주장을 정당화할 수 있습니다.

학습 목표 LIM-1.C: 함수의 극한을 추정합니다.

LIM-1.C.5 수치 정보를 사용하여 극한을 추정할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

When you have data or a formula but no picture, a table 表格 of values estimates a limit numerically. Choose inputs that creep toward $c$ from both sides and watch the outputs.

For example, to estimate $\displaystyle \lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ (which is $\tfrac{0}{0}$ at $x=2$):

$x$	$1.9$	$1.99$	$1.999$	$\to 2 \leftarrow$	$2.001$	$2.01$	$2.1$
$f(x)$	$3.9$	$3.99$	$3.999$	?	$4.001$	$4.01$	$4.1$

Both sides march toward $4$, so we estimate the limit is $4$. A table only suggests a value – it is a numerical estimate, not a proof.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
table/ˈteɪbl/	표

1.5

Determining Limits Using Algebraic Properties of Limits

Syllabus

English

Enduring Understanding (LIM-1): Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Learning Objective LIM-1.D: Determine the limits of functions using limit theorems.

LIM-1.D.1 One-sided limits can be determined analytically or graphically.
LIM-1.D.2 Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.

한국어

지속적 이해(LIM-1): 정의, 정리, 성질을 이용한 추론을 통해 극한에 대한 주장을 정당화할 수 있습니다.

학습 목표 LIM-1.D: 극한 정리를 사용하여 함수의 극한을 결정합니다.

LIM-1.D.1 양측 극한은 해석적 또는 기하적으로 결정할 수 있습니다.
LIM-1.D.2 합, 차, 곱, 상, 합성함수의 극한은 극한 정리를 통해 구할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Most limits are found analytically using limit theorems 极限定理. If $\lim_{x\to c} f(x)$ and $\lim_{x\to c} g(x)$ both exist, the limit of a combination is the same combination of the limits:

Sum / difference: $\displaystyle \lim_{x\to c}\big[f(x)\pm g(x)\big] = \lim_{x\to c}f(x) \pm \lim_{x\to c}g(x)$
Product: $\displaystyle \lim_{x\to c}\big[f(x)\,g(x)\big] = \lim_{x\to c}f(x)\cdot\lim_{x\to c}g(x)$
Quotient: $\displaystyle \lim_{x\to c}\frac{f(x)}{g(x)} = \frac{\lim_{x\to c}f(x)}{\lim_{x\to c}g(x)}$, provided the bottom limit is not $0$.
Composite 复合函数: if $g$ is continuous at $\lim_{x\to c} f(x)$, then $\displaystyle \lim_{x\to c} g\big(f(x)\big) = g\!\left(\lim_{x\to c} f(x)\right)$.

The practical rule: for a function built from polynomials, roots, and the like, first try direct substitution 直接代入 – put $x = c$ in. If you get a real number, that is the limit. One-sided limits obey the same theorems, read from one direction only.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
limit theorems/ˈlɪmɪt ˈθɪərəmz/	극한 정리
Composite/ˈkɒmpəzɪt/	합성체
direct substitution/daɪˈrekt ˌsʌbstɪˈtjuːʃn/	직접 치환

1.6

Determining Limits Using Algebraic Manipulation

Syllabus

English

Enduring Understanding (LIM-1): Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Learning Objective LIM-1.E: Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.

LIM-1.E.1 It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.
- Illustrative examples for LIM-1.E.1:
  - Factoring and dividing common factors of rational functions
  - Multiplying by an expression involving the conjugate of a sum or difference in order to simplify functions involving radicals
  - Using alternate forms of trigonometric functions

한국어

지속적 이해(LIM-1): 정의, 정리, 성질을 이용한 추론을 통해 극한에 대한 주장을 정당화할 수 있습니다.

학습 목표 LIM-1.E: 함수의 등가 표현이나挟定理(Squeeze Theorem)을 사용하여 함수의 극한을 결정합니다.

LIM-1.E.1 극한을 평가하기 전에 식을 등가 형태로 재배열해야 하거나 도움이 될 수 있습니다.
- LIM-1.E.1의 예시:
  - 유리함수의 공통 인수를 인수분해하여 나누기
  - 근호를 포함하는 함수를 단순화하기 위해 합이나 차의 켤레를 포함한 식을 곱하기
  - 삼각함수의 대안적 형식 사용

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Direct substitution sometimes gives the indeterminate form 未定式 $\tfrac{0}{0}$. This does not mean the limit fails – it means you must rewrite the function into an equivalent form 等价形式 that removes the trouble, then substitute. Three standard moves:

Factor and cancel 因式分解并约分 a rational function 有理函数. Example: $\displaystyle \lim_{x\to 2}\frac{x^2-4}{x-2} = \lim_{x\to 2}\frac{(x-2)(x+2)}{x-2} = \lim_{x\to 2}(x+2) = 4$.
Multiply by the conjugate 共轭 to simplify a radical 根式. Example: $\displaystyle \lim_{x\to 0}\frac{\sqrt{x+1}-1}{x} = \lim_{x\to 0}\frac{x}{x\big(\sqrt{x+1}+1\big)} = \frac{1}{2}$.
Use alternate forms of trigonometric functions (identities) to simplify.

The cancelled factor is why the original graph had a hole: the two functions agree everywhere except at $x=c$, so they share the same limit there.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
indeterminate form/ˌɪndɪˈtɜːmɪnət fɔːm/	부정정 형식
equivalent form/ɪˈkwɪvələnt fɔːm/	등가 형태
Factor and cancel/ˈfæktə ænd ˈkænsl/	인수분해 및 약분
rational function/ˈræʃənl ˈfʌŋkʃn/	유계 함수
conjugate/ˈkɒndʒuːɡeɪt/	상쇄 산/염기
radical/ˈrædɪkl/	급진적

1.7

Selecting Procedures for Determining Limits

Syllabus

English

This topic is intended to focus on the skill of selecting an appropriate procedure for determining limits. Students should be given opportunities to practice when and how to apply all learning objectives relating to determining limits.

한국어

이 주제는 적절한 극한 결정 절차를 선택하는 기술에 집중하도록 설계되었습니다. 학생들은 극한 결정과 관련된 모든 학습 목표를 언제와 어떻게 적용할지 연습할 기회를 제공받아야 합니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

This is a skill topic, not new content: choose the right tool for the limit in front of you.

Try direct substitution first. A real answer means you are done.
Getting $\tfrac{0}{0}$? Rewrite – factor and cancel, or use the conjugate, or a trig identity – then substitute.
A non-zero number over $0$ (like $\tfrac{5}{0}$)? The limit is infinite or DNE – check the sign from each side (see vertical asymptotes below).
As $x\to\pm\infty$? Compare the dominant 主导 terms (see limits at infinity).
Trapped between two functions? The squeeze theorem may apply.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
dominant/ˈdɒmɪnənt/	주요(dominant)

1.8

Determining Limits Using the Squeeze Theorem

Syllabus

English

Enduring Understanding (LIM-1): Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Learning Objective LIM-1.E: Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.

LIM-1.E.2 The limit of a function may be found by using the squeeze theorem.
- Illustrative examples for LIM-1.E.2: The squeeze theorem can be used to show $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ and $\lim_{x \to 0} \dfrac{1 - \cos x}{x} = 0$.

한국어

지속적 이해(LIM-1): 정의, 정리, 성질을 이용한 추론을 통해 극한에 대한 주장을 정당화할 수 있습니다.

학습 목표 LIM-1.E: 함수의 등가 표현이나挟定理(Squeeze Theorem)을 사용하여 함수의 극한을 결정합니다.

LIM-1.E.2 함수의 극한은挟定理를 사용하여 구할 수 있습니다.
- LIM-1.E.2의 예시: 挟定理를 사용하여 $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$과 $\lim_{x \to 0} \dfrac{1 - \cos x}{x} = 0$임을 보여줄 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The squeeze theorem 夹逼定理 (also called the sandwich theorem) finds a limit by trapping the function between two others. If $g(x) \le f(x) \le h(x)$ near $c$, and

$$\lim_{x\to c} g(x) = \lim_{x\to c} h(x) = L,$$

then $f$ is squeezed to the same place: $\displaystyle \lim_{x\to c} f(x) = L$.

The two famous results proved this way, both used throughout calculus, are:

$$\lim_{x\to 0}\frac{\sin x}{x} = 1 \qquad\text{and}\qquad \lim_{x\to 0}\frac{1-\cos x}{x} = 0.$$

The Squeeze Theorem traps x squared sin(1/x) between minus x squared and x squared

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
squeeze theorem/skwiːz ˈθɪərəm/	sandwich 정리

1.9

Connecting Multiple Representations of Limits

Syllabus

English

This topic is intended to focus on connecting representations. Students should be given opportunities to practice when and how to apply all learning objectives relating to limits and translating mathematical information from a single representation or across multiple representations.

한국어

이 주제는 표현 방식 간의 연결에 집중하도록 설계되었습니다. 학생들은 극한과 관련된 모든 학습 목표를 언제와 어떻게 적용하고, 단일 표현 또는 여러 표현 간에 있는 수학 정보를 변환하는 법을 연습할 기회를 제공받아야 합니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Another skill topic: the same limit lives in a graph, a table, and an algebraic form, and you should be able to translate between them. A graph shows the shape and any holes or jumps; a table gives numerical evidence; algebra gives an exact value and a reason. Strong answers use one representation to confirm another.

1.10

Exploring Types of Discontinuities

Syllabus

English

Enduring Understanding (LIM-2): Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Learning Objective LIM-2.A: Justify conclusions about continuity at a point using the definition.

LIM-2.A.1 Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

한국어

지속적 이해 (LIM-2): 정의, 정리, 성질을 활용한 추론을 통해 연속에 대한 주장을 정당화할 수 있다.

학습 목표 LIM-2.A: 정의에 근거하여 특정 점에서의 연속성에 대한 결론을 정당화한다.

LIM-2.A.1 불연속의 종류로는 제거 가능한 불연속, 점프 불연속, 수직 asymptote로 인한 불연속이 포함된다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A function is discontinuous 间断 at $c$ when its graph "breaks" there. There are three types:

Removable discontinuity 可去间断 – a single hole. The two-sided limit exists, but the point is missing or misplaced.
Jump discontinuity 跳跃间断 – the two one-sided limits exist but disagree, so the curve jumps.
Infinite discontinuity 无穷间断 – the function blows up to $\pm\infty$ at a vertical asymptote 垂直渐近线.

The three types of discontinuity: removable, jump, and infinite

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
discontinuous/dɪskənˈtɪnjuːəs/	불연속적
Removable discontinuity/rɪˈmuːvəbl dɪskɒntɪˈnjuːɪti/	제거 가능한 불연속점
Jump discontinuity/dʒʌmp dɪskɒntɪˈnjuːɪti/	약점 불연속점
Infinite discontinuity/ˈɪnfɪnət dɪskɒntɪˈnjuːɪti/	무한 불연속점
vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/	수직 점근선

1.11

Defining Continuity at a Point

Syllabus

English

Enduring Understanding (LIM-2): Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Learning Objective LIM-2.A: Justify conclusions about continuity at a point using the definition.

LIM-2.A.2 A function $f$ is continuous at $x = c$ provided that $f(c)$ exists, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$.

한국어

지속적 이해 (LIM-2): 정의, 정리, 성질을 활용한 추론을 통해 연속에 대한 주장을 정당화할 수 있다.

학습 목표 LIM-2.A: 정의에 근거하여 특정 점에서의 연속성에 대한 결론을 정당화한다.

LIM-2.A.2 함수 $f$가 $x = c$에서 연속이 되기 위해서는 $f(c)$가 존재하고, $\lim_{x \to c} f(x)$가 존재하며, $\lim_{x \to c} f(x) = f(c)$이어야 한다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Continuity is defined by a three-part test. A function $f$ is continuous 连续 at $x=c$ exactly when all three hold:

$$\boxed{\;f(c)\text{ exists}\quad\text{and}\quad \lim_{x\to c} f(x)\text{ exists}\quad\text{and}\quad \lim_{x\to c} f(x) = f(c)\;}$$

In words: the point is there, the limit is there, and the two agree. If any one fails, $f$ is discontinuous at $c$. This test is the backbone of nearly every continuity question, so learn it as a checklist.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
continuous/kənˈtɪnjuːəs/	연속적

1.12

Confirming Continuity over an Interval

Syllabus

English

Enduring Understanding (LIM-2): Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Learning Objective LIM-2.B: Determine intervals over which a function is continuous.

LIM-2.B.1 A function is continuous on an interval if the function is continuous at each point in the interval.
LIM-2.B.2 Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.

한국어

지속적 이해 (LIM-2): 정의, 정리, 성질을 활용한 추론을 통해 연속에 대한 주장을 정당화할 수 있다.

학습 목표 LIM-2.B: 함수가 연속인 구間을 결정한다.

LIM-2.B.1 함수가 어떤 구간의 모든 점에서 연속이라면, 그 구간 전체에서 연속이라고 한다.
LIM-2.B.2 다항함수,有理함수, 지수함수, 로그함수, 삼각함수는 도메인 내 모든 점에서 연속이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A function is continuous on an interval 在区间上连续 if it is continuous at every point of that interval. You rarely check point by point, because whole families are continuous on their domains:

Polynomial, rational, power, exponential 指数, logarithmic 对数, and trigonometric 三角 functions are continuous at every point of their domains.

So a rational function is continuous everywhere except where its denominator is zero; $\ln x$ is continuous for $x>0$; and so on. Knowing this lets you declare continuity quickly and correctly.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
continuous on an interval/kənˈtɪnjuːəs ɒn ən ˈɪntəvl/	구간에 대해 연속함
exponential/ˌekspəˈnenʃl/	지수Unless exponential function.
logarithmic/ˌlɒɡəˈrɪθmɪk/	로그함수
trigonometric/ˌtrɪɡənəʊˈmetrɪk/	삼각함수

1.13

Removing Discontinuities

Syllabus

English

Enduring Understanding (LIM-2): Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Learning Objective LIM-2.C: Determine values of $x$ or solve for parameters that make discontinuous functions continuous, if possible.

LIM-2.C.1 If the limit of a function exists at a discontinuity in its graph, then it is possible to remove the discontinuity by defining or redefining the value of the function at that point, so it equals the value of the limit of the function as $x$ approaches that point.
LIM-2.C.2 In order for a piecewise-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on one side of the boundary must equal the value of the expression defining the other side of the boundary, as well as the value of the function at the boundary.

한국어

지속적 이해 (LIM-2): 정의, 정리, 성질을 활용한 추론을 통해 연속에 대한 주장을 정당화할 수 있다.

학습 목표 LIM-2.C: $x$의 값을 결정하거나, 불연속인 함수를 연속으로 만들 수 있는 매개변수를 구한다(가능할 경우).

LIM-2.C.1 함수의 그래프 상에서 불연속점에서의 함수의 극한이 존재할 경우, 그 점에서의 함수 값을 정의하거나 재정의하여 극한의 값과 같게 함으로써 불연속성을 제거할 수 있다. 이때 $x$이 그 점에 접근할 때의 함수의 극한 값과 같아진다.
LIM-2.C.2分区-defined 함수가 도메인 partition의 경계에서 연속이 되려면, 경계의 한 측면을 정의하는 식의 값이 다른 측면을 정의하는 식의 값 및 경계에서의 함수 값과 모두 같아야 한다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

If the limit exists at a hole, the discontinuity is removable: redefine the function at that one point to equal the limit, and the graph is repaired. Formally, set the missing value to $\displaystyle \lim_{x\to c} f(x)$.

For a piecewise-defined function 分段函数, continuity at a boundary $x=c$ needs the two pieces to meet: the left piece's value, the right piece's value, and $f(c)$ must all be equal. This is a common exam setup – you solve for a parameter 参数 (an unknown constant) that makes the pieces match:

$$\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = f(c).$$

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
piecewise-defined function/ˈpiːswaɪz dɪˈfaɪnd ˈfʌŋkʃn/	부분 정의 함수
parameter/pəˈræmɪtə/	파라미터

1.14

Connecting Infinite Limits and Vertical Asymptotes

Syllabus

English

Enduring Understanding (LIM-2): Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Learning Objective LIM-2.D: Interpret the behavior of functions using limits involving infinity.

LIM-2.D.1 The concept of a limit can be extended to include infinite limits.
LIM-2.D.2 Asymptotic and unbounded behavior of functions can be described and explained using limits.

한국어

지속적 이해 (LIM-2): 정의, 정리, 성질을 활용한 추론을 통해 연속에 대한 주장을 정당화할 수 있다.

학습 목표 LIM-2.D: 무한을 포함한 극한을 사용하여 함수의 거동을 해석한다.

LIM-2.D.1 극한의 개념은 무한극한을 포함하도록 확장될 수 있다.
LIM-2.D.2 함수의 점근적 거동 및 비유계 거동은 극한을 사용하여 설명하고 해석할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The idea of a limit extends to infinite limits 无穷极限. When a function grows without bound near $x=c$, we write $\lim_{x\to c} f(x) = \pm\infty$. This describes a vertical asymptote at $x=c$: the graph hugs the vertical line $x=c$ and shoots off toward $\pm\infty$.

This happens where a non-zero number is divided by something approaching $0$, such as at a zero of a denominator that does not cancel. Always check each side separately – the two sides can shoot opposite ways (one to $+\infty$, one to $-\infty$).

An infinite limit at a vertical asymptote x = c, where the two sides shoot to opposite infinities — Near a vertical asymptote the graph hugs the line $x=c$, and the two sides can shoot to opposite infinities.

Explore · ⁨탐색하기⁩

Explore an infinite limit at a vertical asymptote

y = a/(x − b) + c

As $x \to 0$ the curve $y=\frac{1}{x}$ shoots to $+\infty$ from the right and $-\infty$ from the left — the line $x=0$ is a vertical asymptote the graph hugs but never touches.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
infinite limits/ˈɪnfɪnət ˈlɪmɪts/	무한 극한

1.15

Connecting Limits at Infinity and Horizontal Asymptotes

Syllabus

English

Enduring Understanding (LIM-2): Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Learning Objective LIM-2.D: Interpret the behavior of functions using limits involving infinity.

LIM-2.D.3 The concept of a limit can be extended to include limits at infinity.
LIM-2.D.4 Limits at infinity describe end behavior.
LIM-2.D.5 Relative magnitudes of functions and their rates of change can be compared using limits.

한국어

지속적 이해 (LIM-2): 정의, 정리, 성질을 활용한 추론을 통해 연속에 대한 주장을 정당화할 수 있다.

학습 목표 LIM-2.D: 무한을 포함한 극한을 사용하여 함수의 거동을 해석한다.

LIM-2.D.3 극한의 개념은 무한에서의 극한을 포함하도록 확장될 수 있다.
LIM-2.D.4 무한에서의 극한은 끝행동을 설명합니다.
LIM-2.D.5 함수와 그 변화율의 상대적 크기는 극한을 사용하여 비교할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

We can also let the input grow: limits at infinity 无穷远处的极限 describe the end behavior 末端行为 of a function as $x\to\pm\infty$. If the outputs settle toward a finite value $L$, then $y=L$ is a horizontal asymptote 水平渐近线.

For a rational function, compare the degrees 次数 of the top and bottom:

top degree < bottom degree $\Rightarrow$ limit is $0$ (asymptote $y=0$);
top degree = bottom degree $\Rightarrow$ limit is the ratio of the leading coefficients 首项系数之比;
top degree > bottom degree $\Rightarrow$ the function is unbounded (no horizontal asymptote).

More generally, we compare the relative magnitudes 相对大小 (relative growth rates) of functions: far out, an exponential beats any polynomial, and a polynomial beats any logarithm. On the exam, "as $t\to\infty$, which quantity is larger/where does the rate settle?" is answered with a limit at infinity.

A limit at infinity produces a horizontal asymptote

Explore · ⁨탐색하기⁩

Explore end behavior and a horizontal asymptote

y = a/(x − b) + c

Far out to the left and right the curve levels off toward $y=\mathbf{c}$ — that is $\lim_{x\to\pm\infty}f(x)$, the horizontal asymptote. Change $\mathbf{c}$ to move the level it settles at.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
limits at infinity/ˈlɪmɪts æt ɪnˈfɪnɪti/	무한에서의 한계
end behavior/end bɪˈheɪvjə/	말단挙動
horizontal asymptote/ˌhɒrɪˈzɒntl ˈæsɪmptəʊt/	수평 점근선
degrees/dɪˈɡriːz/	도
ratio of the leading coefficients/ˈreɪʃɪəʊ ɒvðə ˈliːdɪŋ ˌkəʊɪˈfɪʃənts/	최고차 계수의 비율
relative magnitudes/ˈrelətɪv ˈmæɡnɪtjuːdz/	상대 크기

1.16

Working with the Intermediate Value Theorem (IVT)

Syllabus

English

Enduring Understanding (FUN-1): Existence theorems allow us to draw conclusions about a function's behavior on an interval without precisely locating that behavior.

Learning Objective FUN-1.A: Explain the behavior of a function on an interval using the Intermediate Value Theorem.

FUN-1.A.1 If $f$ is a continuous function on the closed interval $[a, b]$ and $d$ is a number between $f(a)$ and $f(b)$, then the Intermediate Value Theorem guarantees that there is at least one number $c$ between $a$ and $b$, such that $f(c) = d$.

한국어

지속적 이해(FUN-1): 존재 정리는 특정 구간에서 함수의 행동을 정확히 locating하지 않고도 결론을 내릴 수 있게 해줍니다.

학습 목표 FUN-1.A: 중간값 정리를 사용하여 한 구간에서 함수의 행동을 설명합니다.

FUN-1.A.1 $f$가 닫힌 구간 $[a, b]$에서 연속인 함수이고, $d$가 $f(a)$와 $f(b)$ 사이의 실수라면, 중간값 정리는 적어도 하나의 숫자 $c$이 $a$과 $b$ 사이에 존재하여 $f(c) = d$이 성립함을 보장합니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The Intermediate Value Theorem 介值定理 is an existence theorem 存在性定理 – it guarantees a value exists without telling you where:

Opposite signs of f(a) and f(b) trap a root between a and b

If $f$ is continuous on the closed interval $[a,b]$, and $d$ is any number between $f(a)$ and $f(b)$, then there is at least one number $c$ in $(a,b)$ with $f(c)=d$.

An unbroken curve cannot skip a height between its endpoints – it must pass through every one.

Exam skill – how to justify with the IVT. These questions appear almost every year (for example, "Must there be a value $c$ with $R(c)=155$?" or "Is there a time when $r'(t)=-6$?"). A full-credit justification has three moves:

State continuity. Say the function is continuous on $[a,b]$ (often because it is differentiable, or given continuous).
Show $d$ is trapped. Compute the two endpoint values and show the target $d$ lies between them, e.g. $f(a) < d < f(b)$.
Conclude by name. "By the Intermediate Value Theorem, there is a $c$ in $(a,b)$ with $f(c)=d$."

Skipping the continuity statement, or not showing $d$ is between the endpoints, loses the point – the theorem requires both conditions.

Worked example. Evaluate $\lim_{x\to\infty}\dfrac{3x^2-5}{2x^2+x}$. Divide top and bottom by the highest power, $x^2$: $\dfrac{3-5/x^2}{2+1/x}\to\dfrac{3-0}{2+0}=\dfrac{3}{2}$. Because the limit is a finite number, the line $y=\tfrac{3}{2}$ is a horizontal asymptote of the graph.

The Intermediate Value Theorem: a continuous curve hits every height between f(a) and f(b) — A continuous curve from $(a,f(a))$ to $(b,f(b))$ must cross every height $d$ in between at least once.

Explore · ⁨탐색하기⁩

Why a continuous curve can't skip a value

y = ax³ + bx² + cx + d

The Intermediate Value Theorem: a function continuous on $[a,b]$ takes every $y$ between $f(a)$ and $f(b)$ at some point inside. An unbroken curve cannot leap over a height — it must pass through it.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Intermediate Value Theorem/ˌɪntəˈmiːdɪət ˈvæljuː ˈθɪərəm/	중간값 정리
existence theorem/eɡˈzɪstəns ˈθɪərəm/	존재 정리

1.16

Exam tips

A limit describes what $f(x)$ approaches, which need not equal $f(a)$ — the two-sided limit exists only if both sides agree.
Try direct substitution first; for a $\tfrac00$ form, factor and cancel or rationalise before substituting.
A function is continuous at $a$ when the limit exists, $f(a)$ is defined, and they are equal.
Use the Intermediate Value Theorem to guarantee a root: a continuous function that changes sign on $[a,b]$ takes every value between.
Read horizontal asymptotes from end behaviour (limits at $\pm\infty$) and vertical asymptotes where the denominator (not the numerator) is zero.

Differentiation: Definition and Fundamental Properties · ⁨미분: 정의 및 기본 성질⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

2.1

Average and Instantaneous Rates of Change at a Point

Syllabus

English

Enduring Understanding (CHA-2): Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

Learning Objective CHA-2.A: Determine average rates of change using difference quotients.

CHA-2.A.1 The difference quotients $\dfrac{f(a+h)-f(a)}{h}$ and $\dfrac{f(x)-f(a)}{x-a}$ express the average rate of change of a function over an interval.

Learning Objective CHA-2.B: Represent the derivative of a function as the limit of a difference quotient.

CHA-2.B.1 The instantaneous rate of change of a function at $x=a$ can be expressed by $\lim\limits_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$ or $\lim\limits_{x\to a}\dfrac{f(x)-f(a)}{x-a}$, provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted $f'(a)$.

한국어

지속적 이해 (CHA-2): 미분은 구간 내의 변화율에 대한 지식을 극한에 적용함으로써 순간의 변화율을 결정할 수 있게 해줍니다.

학습 목표 CHA-2.A: 차분商(difference quotients)을 사용하여 평균 변화율을 결정합니다.

CHA-2.A.1 차분商 $\dfrac{f(a+h)-f(a)}{h}$과 $\dfrac{f(x)-f(a)}{x-a}$은 한 구간에서 함수의 평균 변화율을 나타냅니다.

학습 목표 CHA-2.B: 함수의 미분을 차분商의 극한으로 표현합니다.

CHA-2.B.1 함수의 $x=a$에서의 순간 변화율은 극한이 존재할 때 $\lim\limits_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$ 또는 $\lim\limits_{x\to a}\dfrac{f(x)-f(a)}{x-a}$로 표현할 수 있습니다. 이들은 미분의 정의의 등가 형태이며 $f'(a)$로 표시됩니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A roller coaster on a lift hill: the derivative measures instantaneous rate of change

The derivative from first principles

Unit 1 built the limit. Unit 2 uses it to define the derivative 导数 – the exact rate of change at a point.

The instantaneous rate of change is the gradient of the tangent at a point

Over an interval, the average rate of change is a difference quotient 差商. Two equivalent forms appear:

$$\frac{f(a+h)-f(a)}{h} \qquad\text{and}\qquad \frac{f(x)-f(a)}{x-a}.$$

The first uses a step of size $h$ from $a$; the second uses two points $x$ and $a$. Both compute $\dfrac{\text{change in output}}{\text{change in input}}$ over the interval.

The instantaneous 瞬时 rate of change at $x=a$ is what the difference quotient approaches as the interval shrinks to zero. This limit is the derivative at $a$, written $f'(a)$:

$$f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h} = \lim_{x\to a}\frac{f(x)-f(a)}{x-a},$$

provided the limit exists.

Explore · ⁨탐색하기⁩

From average rate to instantaneous rate

y = ax³ + bx² + cx + d

Slide the point: the secant through two nearby points tips toward the tangent as they merge. The tangent's slope is the derivative — the instantaneous rate of change.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
derivative/dɪˈrɪvətɪv/	미분Unless derivative.
difference quotient/ˈdɪfrəns ˈkwəʊʃənt/	차분 비
instantaneous/ˌɪnstənˈteɪnɪəs/	순간
first principles/fɜːst ˈprɪnsɪplz/	기본 원리

2.2

Defining the Derivative and Reading Its Notation

Syllabus

English

Enduring Understanding (CHA-2): Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

Learning Objective CHA-2.B: Represent the derivative of a function as the limit of a difference quotient.

CHA-2.B.2 The derivative of $f$ is the function whose value at $x$ is $\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, provided this limit exists.
CHA-2.B.3 For $y=f(x)$, notations for the derivative include $\dfrac{dy}{dx}$, $f'(x)$, and $y'$.
CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verbally.

Learning Objective CHA-2.C: Determine the equation of a line tangent to a curve at a given point.

CHA-2.C.1 The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.

한국어

지속적 이해 (CHA-2): 미분은 구간 내의 변화율에 대한 지식을 극한에 적용함으로써 순간의 변화율을 결정할 수 있게 해줍니다.

학습 목표 CHA-2.B: 함수의 미분을 차분商의 극한으로 표현합니다.

CHA-2.B.2 $f$의 미분은 극한이 존재할 때 $x$에서의 값이 $\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$인 함수입니다.
CHA-2.B.3 $y=f(x)$에 대한 미분 표기법에는 $\dfrac{dy}{dx}$, $f'(x)$, $y'$이 포함됩니다.
CHA-2.B.4 미분은 기하학적, 수치적, 해석적, 서술적으로 표현될 수 있습니다.

학습 목표 CHA-2.C: 주어진 점에서 곡선에 접선인 직선의 방정식을 결정합니다.

CHA-2.C.1 점에서의 함수의 미분은 해당 점에서의 함수 그래프에 접선인 직선의 기울기입니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Let the point $a$ vary and the derivative becomes a new function:

$$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.$$

This is the definition of the derivative (sometimes called differentiating "by first principles" 用定义求导). Its value at each $x$ is the instantaneous rate of change there.

Common notations 记号 for the derivative of $y=f(x)$ are:

$$\frac{dy}{dx}, \qquad f'(x), \qquad y'.$$

The derivative can be represented graphically, numerically, analytically, and verbally – be ready to move between them.

Geometric meaning. The derivative at a point is the slope 斜率 of the tangent line 切线 to the graph there. So the tangent line at $x=a$ passes through $\big(a, f(a)\big)$ with slope $f'(a)$:

$$y - f(a) = f'(a)\,(x-a).$$

Writing this line is a routine exam task, so keep the point-slope form ready.

Secant slopes approach the tangent slope: the derivative is the limit of average rates

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
notations/nəʊˈteɪʃnz/	표기법
slope/sləʊp/	기울기
tangent line/ˈtændʒənt laɪn/	접선

2.3

Estimating a Derivative at a Point

Syllabus

English

Enduring Understanding (CHA-2): Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

Learning Objective CHA-2.D: Estimate derivatives.

CHA-2.D.1 The derivative at a point can be estimated from information given in tables or graphs.
CHA-2.D.2 Technology can be used to calculate or estimate the value of a derivative of a function at a point.

한국어

지속적 이해 (CHA-2): 미분은 구간 내의 변화율에 대한 지식을 극한에 적용함으로써 순간의 변화율을 결정할 수 있게 해줍니다.

학습 목표 CHA-2.D: 도함수를 추정합니다.

CHA-2.D.1 표나 그래프에 제공된 정보에서 점에서의 도함수를 추정할 수 있습니다.
CHA-2.D.2 기술을 사용하여 점에서의 함수의 도함수 값을 계산하거나 추정할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

You do not always have a formula. When a function is given by a table 表格 or a graph, estimate the derivative $f'(a)$ with a difference quotient over a small interval around $a$. A table with values on both sides of $a$ gives the best estimate:

$$f'(a) \approx \frac{f(b)-f(c)}{b-c},\qquad \text{where } c < a < b \text{ are the closest table inputs}.$$

Technology (a calculator) can also estimate a derivative at a point.

Exam skill (appears almost every year). Questions such as "Approximate $M'(7.5)$ using the average rate of change of $M$ over the interval $5 \le t \le 10$" ask for exactly this difference quotient. Show the setup:

$$M'(7.5) \approx \frac{M(10)-M(5)}{10-5}.$$

Full credit needs the numbers plugged in and the correct units 单位 (output units per input unit), since these come from real-world models.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
table/ˈteɪbl/	표
units/ˈjuːnɪts/	단위

2.4

Differentiability and Continuity: When a Derivative Exists

Syllabus

English

Enduring Understanding (FUN-2): Recognizing that a function's derivative may also be a function allows us to develop knowledge about the related behaviors of both.

Learning Objective FUN-2.A: Explain the relationship between differentiability and continuity.

FUN-2.A.1 If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of $f$, then it is not in the domain of $f'$.
FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain.
- Illustrative examples for FUN-2.A.2:
  - The left hand and right hand limits of the difference quotient are not equal, as in $f(x)=|x|$ at $x=0$.
  - The tangent line is vertical and has no slope, as in $f(x)=\sqrt[3]{x}$ at $x=0$.

한국어

지속적 이해 (FUN-2): 함수의 미분도 하나의 함수일 수 있음을 인식하면 두 함수의 관련된 행동에 대한 지식을 개발할 수 있습니다.

학습 목표 FUN-2.A: 가미분 가능성과 연속성 사이의 관계를 설명합니다.

FUN-2.A.1 함수가 한 점에서 가미분 가능하면 그 점에서 연속입니다. 특히, $f$의 정의역에 없는 점은 $f'$의 정의역에도 없습니다.
FUN-2.A.2 연속 함수라도 정의역 내의 한 점에서 가미분 불가능할 수 있습니다.
- FUN-2.A.2의 예시:
  - 차분商의 좌극한과 우극한이 서로 다른 경우, 예를 들어 $f(x)=|x|$에서 $x=0$인 경우
  - 접선이 수직이고 기울기가 없는 경우, 예를 들어 $f(x)=\sqrt[3]{x}$에서 $x=0$인 경우

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Differentiability is stronger than continuity. The key relationship:

If $f$ is differentiable 可导 at a point, then $f$ is continuous 连续 there.

So differentiability implies continuity. The reverse is false: a continuous function can fail to be differentiable. Two ways this happens:

A corner 尖点: the left and right difference-quotient limits disagree, as with $f(x)=|x|$ at $x=0$.
A vertical tangent 垂直切线: the slope is infinite (no real number), as with $f(x)=\sqrt[3]{x}$ at $x=0$.

Two ways a continuous function is not differentiable: a corner and a vertical tangent

Also, a point outside the domain of $f$ cannot be in the domain of $f'$. Use the contrapositive on the exam: if $f$ is not continuous at $a$, then $f$ is not differentiable at $a$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
differentiable/ˈdɪfərenʃɪəbl/	미분 가능해야 함
continuous/kənˈtɪnjuːəs/	연속적
corner/ˈkɔːnə/	모서리
vertical tangent/ˈvɜːtɪkl ˈtændʒənt/	수직 접선

2.5

The Power Rule

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.

FUN-3.A.1 Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form $f(x)=x^{r}$.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.A: 친숙한 함수의 도함수를 계산합니다.

FUN-3.A.1 미분의 정의와 특정 규칙을 직접 적용하여 $f(x)=x^{r}$ 형태의 함수에 대한 미분값을 계산할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

From here we use rules instead of the limit definition each time. The power rule 幂法则 handles any power of $x$:

$$\frac{d}{dx}\,x^{r} = r\,x^{\,r-1}\qquad\text{for any real } r.$$

It works for whole-number powers, negative powers ($\tfrac{1}{x}=x^{-1}$), and roots ($\sqrt{x}=x^{1/2}$) – rewrite as a power first, then apply the rule.

Explore · ⁨탐색하기⁩

A power function and its steepening slope

y = ax³ + bx² + cx + d

The power rule $\frac{d}{dx}x^n = nx^{n-1}$ drops the exponent as a factor. For $x^3$ the slope grows quickly as $x$ leaves 0 — the curve steepens.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
power rule/ˈpaʊə ruːl/	거듭제곱 법칙

2.6

Constant, Sum, Difference, and Constant Multiple Rules

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.

FUN-3.A.2 Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
FUN-3.A.3 The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.A: 친숙한 함수의 도함수를 계산합니다.

FUN-3.A.2 함수의 합, 차이 및 상수배는 미분 규칙을 사용하여 미분할 수 있다.
FUN-3.A.3 지수 법칙을 합, 차이 및 상수배의 성질과 결합하여 다항함수에 대한 미분값을 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

These rules let you differentiate term by term:

Constant: $\dfrac{d}{dx}\,k = 0$ (a constant does not change).
Constant multiple 常数倍: $\dfrac{d}{dx}\big[k\,f(x)\big] = k\,f'(x)$.
Sum / difference: $\dfrac{d}{dx}\big[f(x)\pm g(x)\big] = f'(x)\pm g'(x)$.

Combined with the power rule, they differentiate any polynomial 多项式 term by term. Example:

$$\frac{d}{dx}\big(4x^3 - 5x + 7\big) = 12x^2 - 5.$$

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Constant multiple/ˈkɒnstənt ˈmʌltɪpl/	상수 곱
polynomial/ˌpɒlɪˈnəʊmɪəl/	다항함수

2.7

Derivatives of cos x, sin x, e^x, and ln x

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.A: Calculate derivatives of familiar functions.

FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.

Enduring Understanding (LIM-3): Reasoning with definitions, theorems, and properties can be used to determine a limit.

Learning Objective LIM-3.A: Interpret a limit as a definition of a derivative.

LIM-3.A.1 In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.A: 친숙한 함수의 도함수를 계산합니다.

FUN-3.A.4 특정 규칙을 사용하여 삼각함수, 지수함수, 로그함수의 미분값을 구할 수 있다.

지속적 이해 (LIM-3): 정의, 정리, 성질을 이용한 추론으로 극한을 결정할 수 있다.

학습 목표 LIM-3.A: 미분의 정의로서 극함을 해석한다.

LIM-3.A.1 일부 경우, 미분값이 알려진 함수의 미분 정의 식을 인식하는 것이 극한을 결정하기 위한 전략이 될 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Learn these four building-block derivatives by heart:

$$\frac{d}{dx}\sin x = \cos x, \qquad \frac{d}{dx}\cos x = -\sin x,$$

$$\frac{d}{dx}e^{x} = e^{x}, \qquad \frac{d}{dx}\ln x = \frac{1}{x}\ \ (x>0).$$

Note the minus sign on the derivative of cosine, and that $e^{x}$ is its own derivative.

A limit that is really a derivative (LIM-3.A.1). Sometimes a limit is secretly the definition of a known derivative. If you recognize

$$\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$

for a function $f$ whose derivative you know, just evaluate $f'(a)$. For example, $\displaystyle \lim_{h\to 0}\frac{\sin\!\big(\tfrac{\pi}{2}+h\big)-1}{h} = \left.\frac{d}{dx}\sin x\right|_{x=\pi/2} = \cos\tfrac{\pi}{2} = 0$.

Explore · ⁨탐색하기⁩

The shape of sin x (whose slope is cos x)

y = asin(bx + c) + d

The derivative of $\sin x$ is $\cos x$: the slope of the sine curve is largest where sine crosses zero and zero at its peaks. Watch the curve to feel where its slope is steep or flat.

2.8

The Product Rule

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.B: Calculate derivatives of products and quotients of differentiable functions.

FUN-3.B.1 Derivatives of products of differentiable functions can be found using the product rule.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.B: 가미분 가능한 함수의 곱과 상의 도함수를 계산합니다.

FUN-3.B.1 가미분 가능한 함수의 곱에 대한 미분값은 곱의 미분법을 통해 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A product of two functions is not differentiated by multiplying the derivatives. Use the product rule 乘积法则:

$$\frac{d}{dx}\big[u\,v\big] = u'v + uv'.$$

"Derivative of the first times the second, plus the first times the derivative of the second." Example:

$$\frac{d}{dx}\big(x^2 e^{x}\big) = 2x\,e^{x} + x^2 e^{x}.$$

Exam questions often build a new function from given pieces, e.g. $k'(x) = \big(f(x)\big)^2 g(x)$, and ask you to combine rules while reading values from a table.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
product rule/ˈprɒdʌkt ruːl/	곱의 법칙

2.9

The Quotient Rule

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.B: Calculate derivatives of products and quotients of differentiable functions.

FUN-3.B.2 Derivatives of quotients of differentiable functions can be found using the quotient rule.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.B: 가미분 가능한 함수의 곱과 상의 도함수를 계산합니다.

FUN-3.B.2 가미분 가능한 함수의商에 대한 미분값은商的 미분법을 통해 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

For a quotient, use the quotient rule 商法则:

$$\frac{d}{dx}\!\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^{2}}.$$

"Bottom times derivative of top, minus top times derivative of bottom, all over bottom squared." The order matters because of the minus sign, so write the numerator carefully. Example:

$$\frac{d}{dx}\!\left(\frac{x}{\cos x}\right) = \frac{1\cdot\cos x - x\cdot(-\sin x)}{\cos^2 x} = \frac{\cos x + x\sin x}{\cos^2 x}.$$

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
quotient rule/ˈkwəʊʃənt ruːl/	비율의 법칙

2.10

Derivatives of Tangent, Cotangent, Secant, and Cosecant

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.B: Calculate derivatives of products and quotients of differentiable functions.

FUN-3.B.3 Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.B: 가미분 가능한 함수의 곱과 상의 도함수를 계산합니다.

FUN-3.B.3 항등식을 사용하여 탄젠트, 코탄젠트, 섹cant, 코섹ant 함수를 재배열하면 미분 규칙을 사용하여 미분할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The remaining trigonometric derivatives are not memorized separately – you rewrite them with identities 恒等式 and apply the quotient (or product) rule. For instance, $\tan x = \dfrac{\sin x}{\cos x}$, so the quotient rule gives

$$\frac{d}{dx}\tan x = \frac{\cos x\cos x - \sin x(-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x.$$

The same method (writing $\cot x=\tfrac{\cos x}{\sin x}$, $\sec x=\tfrac{1}{\cos x}$, $\csc x=\tfrac{1}{\sin x}$) gives $-\csc^2 x$, $\sec x\tan x$, and $-\csc x\cot x$.

Higher-order derivatives. Differentiating $f'$ again gives the second derivative 二阶导数 $f''(x)$ (or $\tfrac{d^2y}{dx^2}$) – the rate of change of the rate of change. An exam part like "Find $k''(3)$" just means differentiate twice, then substitute. You can also estimate a second derivative from a table by applying the average-rate-of-change method to the $f'$ values.

Worked example. Differentiate $g(x)=\dfrac{\sin x}{x}$ with the quotient rule $\left(\tfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}$: take $u=\sin x$, $v=x$, giving $g'(x)=\dfrac{x\cos x-\sin x}{x^2}$. Keep the order $u'v-uv'$ in the numerator — swapping the terms flips the sign and loses the mark.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
identities/aɪˈdentɪtiz/	항등식
second derivative/ˈsekənd dɪˈrɪvətɪv/	이계 도함수Unless second derivative.

2.10

Exam tips

The derivative is the slope of the tangent — the limit of the secant slope $\tfrac{f(x+h)-f(x)}{h}$ as $h\to0$.
Memorise the rules: power, product, quotient, and the derivatives of $\sin$, $\cos$, $e^x$, and $\ln x$.
Differentiability implies continuity, but not the reverse (a corner or cusp is continuous yet not differentiable).
Distinguish average rate of change (secant slope over an interval) from instantaneous rate (the derivative at a point).
Give a tangent-line equation as $y-f(a)=f'(a)(x-a)$.

Differentiation: Composite, Implicit, and Inverse Functions · ⁨미분: 합성함수, 암시함수 및 역함수⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

3.1

The Chain Rule

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.C: Calculate derivatives of compositions of differentiable functions.

FUN-3.C.1 The chain rule provides a way to differentiate composite functions.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.C: 가미분 가능한 함수의 합성函数的 미분값을 계산한다.

FUN-3.C.1 연쇄 법칙은 합성함수를 미분하는 방법을 제공한다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A mountain path: composite functions nest rates of change (chain rule)

The chain rule

Unit 2 differentiated single functions. Unit 3 differentiates functions built inside other functions. The chain rule 链式法则 differentiates a composite function 复合函数 $f\big(g(x)\big)$:

$$\frac{d}{dx}\,f\big(g(x)\big) = f'\big(g(x)\big)\cdot g'(x).$$

"Derivative of the outer function (leaving the inside alone), times the derivative of the inside." The inner derivative $g'(x)$ is the piece students forget, so always ask "what is the inside, and what is its derivative?" Example:

$$\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x.$$

In Leibniz notation, with $y=f(u)$ and $u=g(x)$, the rule reads $\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}$ – the intermediate $du$ appears to "cancel." Exam questions often give a table for $f$, $g$, $f'$, $g'$ and ask for $h'(a)$ where $h(x)=f\big(g(x)\big)$; evaluate $f'\big(g(a)\big)\cdot g'(a)$ by reading values.

Worked example. Differentiate $h(x)=(2x^2+1)^5$. The outer function is "(something)$^5$" and the inner is $2x^2+1$:

$$h'(x)=5(2x^2+1)^4\cdot 4x=20x(2x^2+1)^4.$$

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
chain rule/tʃeɪn ruːl/	연쇄 법칙
composite function/ˈkɒmpəzɪt ˈfʌŋkʃn/	합성 함수

3.2

Implicit Differentiation

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.D: Calculate derivatives of implicitly defined functions.

FUN-3.D.1 The chain rule is the basis for implicit differentiation.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.D: 암시적으로 정의된 함수의 미분값을 계산한다.

FUN-3.D.1 연쇄 법칙은 암시적 미분의 기초이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Some curves are defined implicitly – by an equation in $x$ and $y$ that is not solved for $y$, such as $x^2+y^2=25$. Implicit differentiation 隐函数求导 finds $\dfrac{dy}{dx}$ without solving for $y$ first. It is just the chain rule, treating $y$ as a function of $x$.

The method: differentiate both sides with respect to $x$; every time you differentiate a $y$-term, multiply by $\dfrac{dy}{dx}$ (the chain rule); then solve algebraically for $\dfrac{dy}{dx}$. For $x^2+y^2=25$:

$$2x + 2y\frac{dy}{dx}=0 \;\Longrightarrow\; \frac{dy}{dx} = -\frac{x}{y}.$$

Worked example. Find the tangent to $x^2+y^2=25$ at $(3,4)$. Here $\dfrac{dy}{dx}=-\dfrac{3}{4}$, so the tangent line is $y-4=-\tfrac{3}{4}(x-3)$ – perpendicular to the radius, as geometry predicts.

Implicit differentiation gives the tangent to a circle, perpendicular to the radius

Exam skill – "Show that $\dfrac{dy}{dx}=\ldots$". This exact prompt appears most years (e.g. "Show that $\dfrac{dy}{dx}=\dfrac{2y}{y^2-2x}$"). Because the target is given, you must show every algebra step cleanly: differentiate both sides, use the product/chain rules on mixed $xy$ terms, collect all $\dfrac{dy}{dx}$ terms on one side, factor, and divide. A correct final line that skips the algebra earns little. Follow-up parts then ask for a tangent line, or where the tangent is horizontal ($\tfrac{dy}{dx}=0$, so the numerator is $0$) or vertical (the denominator is $0$).

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Implicit differentiation/ɪmˈplɪsɪt ˌdɪfəˌrenʃɪˈeɪʃn/	은밀 미분법

3.3

Differentiating Inverse Functions

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.E: Calculate derivatives of inverse and inverse trigonometric functions.

FUN-3.E.1 The chain rule and definition of an inverse function can be used to find the derivative of an inverse function, provided the derivative exists.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.E: 역함수와 역삼각함수의 미분값을 계산한다.

FUN-3.E.1 연쇄 법칙과 역함수의 정의를 사용하여 역함수의 미분값을 구할 수 있으며, 이는 미분값이 존재할 때만 가능하다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

If $g$ is the inverse function 反函数 of $f$ (so $f(g(x))=x$), the chain rule links their derivatives:

$$g'(x) = \frac{1}{f'\big(g(x)\big)},\qquad\text{provided } f'\big(g(x)\big)\neq 0.$$

In words: the derivative of the inverse at a point is the reciprocal 倒数 of the derivative of the original function at the matching point. A common exam setup gives a table and a point $(a,b)$ on $f$ (so $(b,a)$ is on $g$), then asks for $g'(b)=\dfrac{1}{f'(a)}$.

Worked example. If $f(2)=5$ and $f'(2)=3$, and $g$ is the inverse of $f$, then $(5,2)$ lies on $g$ and $g'(5)=\dfrac{1}{f'(2)}=\dfrac{1}{3}$.

The inverse function is the mirror image of the function in the line y = x

Explore · ⁨탐색하기⁩

An exponential and its inverse the log · ⁨지수 함수와 그 역함수인 로그⁩

y = a·e^(bx) + c

Inverse functions mirror across $y=x$ and their slopes are reciprocals. Where one is steep, its inverse is shallow. · ⁨역함수는 $y=x$를 기준으로 대칭이며 기울기는 서로 역수입니다. 한쪽이 가파르면 역함수는 완만합니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
inverse function/ɪnˈvɜːs ˈfʌŋkʃn/	역함수
reciprocal/rɪˈsɪprəkl/	역수

3.4

Differentiating Inverse Trigonometric Functions

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.E: Calculate derivatives of inverse and inverse trigonometric functions.

FUN-3.E.2 The chain rule applied with the definition of an inverse function, or the formula for the derivative of an inverse function, can be used to find the derivatives of inverse trigonometric functions.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.E: 역함수와 역삼각함수의 미분값을 계산한다.

FUN-3.E.2 연쇄 법칙을 역함수의 정의에 적용하거나, 역함수의 미분 공식을 사용하여 역삼각함수의 미분값을 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The same idea gives the derivatives of the inverse trigonometric functions 反三角函数. The three you should know:

$$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan x = \frac{1}{1+x^2}, \quad \frac{d}{dx}\text{arcsec}\,x = \frac{1}{|x|\sqrt{x^2-1}}.$$

Combine these with the chain rule when the input is itself a function, e.g. $\dfrac{d}{dx}\arctan(3x)=\dfrac{3}{1+9x^2}$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
inverse trigonometric functions/ɪnˈvɜːs ˌtrɪɡənəʊˈmetrɪk ˈfʌŋkʃnz/	역삼각함수

3.5

Selecting Procedures for Calculating Derivatives

Syllabus

English

This topic is intended to focus on the skill of selecting an appropriate procedure for calculating derivatives. Students should be given opportunities to practice when and how to apply all learning objectives relating to calculating derivatives.

한국어

이 주제는 적절한 미분 계산 절차를 선택하는 기술에 집중하도록 설계되었다. 학생들은 미분값 계산과 관련된 모든 학습 목표를 언제 및 어떻게 적용할지 연습할 기회가 주어져야 한다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A skill topic: real derivatives mix several rules, so read the structure of the expression first, from the outside in.

Is it a sum? Differentiate term by term.
A product or quotient? Apply that rule, and expect to use the chain rule inside.
A composite (something inside something)? Chain rule.
Given implicitly? Implicit differentiation.

Name the outermost operation, apply its rule, and recurse inward. Neatness prevents the sign and bookkeeping errors that cost marks.

3.6

Calculating Higher-Order Derivatives

Syllabus

English

Enduring Understanding (FUN-3): Recognizing opportunities to apply derivative rules can simplify differentiation.

Learning Objective FUN-3.F: Determine higher order derivatives of a function.

FUN-3.F.1 Differentiating $f'$ produces the second derivative $f''$, provided the derivative of $f'$ exists; repeating this process produces higher-order derivatives of $f$.
FUN-3.F.2 Higher-order derivatives are represented with a variety of notations. For $y = f(x)$, notations for the second derivative include $\dfrac{d^2 y}{dx^2}$, $f''(x)$, and $y''$. Higher-order derivatives can be denoted $\dfrac{d^n y}{dx^n}$ or $f^{(n)}(x)$.

한국어

지속적 이해 (FUN-3): 미분 규칙을 적용할 수 있는 상황을 파악하면 미분 과정이 단순화됩니다.

학습 목표 FUN-3.F: 함수의 고계 미분값을 결정한다.

FUN-3.F.1 $f'$를 미분하면 $f''$인 이계 미분값을 얻으며, 이는 $f'$의 미분값이 존재할 때 가능하고, 이 과정을 반복하면 $f$의 고계 미분값을 얻을 수 있다.
FUN-3.F.2 고계 미분값은 다양한 표기법으로 표현된다. $y = f(x)$에 대해 이계 미분값의 표기법은 $\dfrac{d^2 y}{dx^2}$, $f''(x)$, $y''$가 포함된다. 고계 미분값은 $\dfrac{d^n y}{dx^n}$ 또는 $f^{(n)}(x)$로 나타낼 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Differentiating $f'$ produces the second derivative 二阶导数 $f''$; repeating gives higher-order derivatives. The notations:

$$f''(x)=\frac{d^2y}{dx^2}=y'', \qquad\text{and in general}\qquad f^{(n)}(x)=\frac{d^n y}{dx^n}.$$

The second derivative measures how the slope is changing; it drives concavity 凹凸性 and acceleration in later units. To find $f''$ implicitly, differentiate the expression for $\dfrac{dy}{dx}$ again (with the quotient and chain rules), then substitute $\dfrac{dy}{dx}$ back in.

Explore · ⁨탐색하기⁩

The second derivative is the slope of the slope · ⁨이계도함수는 기울기의 기울기입니다⁩

y = ax³ + bx² + cx + d

Differentiating again gives $f''$, the rate the slope changes. Where the slope is increasing the curve bends upward. · ⁨다시 미분하면 $f''$를 얻는데, 이는 기울기가 변하는 속도입니다. 기울기가 증가하는 곳에서는 곡선이 위로 볼록하게 굽습니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
second derivative/ˈsekənd dɪˈrɪvətɪv/	이계 도함수Unless second derivative.
concavity/kənˈkævɪti/	볼록성

3.6

Exam tips

Use the chain rule for composite functions — differentiate the outside, then multiply by the derivative of the inside (the most-forgotten factor).
For implicit differentiation, differentiate both sides with respect to $x$ and attach $\tfrac{dy}{dx}$ each time $y$ is differentiated, then solve.
Get the second derivative by differentiating twice (velocity → acceleration).
Combine rules carefully in layered expressions (chain inside product, etc.).
The inverse function's graph is the reflection in $y=x$; its slope is the reciprocal of the original's at the matching point.

Contextual Applications of Differentiation · ⁨미분의 맥락적 적용⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

4.1

Interpreting the Meaning of the Derivative in Context

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.A: Interpret the meaning of a derivative in context.

CHA-3.A.1 The derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable.
CHA-3.A.2 The derivative can be used to express information about rates of change in applied contexts.
CHA-3.A.3 The unit for $f'(x)$ is the unit for $f$ divided by the unit for $x$.

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.A: 맥락에서 미분의 의미를 해석한다.

CHA-3.A.1 함수의 미분은 독립변수에 대한 순간 변화율로 해석될 수 있다.
CHA-3.A.2 미분은 적용 맥락에서 변화율에 관한 정보를 표현하는 데 사용될 수 있다.
CHA-3.A.3 $f'(x)$의 단위는 $f$의 단위에서 $x$의 단위를 나눈 값이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Once you can compute derivatives, you use them to describe the real world. The derivative $f'(x)$ is the instantaneous rate of change of $f$ with respect to its input. Reading and reporting this rate correctly is a graded skill.

Units matter. The unit of $f'(x)$ is the unit of $f$ divided by the unit of $x$. If $C(t)$ is a number of acres and $t$ is in weeks, then $C'(t)$ is in acres per week. On the exam, "Using correct units, interpret the meaning of $g'(140)$" wants a full sentence: the value, the quantity, the rate word "per", and the moment. For example: "$g'(140)=2.3$ means that at $x=140$, the quantity is increasing at about $2.3$ units per unit of $x$."

4.2

Straight-Line Motion: Position, Velocity, and Acceleration

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.B: Calculate rates of change in applied contexts.

CHA-3.B.1 The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration.

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.B: 적용 맥락에서 변화율을 계산한다.

CHA-3.B.1 미분은 위치, 속력, 속도, 가속도와 관련된 직선 운동 문제를 해결하는 데 사용될 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

For a particle moving on a line, three functions of time are linked by differentiation:

On a velocity-time graph, the area is displacement and the gradient is acceleration

position 位置 $s(t)$;
velocity 速度 $v(t)=s'(t)$ – signed; its sign gives direction;
acceleration 加速度 $a(t)=v'(t)=s''(t)$.

Key readings (frequent exam parts):

The particle is at rest 静止 when $v(t)=0$.
It moves right/up when $v(t)>0$ and left/down when $v(t)<0$; it changes direction where $v$ changes sign.
Speed 速率 is $|v(t)|$. Speed is increasing when $v$ and $a$ have the same sign (the particle is speeding up), and decreasing when they have opposite signs.

Distinguish carefully between velocity (has direction) and speed (does not) – the exam tests this exact difference.

Worked example. A particle moves with $s(t)=t^3-6t^2+9t$. Then $v(t)=3(t-1)(t-3)$, so it is at rest at $t=1$ and $t=3$ and changes direction at each. At $t=2$, $v=-3<0$ and $a(2)=6(2)-12=0$; just after, $a>0$ while $v<0$, so the particle is slowing down there.

Explore · ⁨탐색하기⁩

Velocity is the slope of position · ⁨속도는 위치의 기울기입니다⁩

y = ax³ + bx² + cx + d

For straight-line motion, velocity is the derivative (slope) of position and acceleration the derivative of velocity. Slide the point to read the instantaneous velocity. · ⁨직선 운동에서 속도는 위치의 도함수(기울기)이고, 가속도는 속도의 도함수입니다. 점을 밀어서 순간 속도를 확인하세요.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
position/pəˈzɪʃn/	위치
velocity/vəˈlɒsɪti/	속도
acceleration/əkˌseləˈreɪʃn/	가속도
at rest/æt rest/	정지 상태임
Speed/spiːd/	속도

4.3

Rates of Change in Applied Contexts Other Than Motion

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.C: Interpret rates of change in applied contexts.

CHA-3.C.1 The derivative can be used to solve problems involving rates of change in applied contexts.

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.C: 적용 맥락에서 변화율을 해석한다.

CHA-3.C.1 미분은 적용 맥락에서의 변화율 문제를 해결하는 데 사용될 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The same derivative idea models any changing quantity: a draining tank, a spreading population, a cooling cup. Whenever a problem says "the rate at which...", it is describing a derivative. Read the units to know which quantity's rate you have, then interpret in context.

4.4

Introduction to Related Rates

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.D: Calculate related rates in applied contexts.

CHA-3.D.1 The chain rule is the basis for differentiating variables in a related rates problem with respect to the same independent variable.
CHA-3.D.2 Other differentiation rules, such as the product rule and the quotient rule, may also be necessary to differentiate all variables with respect to the same independent variable.

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.D: 적용 맥락에서 종속 변수의 변화율을 계산한다.

CHA-3.D.1 연쇄 법칙은 종속 변수의 변화율 문제에서 변수를 동일한 독립변수에 대해 미분하는 기초이다.
CHA-3.D.2 곱의 미분법이나商的 미분법과 같은 다른 미분 규칙도 동일한 독립변수에 대해 모든 변수를 미분하는 데 필요할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A roller coaster: related rates connect how fast linked quantities change together

In a related rates 相关变化率 problem, several quantities change together over time, and you know some rates but want another. The engine is the chain rule: differentiate a relationship with respect to time $t$. Every variable becomes a function of $t$, so each derivative picks up a "$\,/\,dt$" factor. Product and quotient rules may also be needed.

A rising balloon: related rates link how fast radius, volume and height change together

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
related rates/rɪˈleɪtɪd reɪts/	연관 변수율

4.5

Solving Related Rates Problems

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.E: Interpret related rates in applied contexts.

CHA-3.E.1 The derivative can be used to solve related rates problems; that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known.

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.E: 적용 맥락에서 종속 변수의 변화율을 해석한다.

CHA-3.E.1 미분은 종속 변수의 변화율 문제를 해결하는 데 사용될 수 있다; 즉, 한 양의 변화율을 알고 있는 다른 양들과 연관시켜 그 양의 변화율을 찾는 것이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A reliable procedure – and a full-credit template on the exam:

Name the variables and write down the given rates and the unknown rate (e.g. "$\dfrac{dh}{dt}=-2$ cm/day, find $\dfrac{dV}{dt}$").
Write an equation relating the quantities (often a geometric or volume formula).
Differentiate both sides with respect to $t$ (chain rule) – before substituting numbers.
Substitute the known values at the instant of interest, and solve for the unknown rate.
State the answer with units and the correct sign (a decreasing quantity has a negative rate).

Substituting numbers too early is the classic error: differentiate the general relationship first, then plug in.

Worked example. A spherical balloon's volume grows at $\dfrac{dV}{dt}=100\ \text{cm}^3/\text{s}$. From $V=\tfrac43\pi r^3$, differentiate first: $\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$. At $r=5$, $100=4\pi(25)\dfrac{dr}{dt}$, so $\dfrac{dr}{dt}=\dfrac{1}{\pi}\approx0.32\ \text{cm/s}$.

An inflating balloon links dV/dt and dr/dt through the chain rule

4.6

Approximating Values Using Local Linearity and Linearization

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.F: Approximate a value on a curve using the equation of a tangent line.

CHA-3.F.1 The tangent line is the graph of a locally linear approximation of the function near the point of tangency.
CHA-3.F.2 For a tangent line approximation, the function's behavior near the point of tangency may determine whether a tangent line value is an underestimate or an overestimate of the corresponding function value.

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.F: 접선의 방정식을 사용하여 곡선 위의 값을 근사한다.

CHA-3.F.1 접선은 접점 근처에서 함수의 국소 선형 근사의 그래프이다.
CHA-3.F.2 접선 근사 approximation에서, 접점 근처에서의 함수의 거동은 해당 접선 값이 대응하는 함수 값에 대한 과소평가인지 과대평가인지를 결정할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Local linearity and linearisation

Near a point of tangency, a smooth curve looks like its tangent line – this is local linearity 局部线性. So the tangent line gives a linear approximation 线性近似 (linearization) of the function near that point:

$$f(x) \approx L(x) = f(a) + f'(a)(x-a).$$

Use it to estimate $f$ at an $x$ close to $a$.

Over- or underestimate? The answer follows from concavity 凹凸性. If the graph is concave up near $a$ (it curves above its tangent), the tangent-line value is an underestimate 低估. If it is concave down, the tangent line lies above the curve, giving an overestimate 高估. Exam parts test this reasoning, so justify with the sign of $f''$.

The tangent line is a local linear approximation; concavity fixes over- or under-estimate

Explore · ⁨탐색하기⁩

Approximate a curve with its tangent line · ⁨접선을 이용해 곡선 근사하기⁩

y = ax³ + bx² + cx + d

Local linearity: near a point a smooth curve looks like its tangent line, so the tangent gives a good linear approximation of nearby values. · ⁨국소 선형성: 점 근처에서 매끄러운 곡선은 접선과 비슷해 보이므로, 접선은 주변 값들의 좋은 선형 근사를 제공합니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
local linearity/ˈləʊkl lɪˈnɪərɪti/	국소 선형성
linear approximation/ˈlɪnɪə əˌprɒksɪˈmeɪʃn/	선형 근사
concavity/kənˈkævɪti/	볼록성
underestimate/ˌʌndəˈrestɪmət/	과소평가(underestimate)
overestimate/ˌəʊvəˈrestɪmət/	과대평가(overestimate)

4.7

Using L'Hospital's Rule for Indeterminate Forms

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.	LIM-4.A Determine limits of functions that result in indeterminate forms.	LIM-4.A.1 When the ratio of two functions tends to $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ in the limit, such forms are said to be indeterminate. Exclusion statement: There are many other indeterminate forms, such as $\infty - \infty$, for example, but these will not be assessed on either the AP Calculus AB or BC Exam. However, teachers may include these topics, if time permits. LIM-4.A.2 Limits of the indeterminate forms $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ may be evaluated using L'Hospital's Rule.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

When direct substitution in a quotient of limits gives the indeterminate form 未定式 $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, you may use L'Hospital's Rule 洛必达法则:

$$\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)},$$

provided the right-hand limit exists. Differentiate the top and bottom separately (this is not the quotient rule), then try the limit again. First confirm the form really is $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ – applying the rule to any other form is a mistake.

Worked example. $\displaystyle\lim_{x\to0}\frac{\sin x}{x}$ gives $\tfrac00$, so differentiate top and bottom: $\displaystyle\lim_{x\to0}\frac{\cos x}{1}=1$. And $\displaystyle\lim_{x\to0}\frac{e^{2x}-1}{x}$ is also $\tfrac00$; it becomes $\displaystyle\lim_{x\to0}\frac{2e^{2x}}{1}=2$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
indeterminate form/ˌɪndɪˈtɜːmɪnət fɔːm/	부정정 형식
L'Hospital's Rule/ˈelhɒspɪtlz ruːl/	L'Hospital 법칙

4.7

Exam tips

In motion problems: velocity is the derivative of position, acceleration the derivative of velocity; speed increases when velocity and acceleration share a sign.
For related rates, differentiate the relating equation with respect to time, then substitute the given values last.
Use the tangent line for a linear approximation near a known point; it is accurate only close by.
Read the sign of a rate: positive means the quantities move together, negative means opposite.
Always state units and interpret the answer in context.

Analytical Applications of Differentiation · ⁨미분의 분석적 적용⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

5.1

Using the Mean Value Theorem

Syllabus

English

Enduring Understanding (FUN-1): Existence theorems allow us to draw conclusions about a function's behavior on an interval without precisely locating that behavior.

Learning Objective FUN-1.B: Justify conclusions about functions by applying the Mean Value Theorem over an interval.

FUN-1.B.1 If a function $f$ is continuous over the interval $[a, b]$ and differentiable over the interval $(a, b)$, then the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.

한국어

지속적 이해(FUN-1): 존재 정리는 특정 구간에서 함수의 행동을 정확히 locating하지 않고도 결론을 내릴 수 있게 해줍니다.

학습 목표 FUN-1.B: 평균값 정리를 구간에서 적용하여 함수에 대한 결론을 정당화합니다.

FUN-1.B.1 함수 $f$가 구간 $[a, b]$에서 연속이고 구간 $(a, b)$에서 가미 가능하면, 평균값 정리에 의해 이 개구간 내에 순간 변화율이 구간 전체의 평균 변화율과 같아지는 점이 존재함을 보장합니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The Mean Value Theorem

The Mean Value Theorem 中值定理 (MVT) links the average rate of change to an instantaneous one:

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is at least one point $c$ in $(a,b)$ where
$$f'(c) = \frac{f(b)-f(a)}{b-a}.$$

In words: somewhere inside the interval, the instantaneous rate equals the average rate. Geometrically, some tangent line is parallel to the line joining the endpoints.

The Mean Value Theorem: some tangent is parallel to the secant over the interval

Exam skill. Like the IVT, the MVT is an existence theorem, and questions ask you to justify. Full credit needs: (1) state $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$; (2) compute the average rate $\frac{f(b)-f(a)}{b-a}$; (3) conclude "by the MVT there is a $c$ in $(a,b)$ with $f'(c)$ equal to that value." Both hypotheses must be named.

Worked example. For $f(x)=x^2$ on $[1,3]$ the average rate is $\dfrac{9-1}{2}=4$; setting $f'(c)=2c=4$ gives $c=2$, which lies in $(1,3)$ – the guaranteed point.

Explore · ⁨탐색하기⁩

The Mean Value Theorem in action

y = ax³ + bx² + cx + d

The Mean Value Theorem guarantees a point where the tangent is parallel to the secant across an interval — the instantaneous rate equals the average rate somewhere inside.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Mean Value Theorem/miːn ˈvæljuː ˈθɪərəm/	평균값 정리

5.2

Extreme Values, Global vs Local Extrema, and Critical Points

Syllabus

English

Enduring Understanding (FUN-1): Existence theorems allow us to draw conclusions about a function's behavior on an interval without precisely locating that behavior.

Learning Objective FUN-1.C: Justify conclusions about functions by applying the Extreme Value Theorem.

FUN-1.C.1 If a function $f$ is continuous over the interval $(a, b)$, then the Extreme Value Theorem guarantees that $f$ has at least one minimum value and at least one maximum value on $[a, b]$.
FUN-1.C.2 A point on a function where the first derivative equals zero or fails to exist is a critical point of the function.
FUN-1.C.3 All local (relative) extrema occur at critical points of a function, though not all critical points are local extrema.

한국어

지속적 이해(FUN-1): 존재 정리는 특정 구간에서 함수의 행동을 정확히 locating하지 않고도 결론을 내릴 수 있게 해줍니다.

학습 목표 FUN-1.C: 극대값 정리를 적용하여 함수에 대한 결론을 정당화합니다.

FUN-1.C.1 함수 $f$가 구간 $(a, b)$에서 연속이면, 극한값 정리에 의해 $f$는 구간 $[a, b]$에서 최소값과 최대값이 각각 적어도 하나씩 존재함을 보장합니다.
FUN-1.C.2 함수 위에서 첫 번째 도함수가 0이거나 존재하지 않는 점은 해당 함수의 임계점입니다.
FUN-1.C.3 모든 국소(상대) 극값은 함수의 임계점에서 발생하지만, 모든 임계점이 국소 극값인 것은 아닙니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A mountain peak: extrema and critical points mark highest and lowest values

The Extreme Value Theorem 极值定理 (EVT) guarantees extremes exist: a function continuous on a closed interval $[a,b]$ attains both an absolute maximum and an absolute minimum on it.

At a maximum or minimum the derivative is zero

A critical point 临界点 is an interior point where $f'(x)=0$ or $f'(x)$ does not exist. All local (relative) extrema 局部极值 occur at critical points – but not every critical point is an extremum. So critical points are the candidates; you must test each.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Extreme Value Theorem/ekˈstriːm ˈvæljuː ˈθɪərəm/	극한값 정리
critical point/ˈkrɪtɪkl pɔɪnt/	임계점
local (relative) extrema/ˈləʊkl ekˈstremə/	국소(상대) 극값

5.3

Where a Function Increases or Decreases

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.A: Justify conclusions about the behavior of a function based on the behavior of its derivatives.

FUN-4.A.1 The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.A: 도함수의 거동에 기반하여 함수의 거동에 대한 결론을 정당화합니다.

FUN-4.A.1 함수의 첫 번째 도함수는 함수와 그 그래프에 대한 정보를 제공하며,其中包括函数增加或减少的区间。

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The first derivative tells you where $f$ rises or falls:

$f'(x) > 0$ on an interval $\Rightarrow$ $f$ is increasing 递增 there;
$f'(x) < 0$ $\Rightarrow$ $f$ is decreasing 递减.

On the exam, "find the intervals where $f$ is increasing" means: find the critical points, then test the sign of $f'$ between them, and justify with the sign of $f'$ (a stated reason, not just an interval).

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
increasing/ɪnˈkriːsɪŋ/	증가함
decreasing/ˈdiːkriːsɪŋ/	감소함

5.4

The First Derivative Test for Local Extrema

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.A: Justify conclusions about the behavior of a function based on the behavior of its derivatives.

FUN-4.A.2 The first derivative of a function can determine the location of relative (local) extrema of the function.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.A: 도함수의 거동에 기반하여 함수의 거동에 대한 결론을 정당화합니다.

FUN-4.A.2 함수의 첫 번째 도함수는 함수의 국소(상대) 극값 위치를 결정할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

To classify a critical point $x=c$ as a local max, local min, or neither, check how $f'$ changes sign there:

$f'$ changes $+$ to $-$ at $c$ $\Rightarrow$ local maximum 极大值;
$f'$ changes $-$ to $+$ at $c$ $\Rightarrow$ local minimum 极小值;
$f'$ does not change sign $\Rightarrow$ neither.

Always state the sign change as your justification.

Worked example. For $f(x)=x^3-3x^2$, $f'(x)=3x(x-2)$ is zero at $x=0,2$. Signs give $+,-,+$, so $x=0$ is a local maximum ($f=0$) and $x=2$ a local minimum ($f=-4$).

The sign of f-prime sets where a function increases or decreases and locates its local extrema — Where $f' > 0$ the graph rises and where $f' < 0$ it falls; a local max sits where $f'$ turns $+$ to $-$, a local min where it turns $-$ to $+$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
local maximum/ˈləʊkl ˈmæksɪməm/	국소 최대점
local minimum/ˈləʊkl ˈmɪnɪməm/	국소 최소점

5.5

The Candidates Test for Absolute Extrema

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.A: Justify conclusions about the behavior of a function based on the behavior of its derivatives.

FUN-4.A.3 Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.A: 도함수의 거동에 기반하여 함수의 거동에 대한 결론을 정당화합니다.

FUN-4.A.3 닫힌 구간에서 함수의 전역(전체) 극값은 오직 임계점이나 경계점에서만 발생할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

On a closed interval, absolute (global) extrema occur only at critical points or endpoints. The candidates test:

List all critical points in $[a,b]$ and the two endpoints.
Evaluate $f$ at each candidate.
The largest output is the absolute maximum; the smallest is the absolute minimum.

Show the table of values – the comparison is the argument.

5.6

Concavity and Points of Inflection

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.A: Justify conclusions about the behavior of a function based on the behavior of its derivatives.

FUN-4.A.4 The graph of a function is concave up (down) on an open interval if the function's derivative is increasing (decreasing) on that interval.
FUN-4.A.5 The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.
FUN-4.A.6 The second derivative of a function may be used to locate points of inflection for the graph of the original function.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.A: 도함수의 거동에 기반하여 함수의 거동에 대한 결론을 정당화합니다.

FUN-4.A.4 함수의 도함수가某开区间上递增(递减)时，该函数的图像在该开区间上是凹向上(下)。
FUN-4.A.5 함수의 이阶导数 provides information about the function and its graph, including intervals of upward or downward concavity.
FUN-4.A.6 함수의 이阶导数는 원본 함수의 그래프에서 변곡점을 locating하는 데 사용될 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The second derivative describes bending:

$f'' > 0$ $\Rightarrow$ $f$ is concave up 上凹 (curving like a cup; $f'$ is increasing);
$f'' < 0$ $\Rightarrow$ $f$ is concave down 下凹 ($f'$ is decreasing).

A point of inflection 拐点 is where concavity changes, i.e. where $f''$ changes sign (not merely where $f''=0$). Report its $x$-coordinate and justify with the sign change of $f''$.

Concavity comes from the sign of the second derivative; it flips at an inflection point

Explore · ⁨탐색하기⁩

Find where concavity flips

y = ax³ + bx² + cx + d

Concavity is the sign of the second derivative: concave up where the curve holds water, concave down where it spills. A point of inflection is where it switches.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
concave up/kɒnˈkeɪv ʌp/	오목 위(upward concave)
concave down/kɒnˈkeɪv daʊn/	오목 아래
point of inflection/pɔɪnt ɒv ɪnˈflekʃn/	변곡점

5.7

The Second Derivative Test for Extrema

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.A: Justify conclusions about the behavior of a function based on the behavior of its derivatives.

FUN-4.A.7 The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.
FUN-4.A.8 When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.A: 도함수의 거동에 기반하여 함수의 거동에 대한 결론을 정당화합니다.

FUN-4.A.7 함수의 이阶导数는 임계점이 국소(상대) 극대값 또는 극소값의 위치인지 여부를 결정할 수 있습니다.
FUN-4.A.8 연속 함수가 정의역의 한 구간에서 오직 하나의 임계점만 가지고 있으며, 해당 임계점이 구간에 대한 함수의 상대적(국소) 극값에 대응한다면, 그 임계점은 또한 구간에 대한 함수의 절대적(전체) 극값에도 대응합니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

An alternative way to classify a critical point $c$ where $f'(c)=0$:

$f''(c) > 0$ $\Rightarrow$ concave up $\Rightarrow$ local minimum;
$f''(c) < 0$ $\Rightarrow$ concave down $\Rightarrow$ local maximum;
$f''(c) = 0$ $\Rightarrow$ the test is inconclusive – fall back on the first derivative test.

Special case: if a continuous function has only one critical point on an interval and it is a local extremum, that point is also the absolute extremum there.

5.8

Sketching a Function and Its Derivative

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.A: Justify conclusions about the behavior of a function based on the behavior of its derivatives.

FUN-4.A.9 Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.
FUN-4.A.10 Graphical, numerical, and analytical information from $f'$ and $f''$ can be used to predict and explain the behavior of $f$.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.A: 도함수의 거동에 기반하여 함수의 거동에 대한 결론을 정당화합니다.

FUN-4.A.9 함수와 도함수의 핵심 특징을 식별하고 이를 기하학적, 수치적, 분석적 표현과 연결할 수 있습니다.
FUN-4.A.10 $f'$와 $f''$로부터 얻은 그래프적, 수치적, 분석적 정보를 활용하여 $f$의 거동을 예측하고 설명할 수 있습니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Key features of $f$, $f'$, and $f''$ mirror each other. To sketch or read graphs:

$f$ increasing $\Leftrightarrow$ $f'$ above the axis; $f$ has a local max $\Leftrightarrow$ $f'$ crosses from $+$ to $-$.
$f$ concave up $\Leftrightarrow$ $f'$ increasing $\Leftrightarrow$ $f''$ above the axis; $f$ has an inflection point $\Leftrightarrow$ $f'$ has a local extremum $\Leftrightarrow$ $f''$ crosses zero.

5.9

Connecting $f$, $f'$, and $f''$

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-4 A function's derivative can be used to understand some behaviors of the function.	FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives.	FUN-4.A.11 Key features of the graphs of $f$, $f'$, and $f''$ are related to one another.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

This is the skill of reading one graph to describe another. A very common exam setup gives the graph of $f'$ and asks about $f$: where is $f$ increasing (where $f'>0$), where are $f$'s extrema (where $f'$ crosses zero, with a sign change), where is $f$ concave up (where $f'$ is increasing). Answer questions about $f$ using the height and slope of the $f'$ graph.

Explore · ⁨탐색하기⁩

Read slope and bend off the graph

y = ax³ + bx² + cx + d

Where $f'>0$ the function rises; where $f''>0$ it bends upward. Slide the tangent to connect the shape of $f$ to the signs of its first and second derivatives.

5.10

Introduction to Optimization Problems

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.B: Calculate minimum and maximum values in applied contexts or analysis of functions.

FUN-4.B.1 The derivative can be used to solve optimization problems; that is, finding a minimum or maximum value of a function on a given interval.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.B: 응용 맥락이나 함수 분석에서 최소값 및 최대값을 계산합니다.

FUN-4.B.1 도함수는 최적화 문제를 해결하는 데 사용될 수 있으며, 이는 주어진 구간에서 함수의 최소값 또는 최대값을 찾는 것입니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Optimisation: the best box

Optimization 最优化 uses the derivative to find the largest or smallest value of a quantity on an interval. It is the candidates/derivative-test machinery applied to a real goal.

Fenced enclosures: optimization finds the dimensions that maximise area for a fixed fence length

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Optimization/ˌɒptɪmaɪˈzeɪʃn/	최적화

5.11

Solving Optimization Problems

Syllabus

English

Enduring Understanding (FUN-4): A function's derivative can be used to understand some behaviors of the function.

Learning Objective FUN-4.C: Interpret minimum and maximum values calculated in applied contexts.

FUN-4.C.1 Minimum and maximum values of a function take on specific meanings in applied contexts.

한국어

지속적 이해 (FUN-4): 함수의 도함수를 통해 함수의 일부 거동을 이해할 수 있습니다.

학습 목표 FUN-4.C: 응용 맥락에서 계산된 최소값 및 최대값을 해석합니다.

FUN-4.C.1 함수의 최소값 및 최대값은 응용 맥락에서 구체적인 의미를 가집니다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A dependable procedure:

Write the quantity to optimize as a function of one variable (use a constraint equation to eliminate extras).
State the interval of allowed inputs.
Find critical points ($f'=0$ or undefined) and test them (first- or second-derivative test, or candidates test if the interval is closed).
Answer the question asked, with units and interpretation in context – the maximum area, the minimum cost, etc.

Worked example. With $100\ \text{m}$ of fence for a rectangular pen against a wall (only three sides fenced), let the ends be $x$ and the far side $y=100-2x$. The area $A(x)=x(100-2x)=100x-2x^2$ has $A'(x)=100-4x=0$ at $x=25$; since $A''=-4<0$ this is the maximum, giving $y=50$ and $A=1250\ \text{m}^2$.

5.12

Exploring Behaviors of Implicit Relations

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-4 A function's derivative can be used to understand some behaviors of the function.	FUN-4.D Determine critical points of implicit relations.	FUN-4.D.1 A point on an implicit relation where the first derivative equals zero or does not exist is a critical point of the function.
	FUN-4.E Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.	FUN-4.E.1 Applications of derivatives can be extended to implicitly defined functions. FUN-4.E.2 Second derivatives involving implicit differentiation may be relations of $x$, $y$, and $\dfrac{dy}{dx}$.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

All of this extends to implicitly defined relations. A critical point of an implicit relation is where $\dfrac{dy}{dx}=0$ (horizontal tangent) or is undefined (vertical tangent). Because $\dfrac{dy}{dx}$ is usually a relation in $x$ and $y$, and the second derivative involves $x$, $y$, and $\dfrac{dy}{dx}$, substitute your first-derivative expression back in when finding $\dfrac{d^2y}{dx^2}$, then reason about concavity from its sign.

5.12

Exam tips

$f'>0$ means increasing, $f'<0$ decreasing; candidates for extrema are where $f'=0$ or is undefined.
Classify a critical point with the first-derivative sign change or the second-derivative test ($f''>0$ minimum, $f''<0$ maximum).
$f''>0$ is concave up, $f''<0$ concave down; a point of inflection is where concavity changes ($f''$ changes sign).
For an absolute extremum on a closed interval, also check the endpoints.
Justify every conclusion by citing the sign of $f'$ or $f''$ — the exam demands the reasoning, not just the answer.

Integration and Accumulation of Change · ⁨적분과 변화의 축적⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

6.1

Exploring Accumulations of Change · ⁨변화의 적분 탐구⁩

Syllabus

English

Enduring Understanding (CHA-4): Definite integrals allow us to solve problems involving the accumulation of change over an interval.

Learning Objective CHA-4.A: Interpret the meaning of areas associated with the graph of a rate of change in context.

CHA-4.A.1 The area of the region between the graph of a rate of change function and the $x$ axis gives the accumulation of change.
CHA-4.A.2 In some cases, accumulation of change can be evaluated by using geometry.
CHA-4.A.3 If a rate of change is positive (negative) over an interval, then the accumulated change is positive (negative).
CHA-4.A.4 The unit for the area of a region defined by rate of change is the unit for the rate of change multiplied by the unit for the independent variable.

한국어

지속적 이해 (CHA-4): 정적분은 구간 내에서의 변화량 축적 문제를 해결할 수 있게 해준다.

학습 목표 CHA-4.A: 변화율 그래프와 관련된 면적의 의미를 맥락 속에서 해석한다.

CHA-4.A.1 변화율 함수의 그래프와 $x$축 사이의 영역의 면적은 변화량의 축적을 나타낸다.
CHA-4.A.2 일부 경우, 변화량 축적은 기하학적 방법을 통해 계산할 수 있다.
CHA-4.A.3 어떤 구간에서 변화양이 양수(음수)이면, 축적된 변화량 역시 양수(음수)이다.
CHA-4.A.4 변화율로 정의된 영역의 면적의 단위는 변화율의 단위와 독립 변수의 단위를 곱한 값이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Where a derivative measures a rate, an integral 积分 measures an accumulation 累积 – a total built up from a rate. If a rate of change acts over an interval, the area between its graph and the axis gives the net accumulated change. This "area = total change" idea is the foundation of integral calculus.

한국어

도함수가 **변화율(rate)**을 측정한다면, 적분은 **누적량(accumulation)**을 측정합니다 – 즉, 변화율로부터 쌓여 나간 총량입니다. 어떤 구간 동안 변화율이 작용할 때, 그래프와 축 사이의 면적은 순 누적 변화를 나타냅니다. "면적 = 총 변화"라는 개념은 적분학의 기초입니다.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
accumulation/əˌkjuːmjʊˈleɪʃn/	적분함수(Accumulation function)

6.2

Approximating Areas with Riemann Sums · ⁨리만 합을 이용한 면적 근사⁩

Syllabus

English

Enduring Understanding (LIM-5): Definite integrals can be approximated using geometric and numerical methods.

Learning Objective LIM-5.A: Approximate a definite integral using geometric and numerical methods.

LIM-5.A.1 Definite integrals can be approximated for functions that are represented graphically, numerically, analytically, and verbally.
LIM-5.A.2 Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions.
LIM-5.A.3 Definite integrals can be approximated using numerical methods, with or without technology.
LIM-5.A.4 Depending on the behavior of a function, it may be possible to determine whether an approximation for a definite integral is an underestimate or overestimate for the value of the definite integral.

한국어

지속적 이해 (LIM-5): 정적분은 기하학적 및 수치적 방법을 사용하여 근사할 수 있다.

학습 목표 LIM-5.A: 기하학적 및 수치적 방법을 사용하여 정적분을 근사한다.

LIM-5.A.1 정적분은 그래프, 수치, 분석적, 언어적으로 표현된 함수에 대해 근사할 수 있다.
LIM-5.A.2 정적분은 좌측 리만 합, 우측 리만 합, 중점 리만 합 또는 사다리꼴 합을 사용하여 근사할 수 있으며, 근사는 균등 분할或非等分을 사용하여 계산할 수 있다.
LIM-5.A.3 정적분은 수치적 방법을 사용하여 근사할 수 있으며, 이 과정에서 기술적 도구(Technology)를 사용하거나 사용하지 않을 수 있다.
LIM-5.A.4 함수의 성질에 따라 정적분의 근사값이 실제 정적분 값에 비해 과소평가(underestimate)인지 과대평가(overestimate)인지를 판단할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

The integral as area (Riemann sums)

Trapezoidal sums approximate area

A Riemann sum 黎曼和 estimates the area under a curve by adding the areas of thin rectangles. Split $[a,b]$ into subintervals and use the function's height at the left endpoint, right endpoint, or midpoint of each. A trapezoidal sum 梯形法 uses trapezoids instead, averaging the two endpoint heights – usually more accurate. With more, thinner rectangles the estimate improves.

Exam skill: be able to compute left, right, midpoint, and trapezoidal estimates from a table or graph, and state whether each over- or under-estimates based on whether the function is increasing/decreasing or concave up/down.

한국어

적분을 면적으로 (리만 합)

사다리꼈 합으로 면적을 근사함

**緻密 합(Riemann sum)**은 가늘고 긴 사각형들의 면적을 더하여 곡선 아래 면적을 추정합니다. $[a,b]$을 하구간으로 나누어 각 구간의 좌측 끝점, 우측 끝점, 또는 중간점에서의 함수 높이를 사용하십시오. **사다리꼴 합(trapezoidal sum)**은 사다리꼴을 사용하여 두 끝점의 높이의 평균을 취하므로 일반적으로 더 정확합니다. 사각형이 많고 가늘어질수록 추정치가 개선됩니다.

너비 h의 줄기가 곡선 아래 면적을 근사합니다 — 너비 h의 줄기는 곡선 아래 면적을 근사합니다

시험 기술: 표나 그래프에서 좌측, 우측, 중간점, 사다리꼴 추정치를 계산할 수 있어야 하며, 함수가 증가/감소하거나 위쪽/아래쪽 오목인지에 따라 각 추정치가 과대 또는 과소 추정임을 명시할 수 있어야 합니다.

Explore · ⁨탐색하기⁩

Approximate area with rectangles · ⁨사각형을 이용해 면적 근사하기⁩

y = ax³ + bx² + cx + d

A Riemann sum approximates the area under a curve with rectangles. Add more, thinner rectangles and the estimate converges to the exact definite integral. · ⁨리만 합은 사각형을 사용하여 곡선 아랫면적을 근사합니다. 더 많고 얇은 사각형을 추가하면 추정치가 정확한 정적분으로 수렴합니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Riemann sum/ˈriːmən sʌm/	리만 합
trapezoidal sum/ˈtræpɪzɔɪdl sʌm/	사다리꼴 합

6.3

Riemann Sums, Summation Notation, and Definite Integral Notation · ⁨緻密 합, 합산 기호, 그리고 정적분 기호⁩

Syllabus

English

Enduring Understanding (LIM-5): Definite integrals can be approximated using geometric and numerical methods.

Learning Objective LIM-5.B: Interpret the limiting case of the Riemann sum as a definite integral.

LIM-5.B.1 The limit of an approximating Riemann sum can be interpreted as a definite integral.
LIM-5.B.2 A Riemann sum, which requires a partition of an interval $I$, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.

Learning Objective LIM-5.C: Represent the limiting case of the Riemann sum as a definite integral.

LIM-5.C.1 The definite integral of a continuous function $f$ over the interval $[a, b]$, denoted by $\int_{a}^{b} f(x)\,dx$, is the limit of Riemann sums as the widths of the subintervals approach 0. That is, $\int_{a}^{b} f(x)\,dx = \lim_{\max \Delta x_i \to 0} \sum_{i=1}^{n} f(x_i^*)\Delta x_i$, where $n$ is the number of subintervals, $\Delta x_i$ is the width of the $i$th subinterval, and $x_i^*$ is a value in the $i$th subinterval.
LIM-5.C.2 A definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.

한국어

지속적 이해 (LIM-5): 정적분은 기하학적 및 수치적 방법을 사용하여 근사할 수 있다.

학습 목표 LIM-5.B: 리만 합의 극한을 정적분으로 해석한다.

LIM-5.B.1 근사 리만 합의 극한은 정적분으로 해석할 수 있다.
LIM-5.B.2 리만 합은 구간 $I$의 분할(partition)을 필요로 하며, 각 항은 분할된 하위 구간의 한 점에서의 함수 값과 해당 하위 구간의 길이의 곱으로 이루어진 합이다.

학습 목표 LIM-5.C: 리만 합의 극한을 정적분으로 표현한다.

LIM-5.C.1 연속 함수 $f$의区间 $[a, b]$에서의 정적분, 즉 $\int_{a}^{b} f(x)\,dx$는 하위 구간의 너비가 0에 수렴하는 리만 합의 극한이다. 즉, $\int_{a}^{b} f(x)\,dx = \lim_{\max \Delta x_i \to 0} \sum_{i=1}^{n} f(x_i^*)\Delta x_i$이며, 여기서 $n$는 하위 구간의 개수, $\Delta x_i$는 $i$번째 하위 구간의 너비, $x_i^*$는 $i$번째 하위 구간 안의 임의의 점이다.
LIM-5.C.2 정적분은 관련 리만 합의 극한으로 변환될 수 있으며, 리만 합의 극한은 정적분으로 표기될 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Writing a Riemann sum with summation notation $\sum_{k=1}^{n} f(x_k)\,\Delta x$ and letting the number of rectangles grow without bound gives the exact area – the definite integral 定积分:

$$\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{k=1}^{n} f(x_k)\,\Delta x.$$

The integral is the limit of Riemann sums; $a$ and $b$ are the limits of integration.

한국어

합산 기호(summation notation) $\sum_{k=1}^{n} f(x_k)\,\Delta x$을 사용하여 緻密 합을 쓰고 사각형의 개수가 무한대로 커지도록 하면 정확한 면적, 즉 **정적분(definite integral)**을 얻을 수 있습니다:

$$\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{k=1}^{n} f(x_k)\,\Delta x.$$

적분은 緻密 합의 극한(limit) 입니다; $a$과 $b$은 적분의 한계입니다.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
integral/ˈɪntɪɡrəl/	적분Unless integral.
definite integral/ˈdefɪnət ˈɪntɪɡrəl/	정적분

6.4

The Fundamental Theorem of Calculus and Accumulation Functions · ⁨미적분의 기본 정리와 적분 함수⁩

Syllabus

English

Enduring Understanding (FUN-5): The Fundamental Theorem of Calculus connects differentiation and integration.

Learning Objective FUN-5.A: Represent accumulation functions using definite integrals.

FUN-5.A.1 The definite integral can be used to define new functions.
- Illustrative examples for FUN-5.A.1: $f(x) = \int_{0}^{x} e^{-t^2}\,dt$.
FUN-5.A.2 If $f$ is a continuous function on an interval containing $a$, then $\dfrac{d}{dx}\left( \int_{a}^{x} f(t)\,dt \right) = f(x)$, where $x$ is in the interval.

한국어

지속적 이해 (FUN-5): 미적분학의 기본 정리는 미분과 적분을 연결시킨다.

학습 목표 FUN-5.A: 정적분을 사용하여 적분 함수를 표현한다.

FUN-5.A.1 정적분을 사용하여 새로운 함수를 정의할 수 있다.
- FUN-5.A.1을 위한 예시: $f(x) = \int_{0}^{x} e^{-t^2}\,dt$.
FUN-5.A.2 $f$가 $a$를 포함하는 구간에서 연속 함수라면, $\dfrac{d}{dx}\left( \int_{a}^{x} f(t)\,dt \right) = f(x)$이며, 여기서 $x$는 그 구간 안의 점이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

The Fundamental Theorem of Calculus

An accumulation function 累积函数 $g(x)=\int_a^x f(t)\,dt$ gives the accumulated area from $a$ up to $x$. The Fundamental Theorem of Calculus (FTC) 微积分基本定理 says its derivative is the integrand:

$$\frac{d}{dx}\int_a^x f(t)\,dt=f(x).$$

Differentiation and integration are inverse operations. With a variable upper limit and the chain rule, $\dfrac{d}{dx}\int_a^{u(x)} f(t)\,dt=f(u(x))\,u'(x)$.

한국어

미적분학의 기본 정리

누적 함수(accumulation function) $g(x)=\int_a^x f(t)\,dt$은 $a$부터 $x$까지의 누적 면적을 줍니다. 미적분학의 기본 정리(Fundamental Theorem of Calculus, FTC) 에 따르면 이 함수의 도함수는 피적분함수(integrand)입니다:

$$\frac{d}{dx}\int_a^x f(t)\,dt=f(x).$$

미분과 적분은 상쇄(inverse) 연산입니다. 상한이 변수이고 연쇄 법칙(chain rule)을 적용하면, $\dfrac{d}{dx}\int_a^{u(x)} f(t)\,dt=f(u(x))\,u'(x)$입니다.

적분 함수는 부호 있는 면적을 더합니다; FTC는 그 도함수가 f임을 말함

Explore · ⁨탐색하기⁩

Accumulate area as an integral · ⁨적분을 통해 면적 축적하기⁩

y = ax³ + bx² + cx + d

An accumulation function $\int_a^x f(t)\,dt$ builds up signed area as $x$ moves. The Fundamental Theorem says its derivative is just $f(x)$. · ⁨적분 함수 $\int_a^x f(t)\,dt$는 $x$이 이동함에 따라 부호 있는 면적이 쌓입니다. 기본 정리에 따르면 이 함수의 도함수는 단순히 $f(x)$입니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
accumulation function/əˌkjuːmjʊˈleɪʃn ˈfʌŋkʃn/	적분 함수
Fundamental Theorem of Calculus (FTC)/ˌfʌndəˈmentl ˈθɪərəm ɒv ˈkælkjʊləs/	미적분학 기본 정리 (FTC)
antiderivative/ˌæntɪdɪˈrɪvətɪv/	원함수(alternative integral)
indefinite integral/ɪnˈdefɪnət ˈɪntɪɡrəl/	부정적분
u-substitution/juː ˌsʌbstɪˈtjuːʃn/	u-대치法
Integration by parts/ˌɪntɪˈɡreɪʃn baɪ pɑːts/	부분 적분
Partial fractions/ˈpɑːʃl ˈfrækʃnz/	부분 분수

6.5

Interpreting the Behavior of Accumulation Functions · ⁨누적 함수의 거동을 해석하기⁩

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.	FUN-5.A Represent accumulation functions using definite integrals.	FUN-5.A.3 Graphical, numerical, analytical, and verbal representations of a function $f$ provide information about the function $g$ defined as $g(x) = \int_{a}^{x} f(t)\,dt$.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Because $g'(x)=f(x)$, the graph of $f$ tells you everything about $g$: $g$ increases where $f>0$, decreases where $f<0$, has extrema where $f$ crosses zero, and is concave up where $f$ is increasing. Reading these connections off a graph of $f$ is a classic free-response task.

한국어

$g'(x)=f(x)$이기 때문에, $f$의 그래프는 $g$에 대해 모든 것을 알려줍니다: $g$은 $f>0$인 곳에서 증가하고, $f<0$인 곳에서 감소하며, $f$이 0을 지나는 곳에서 극값을 가지며, $f$이 증가하는 곳에서 위쪽 오목입니다. $f$의 그래프에서 이러한 연결 관계를 읽는 것은 전형적인 서술형 문제입니다.

6.6

Applying Properties of Definite Integrals · ⁨정적분의 성질 활용⁩

Syllabus

English

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Learning Objective FUN-6.A: Calculate a definite integral using areas and properties of definite integrals.

FUN-6.A.1 In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area.
FUN-6.A.2 Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.
FUN-6.A.3 The definition of the definite integral may be extended to functions with removable or jump discontinuities.

한국어

지속적 이해 (FUN-6): 기하학적 지식과 수학적 규칙을 적용할 기회를 파악하면 적분이 단순화될 수 있다.

학습 목표 FUN-6.A: 면적과 정적분의 성질을 사용하여 정적분을 계산한다.

FUN-6.A.1 일부 경우, 정적분은 기하학적 방법과 정적분 및 면적 간의 관계를 사용하여 계산할 수 있다.
FUN-6.A.2 정적분의 성질로는 상수 times 함수의 적분, 두 함수의 합에 대한 적분, 적분 한계의 반전, 인접한 구간에서 한 함수의 적분이 포함된다.
FUN-6.A.3 정적분의 정의는 제거 가능한 불연속점이나 점프 불연속점이 있는 함수로 확장될 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Definite integrals obey useful rules: reversing the limits negates the value ($\int_b^a=-\int_a^b$), an integral over a zero-width interval is $0$, they add over adjacent intervals ($\int_a^c=\int_a^b+\int_b^c$), and constants factor out. Use these to combine or split given integral values.

한국어

정적분은 유용한 규칙을 따릅니다: 적분 한계를 뒤집으면 값의 부호가 반대가 됩니다($\int_b^a=-\int_a^b$), 너비가 0인 구간의 적분은 $0$이며, 인접한 구간에서 적분이 합쳐집니다($\int_a^c=\int_a^b+\int_b^c$), 상수는 outside로 빼낼 수 있습니다. 이를 사용하여 주어진 적분 값을 결합하거나 분리하십시오.

6.7

The Fundamental Theorem of Calculus and Definite Integrals · ⁨미적분학의 기본 정리와 정적분⁩

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-6 Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.	FUN-6.B Evaluate definite integrals analytically using the Fundamental Theorem of Calculus.	FUN-6.B.1 An antiderivative of a function $f$ is a function $g$ whose derivative is $f$. FUN-6.B.2 If a function $f$ is continuous on an interval containing $a$, the function defined by $F(x) = \int_{a}^{x} f(t)\,dt$ is an antiderivative of $f$ for $x$ in the interval. FUN-6.B.3 If $f$ is continuous on the interval $[a, b]$ and $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

The evaluation form of the FTC computes a definite integral from an antiderivative 原函数 $F$ (where $F'=f$):

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

So integrating a rate of change over $[a,b]$ gives the net change in the quantity – the single most-used result in the course.

Worked example. $\displaystyle\int_1^3 (2x+1)\,dx$: an antiderivative is $F(x)=x^2+x$, so the value is $F(3)-F(1)=12-2=10$.

한국어

FTC의 평가(evaluation) 형태는 원시함수(alternative primitive) $F$(여기서 $F'=f$)을 사용하여 정적분을 계산합니다:

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

따라서 $[a,b]$에 대해 변화율을 적분하면 양의 **순 변화(net change)**를 얻게 되는데, 이는 해당 과정에서 가장 많이 사용되는 결과입니다.

해설 예제. $\displaystyle\int_1^3 (2x+1)\,dx$: 원시함수는 $F(x)=x^2+x$이므로, 값은 $F(3)-F(1)=12-2=10$입니다.

6.8

Finding Antiderivatives and Indefinite Integrals · ⁨원시함수와 부정적분 찾기⁩

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-6 Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.	FUN-6.C Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.	FUN-6.C.1 $\int f(x)\,dx$ is an indefinite integral of the function $f$ and can be expressed as $\int f(x)\,dx = F(x) + C$, where $F'(x) = f(x)$ and $C$ is any constant. FUN-6.C.2 Differentiation rules provide the foundation for finding antiderivatives. FUN-6.C.3 Many functions do not have closed-form antiderivatives.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

An indefinite integral 不定积分 $\int f(x)\,dx=F(x)+C$ is the family of all antiderivatives (hence the constant of integration $C$). Reverse each derivative rule: the power rule becomes $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ (for $n\neq-1$), with $\int \frac1x\,dx=\ln|x|+C$, and the antiderivatives of $e^x$, $\sin x$, $\cos x$, and $\sec^2 x$ come straight from their derivatives.

한국어

부정적분 $\int f(x)\,dx=F(x)+C$은 모든 원함수들의 집합(즉, 적분 상수 $C$)입니다. 미분 법칙을 역으로 적용하면: 지수 법칙이 $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$($n\neq-1$에 대해)로 변하며, $\int \frac1x\,dx=\ln|x|+C$를 더하고, $e^x$, $\sin x$, $\cos x$, $\sec^2 x$의 원함수는 각각의 미분 결과에서 바로 얻을 수 있습니다.

6.9

Integrating Using Substitution · ⁨치환법을 이용한 적분⁩

Syllabus

English

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Learning Objective FUN-6.D: For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals.

FUN-6.D.1 Substitution of variables is a technique for finding antiderivatives.
FUN-6.D.2 For a definite integral, substitution of variables requires corresponding changes to the limits of integration.

한국어

지속적 이해 (FUN-6): 기하학적 지식과 수학적 규칙을 적용할 기회를 파악하면 적분이 단순화될 수 있다.

학습 목표 FUN-6.D: 치환이나 동치 형태로 변형이 필요한 피적분식에 대해: (a) 부정적분을 구한다. (b) 정적분을 계산한다.

FUN-6.D.1 변수 치환은 원함수를 구하기 위한 기법이다.
FUN-6.D.2 정적분에 대해 변수 치환을 할 때는 적분 범위의 대응적인 변경도 필요하다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

u-substitution 换元积分 reverses the chain rule: choose $u=g(x)$ so that $g'(x)$ also appears, turning $\int f(g(x))g'(x)\,dx$ into $\int f(u)\,du$. Remember to convert $dx$ to $du$ and, for a definite integral, either change the limits to $u$-values or convert back to $x$ at the end.

한국어

u-대입법은 연쇄 법칙의 역연산입니다: $u=g(x)$을 선택하여 $g'(x)$도 포함되어게 하여 $\int f(g(x))g'(x)\,dx$을 $\int f(u)\,du$로 변환합니다. $dx$를 $du$로 변환하는 것을 잊지 말고, 정적분의 경우 적분 한계를 $u$ 값으로 바꾸거나 최종적으로 다시 $x$로 환원해야 합니다.

6.10

Integrating Using Long Division and Completing the Square · ⁨나눗셈과 제곱완성을 이용한 적분⁩

Syllabus

English

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Learning Objective FUN-6.D: For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals.

FUN-6.D.3 Techniques for finding antiderivatives include rearrangements into equivalent forms, such as long division and completing the square.

한국어

지속적 이해 (FUN-6): 기하학적 지식과 수학적 규칙을 적용할 기회를 파악하면 적분이 단순화될 수 있다.

학습 목표 FUN-6.D: 치환이나 동치 형태로 변형이 필요한 피적분식에 대해: (a) 부정적분을 구한다. (b) 정적분을 계산한다.

FUN-6.D.3 원함수를 찾는 기법에는 등가 형태로 변형하는 방법이 포함되는데, 이를 위해 나눗셈이나 완전제곱식을 사용하기도 한다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

When a rational integrand is "top-heavy" (numerator degree $\ge$ denominator degree), long division rewrites it as a polynomial plus a proper fraction you can integrate. Completing the square in a denominator turns it into a form like $u^2+a^2$, leading to an arctangent antiderivative $\frac1a\arctan\frac{u}{a}+C$.

한국어

유계 함수가 "분자가 분모보다 차수가 높은"(numerator degree $\ge$ denominator degree) 경우, 나눗셈을 통해 다항식과 적분 가능한 진분수로 재표기할 수 있습니다. 분모에서의 제곱완성은 이를 $u^2+a^2$와 같은 형태로 만들어 아크탄젠트 원함수 $\frac1a\arctan\frac{u}{a}+C$를 유도합니다.

6.11

Integrating Using Integration by Parts · ⁨부분적분을 이용한 적분⁩

Syllabus

English

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Learning Objective FUN-6.E: For integrands requiring integration by parts: (a) Determine indefinite integrals. BC ONLY (b) Evaluate definite integrals. BC ONLY

FUN-6.E.1 Integration by parts is a technique for finding antiderivatives. BC ONLY

한국어

지속적 이해 (FUN-6): 기하학적 지식과 수학적 규칙을 적용할 기회를 파악하면 적분이 단순화될 수 있다.

학습 목표 FUN-6.E: 부분적분法이 필요한 피적분식에 대해: (a) 부정적분을 구한다. BC 전공자만 (b) 정적분을 계산한다. BC 전공자만

FUN-6.E.1 부분적분法은的原函数(alternative primitive)을 찾는 기법이다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Integration by parts 分部积分 reverses the product rule:

$$\int u\,dv = uv-\int v\,du.$$

Choose $u$ to simplify when differentiated and $dv$ to be easy to integrate (the LIATE guide: logs, inverse-trig, algebraic, trig, exponential). It handles products like $\int x e^x\,dx$ and $\int x\ln x\,dx$, sometimes applied twice.

Worked example. For $\int x e^x\,dx$, choose $u=x$ ($du=dx$) and $dv=e^x\,dx$ ($v=e^x$):

$$\int x e^x\,dx = x e^x-\int e^x\,dx = x e^x - e^x + C = e^x(x-1)+C.$$

한국어

부분적분은 곱의 미분 법칙의 역연산입니다:

$$\int u\,dv = uv-\int v\,du.$$

미분했을 때 $u$을 단순화하고 적분하기 쉬운 $dv$을 선택하십시오 (LIATE 가이드: 로그, 역삼각함수, 대수함수, 삼각함수, 지수함수). 이는 $\int x e^x\,dx$나 $\int x\ln x\,dx$과 같은 곱에 적용되며, 때로는 두 번 적용되기도 합니다.

작업 예시. $\int x e^x\,dx$에 대해 $u=x$($du=dx$)과 $dv=e^x\,dx$($v=e^x$)을 선택합니다:

$$\int x e^x\,dx = x e^x-\int e^x\,dx = x e^x - e^x + C = e^x(x-1)+C.$$

6.12

Integrating Using Linear Partial Fractions · ⁨선형 편분수를 이용한 적분⁩

Syllabus

English

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Learning Objective FUN-6.F: For integrands requiring integration by linear partial fractions: (a) Determine indefinite integrals. BC ONLY (b) Evaluate definite integrals. BC ONLY

FUN-6.F.1 Some rational functions can be decomposed into sums of ratios of linear, nonrepeating factors to which basic integration techniques can be applied. BC ONLY

한국어

지속적 이해 (FUN-6): 기하학적 지식과 수학적 규칙을 적용할 기회를 파악하면 적분이 단순화될 수 있다.

학습 목표 FUN-6.F: 선형 분수 분해法이 필요한 피적분식에 대해: (a) 부정적분을 구한다. BC 전공자만 (b) 정적분을 계산한다. BC 전공자만

FUN-6.F.1 일부有理函数(rational functions)는 기본 적분 기법을 적용할 수 있는 선형의 비반복 인자의 비율들의 합으로 분해될 수 있다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Partial fractions 部分分式 split a rational function with a factorable denominator into a sum of simpler fractions:

$$\frac{1}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},$$

each of which integrates to a logarithm. This technique is essential for the logistic differential equation in the next unit.

한국어

편분수는 인수분해 가능한 분모를 가진 유계 함수를 더 간단한 분수의 합으로 쪼개어 표현합니다:

$$\frac{1}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},$$

각각의 항은 로그함수로 적분됩니다. 이 기법은 다음 단원의 로지스틱 미분방정식에 필수적입니다.

6.13

Evaluating Improper Integrals · ⁨부등적분 평가하기⁩

Syllabus

English

Enduring Understanding (LIM-6): The use of limits allows us to show that the areas of unbounded regions may be finite.

Learning Objective LIM-6.A: Evaluate an improper integral or determine that the integral diverges. BC ONLY

LIM-6.A.1 An improper integral is an integral that has one or both limits infinite or has an integrand that is unbounded in the interval of integration. BC ONLY
LIM-6.A.2 Improper integrals can be determined using limits of definite integrals. BC ONLY

한국어

지속적 이해(LIM-6): 극한의 사용은 무한 영역의 면적이 유한함을 보여줄 수 있음을 입증한다.

학습 목표 LIM-6.A: 부정적분을 계산하거나 적분이 발산함을 판별한다. BC 전공자만

LIM-6.A.1 부적분은 적분 구간 중 하나 이상의 한계가 무한하거나, 적분구간 내에서 피적분함수가 유계하지 않은 적분이다. BC ONLY
LIM-6.A.2 부적분은 정적분의 극한을 사용하여 결정할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Improper integrals: converge or diverge

An improper integral 反常积分 has an infinite limit of integration or an infinite discontinuity in the integrand. Evaluate it as a limit: $\int_a^\infty f\,dx=\lim_{b\to\infty}\int_a^b f\,dx$. If the limit is a finite number the integral converges 收敛; otherwise it diverges 发散.

Worked example. $\displaystyle\int_1^\infty \frac{1}{x^2}\,dx=\lim_{b\to\infty}\left[-\frac1x\right]_1^b=\lim_{b\to\infty}\left(1-\frac1b\right)=1$, so it converges to $1$. By contrast $\int_1^\infty \frac1x\,dx$ gives $\lim_{b\to\infty}\ln b=\infty$ and diverges – the same integrand-shape can go either way.

한국어

부등적분: 수렴 또는 발산

부등적분은 적분의 무한한 한계나 피적분 함수의 무한한 불연속점을 가집니다. 이를 한계로 평가합니다: $\int_a^\infty f\,dx=\lim_{b\to\infty}\int_a^b f\,dx$. 만약 이 한계가 유한한 수이면 적분은 수렴하며, 그렇지 않으면 발산합니다.

해설 예시. $\displaystyle\int_1^\infty \frac{1}{x^2}\,dx=\lim_{b\to\infty}\left[-\frac1x\right]_1^b=\lim_{b\to\infty}\left(1-\frac1b\right)=1$이므로, 이는 $1$으로 수렴합니다. 반면 $\int_1^\infty \frac1x\,dx$는 $\lim_{b\to\infty}\ln b=\infty$을给出하여 발산합니다 – 동일한 피적분 함수 형태임에도 불구하고 수렴과 발산 중 하나일 수 있습니다.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
improper integral/ɪmˈprɒpə ˈɪntɪɡrəl/	부정적분
converges/kənˈvɜːdʒɪz/	수렴함
diverges/daɪˈvɜːdʒɪz/	발산함

6.14

Selecting Techniques for Antidifferentiation · ⁨원적분 기법 선택⁩

Syllabus

English

This topic is intended to focus on the skill of selecting an appropriate procedure for antidifferentiation. Students should be given opportunities to practice when and how to apply all learning objectives relating to antidifferentiation.

한국어

이 주제는 원함수 구法에 적합한 절차를 선택하는 능력에 초점을 맞춘 것이다. 학생들에게 원함수 구法 관련 모든 학습 목표를 언제, 어떻게 적용해야 하는지 연습할 기회가 제공되어야 한다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

The BC exam expects you to recognize which method fits: basic rules, substitution (a chain-rule pattern), by parts (a product), partial fractions (a factorable rational), or long division/completing the square. Being able to look at an integral and pick the right tool quickly is itself a tested skill.

한국어

BC 시험에서는 어떤 방법이 적합한지 식별하는 능력을 요구합니다: 기본 규칙, 대입법(연쇄 법칙 패턴), 부분적분(곱의 형태), 편분수(인수분해 가능한 유계 함수), 혹은 나눗셈/제곱완성 등입니다. 적분식을 보고 빠르게 적절한 도구를 선택할 수 있는 것 자체가 시험에 출제되는 기술입니다.

6.14

Exam tips · ⁨시험 팁⁩

English

Integration is antidifferentiation; use the power rule $\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C$ and don't forget the $+C$.
The Fundamental Theorem links the two: $\int_a^b f'(x)\,dx=f(b)-f(a)$, and $\tfrac{d}{dx}\int_a^x f(t)\,dt=f(x)$.
Approximate a definite integral with Riemann sums or the trapezoidal rule from a table of values.
A definite integral is a signed area (below the axis counts negative); split at sign changes for total area.
Use u-substitution and remember to change the limits (or back-substitute) accordingly.

한국어

적분은 원적분이므로 지수 법칙 $\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C$을 사용하고 $+C$을 잊지 마십시오.
기본 정리가 두 가지를 연결합니다: $\int_a^b f'(x)\,dx=f(b)-f(a)$, 그리고 $\tfrac{d}{dx}\int_a^x f(t)\,dt=f(x)$.
표의 값으로부터 라이만 합이나 사다리꼴 법칙을 사용하여 정적분을 근사하십시오.
정적분은 부호 있는 면적입니다(축 아래는 음수); 전체 면적을 위해 부호 변경 지점에서 분리하십시오.
u-대입법을 사용하고限界를 변경하거나(또는 역대입하여) accordingly 하십시오.

Differential Equations · ⁨미분방정식⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

7.1

Modeling Situations with Differential Equations · ⁨미분방정식을 이용한 모델링⁩

Syllabus

English

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.A: Interpret verbal statements of problems as differential equations involving a derivative expression.

FUN-7.A.1 Differential equations relate a function of an independent variable and the function's derivatives.

한국어

지속적 이해(FUN-7): 미분 방정식을 풀면 함수를 결정하고 모델을 개발할 수 있다.

학습 목표 FUN-7.A: 도함수 식을 포함하는 미분 방정식으로 문제의 서술을 해석한다.

FUN-7.A.1 미분 방정식은 독립 변수의 함수와 해당 함수의 도함수 사이의 관계를 나타낸다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

A differential equation 微分方程 relates a function to its derivatives. Many real situations are described by a rate: "the population grows at a rate proportional to its size" becomes $\dfrac{dP}{dt}=kP$. Setting up the equation from a verbal description – identifying what changes and what it is proportional to – is the first skill.

한국어

미분방정식은 함수와 그 미분값 사이의 관계를 나타냅니다. 많은 실제 현상은 변화율로 설명되는데, "인구가 크기에 비례하는 속도로 증가한다"는 문장은 $\dfrac{dP}{dt}=kP$로 표현됩니다. 구술 설명에서 방정식을 세우기 위해 어떤 것이 변하고 그것이 무엇에 비례하는지를 식별하는 것이 첫 번째 기술입니다.

7.2

Verifying Solutions for Differential Equations · ⁨미분방정식의 해 검증⁩

Syllabus

English

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.B: Verify solutions to differential equations.

FUN-7.B.1 Derivatives can be used to verify that a function is a solution to a given differential equation.
FUN-7.B.2 There may be infinitely many general solutions to a differential equation.

한국어

지속적 이해(FUN-7): 미분 방정식을 풀면 함수를 결정하고 모델을 개발할 수 있다.

학습 목표 FUN-7.B: 미분 방정식의 해를 검증한다.

FUN-7.B.1 도함수를 사용하여 주어진 함수가 특정 미분 방정식의 해임을 확인할 수 있다.
FUN-7.B.2 미분 방정식의 일반해는 무한히 많을 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

A solution is a function that satisfies the equation. To verify a proposed solution, differentiate it and substitute into the equation, checking that both sides agree. A general solution contains a constant $C$; a particular solution fixes $C$ from a condition.

한국어

해는 방정식을 만족시키는 함수입니다. 제안된 해를 검증하려면 미분하여 방정식에 대입하고 양변이 일치함을 확인합니다. 일반해에는 상수 $C$이 포함되어 있으며, 특이해는 조건으로부터 $C$을 결정합니다.

7.3

Sketching Slope Fields · ⁨기울기 장(slope field) 그리기⁩

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-7 Solving differential equations allows us to determine functions and develop models.	FUN-7.C Estimate solutions to differential equations.	FUN-7.C.1 A slope field is a graphical representation of a differential equation on a finite set of points in the plane. FUN-7.C.2 Slope fields provide information about the behavior of solutions to first-order differential equations.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Slope fields & solution curves

A slope field 斜率场 draws a short line segment at many points, each with the slope $\dfrac{dy}{dx}$ the equation gives there. It pictures the family of solution curves without solving. To sketch one, evaluate the right-hand side at each grid point and draw a segment of that slope.

한국어

기울기장 및 해 곡선

기울기장(slope field) 은 방정식이 해당 지점에서 주는 기울기 $\dfrac{dy}{dx}$을 가진 짧은 선분을 여러 점에 그려서, 해를 풀지 않고도 해 곡선들의 집합을 시각화합니다. 이를 그리려면 각 격자 점에서 우변의 값을 계산하고 해당 기울기의 선분을 그립니다.

Explore · ⁨탐색하기⁩

Read a differential equation as a slope field · ⁨미분방정식을 기울기 장으로 읽기⁩

A slope field draws the slope $dy/dx$ at each point. A solution curve threads through, always tangent to the little segments — you can sketch it by following the flow. · ⁨기울기 장은 각 점에서 기울기 $dy/dx$를 그려줍니다. 해 곡선이 작은 선분들을 항상 접하도록 관통하며, 흐름을 따라 그리면 손으로 스케치할 수 있습니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/	미분 방정식
slope field/sləʊp fiːld/	기울기 장
Euler's method/ˈɔɪləz ˈmeθəd/	오일러 방법
separable/ˈsepərəbl/	분리 가능하다
initial condition/ɪˈnɪʃl kənˈdɪʃn/	초기 조건
exponential growth or decay/ˌekspəˈnenʃl ɡrəʊθ ɔː dɪˈkeɪ/	지수 성장 또는 감쇠
logistic model/ləˈdʒɪstɪk ˈmɒdl/	로지스틱 모델
carrying capacity/ˈkæriɪŋ kəˈpæsɪti/	환경 수용력

7.4

Reasoning Using Slope Fields · ⁨기울기 장을 이용한 추론⁩

Syllabus

English

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.C: Estimate solutions to differential equations.

FUN-7.C.3 Solutions to differential equations are functions or families of functions.

한국어

지속적 이해(FUN-7): 미분 방정식을 풀면 함수를 결정하고 모델을 개발할 수 있다.

학습 목표 FUN-7.C: 미분 방정식의 해를 추정한다.

FUN-7.C.3 미분 방정식의 해는 함수 또는 함수族이다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

A solution curve follows the segments like a boat following a current. From a slope field you can sketch the particular solution through a given point, describe long-run behavior, and locate where solutions level off (slopes near zero) – reasoning about solutions purely from the picture.

한국어

해 곡선 은 선분들을 배가 물결을 타듯 따릅니다. 기울기 장에서 주어진 점을 지나는 특정 해를 그릴 수 있으며, 장기적 거동을 설명하고 해가 평탄해지는 지점(기울기가 0에 가까운 곳)을 찾을 수 있습니다. 이는 그림만으로 해에 대한 추론을 하는 것입니다.

7.5

Approximating Solutions Using Euler's Method · ⁨오일러的方法来用于近似解⁩

Syllabus

English

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.C: Estimate solutions to differential equations.

FUN-7.C.4 Euler's method provides a procedure for approximating a solution to a differential equation or a point on a solution curve. BC ONLY

한국어

지속적 이해(FUN-7): 미분 방정식을 풀면 함수를 결정하고 모델을 개발할 수 있다.

학습 목표 FUN-7.C: 미분 방정식의 해를 추정한다.

FUN-7.C.4 오일러 방법은 미분방정식의 해를 근사하거나 해 곡선 위의 점을 구하는 절차를 제공한다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Euler's method

Euler's method 欧拉方法 approximates a solution numerically by stepping along the slope field. Starting from a known point, take a small step $\Delta x$ and update:

$$y_{\text{new}}=y_{\text{old}}+\frac{dy}{dx}\cdot\Delta x.$$

Repeat for each step. Smaller steps give a better approximation. (This is a BC-only technique.)

Exam skill: be able to carry out two or three Euler steps by hand from a table, and know that Euler's method under- or over-estimates depending on the solution's concavity.

Worked example. Approximate $y(1)$ for $\dfrac{dy}{dx}=x+y$, $y(0)=1$, with step $\Delta x=0.5$. Step 1: slope at $(0,1)$ is $0+1=1$, so $y(0.5)\approx 1+1(0.5)=1.5$. Step 2: slope at $(0.5,1.5)$ is $0.5+1.5=2$, so $y(1)\approx 1.5+2(0.5)=2.5$.

한국어

오일러의 방법

오일러의 방법은 기울기 장을 따라 작은 단계로 이동하여 해를 수치적으로 근사합니다.已知点에서 시작하여 작은 단계 $\Delta x$를 취하고 업데이트합니다:

$$y_{\text{new}}=y_{\text{old}}+\frac{dy}{dx}\cdot\Delta x.$$

각 단계마다 반복합니다. 더 작은 단계일수록 더 정확한 근사가 됩니다. (이것은 BC 과목 전용 기법입니다.)

시험 기술: 표에서 손으로 두 세 번의 오일러 단계를 수행할 수 있어야 하며, 오일러의 방법이 해의 볼록성에 따라 과소 또는 과대 추정함을 알고 있어야 합니다.

연습 문제. $y(1)$에 대해 $\dfrac{dy}{dx}=x+y$, $y(0)=1$를 step $\Delta x=0.5$로 근사하십시오. Step 1: $(0,1)$에서의 기울기는 $0+1=1$이므로, $y(0.5)\approx 1+1(0.5)=1.5$입니다. Step 2: $(0.5,1.5)$에서의 기울기는 $0.5+1.5=2$이므로, $y(1)\approx 1.5+2(0.5)=2.5$입니다.

7.6

Finding General Solutions Using Separation of Variables · ⁨변수분리법을 통한 일반해 구하기⁩

Syllabus

English

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.D: Determine general solutions to differential equations.

FUN-7.D.1 Some differential equations can be solved by separation of variables.
FUN-7.D.2 Antidifferentiation can be used to find general solutions to differential equations.

한국어

지속적 이해(FUN-7): 미분 방정식을 풀면 함수를 결정하고 모델을 개발할 수 있다.

학습 목표 FUN-7.D: 미분 방정식의 일반해를 구한다.

FUN-7.D.1 일부 미분 방정식은 변수 분리법을 통해 풀 수 있다.
FUN-7.D.2 원적분을 이용하여 미분 방정식의 일반해를 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

A separable 可分离 differential equation can be written with all the $y$'s on one side and all the $x$'s on the other, then integrated:

$$\frac{dy}{dx}=g(x)h(y)\ \Rightarrow\ \int\frac{dy}{h(y)}=\int g(x)\,dx.$$

This produces the general solution (with a $+C$), the main analytic method for solving differential equations in this course.

Worked example. Solve $\dfrac{dy}{dx}=xy$ with $y(0)=2$. Separating, $\int\frac{dy}{y}=\int x\,dx$ gives $\ln|y|=\frac{x^2}{2}+C$, so $y=Ae^{x^2/2}$. The condition $y(0)=2$ gives $A=2$, so $y=2e^{x^2/2}$.

한국어

분리 가능한 미분 방정식은 모든 $y$를 한 쪽에, 모든 $x$를 다른 쪽에 배치한 후 적분하여 쓸 수 있습니다:

$$\frac{dy}{dx}=g(x)h(y)\ \Rightarrow\ \int\frac{dy}{h(y)}=\int g(x)\,dx.$$

이것은 상수 $+C$를 포함하는 일반해를 생성하며, 이 과정의 미분 방정식을 푸는 주요 해석적 방법입니다.

연습 문제. $\dfrac{dy}{dx}=xy$를 $y(0)=2$ 조건으로 푼다. 변수 분리 시, $\int\frac{dy}{y}=\int x\,dx$은 $\ln|y|=\frac{x^2}{2}+C$을 주어, 따라서 $y=Ae^{x^2/2}$이다. 조건 $y(0)=2$은 $A=2$을 주어, 따라서 $y=2e^{x^2/2}$이다.

7.7

Finding Particular Solutions Using Initial Conditions · ⁨초기 조건을 사용하여 특정 해 구하기⁩

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-7 Solving differential equations allows us to determine functions and develop models.	FUN-7.E Determine particular solutions to differential equations.	FUN-7.E.1 A general solution may describe infinitely many solutions to a differential equation. There is only one particular solution passing through a given point. FUN-7.E.2 The function $F$ defined by $F(x) = y_0 + \int_a^x f(t)\,dt$ is a particular solution to the differential equation $\dfrac{dy}{dx} = f(x)$, satisfying $F(a) = y_0$. FUN-7.E.3 Solutions to differential equations may be subject to domain restrictions.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

An initial condition 初始条件 (a known point, e.g. $y(0)=5$) pins down the constant $C$. Solve for the general solution, substitute the condition to find $C$, then write the particular solution. Watch the domain – a particular solution is valid only on the interval containing the initial point.

한국어

초기 조건(예: ⟨$y(0)=5$⟩와 같은已知点)은 상수 $C$을 결정합니다. 일반해를 구한 후 조건을 대입하여 $C$을 구한 뒤, 특정 해를 작성합니다. 정의역을 주의하세요 – 특정 해는 초기 점이 포함된 구간에서만 유효합니다.

상수가 곡선의 족을 제공하고, 초기 조건이 하나를 선택합니다 — 상수값이 곡선 가족을 구성하며, 초기 조건이 하나를 선택함

7.8

Exponential Models with Differential Equations · ⁨미분 방정식을 이용한 지수 모델⁩

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-7 Solving differential equations allows us to determine functions and develop models.	FUN-7.F Interpret the meaning of a differential equation and its variables in context.	FUN-7.F.1 Specific applications of finding general and particular solutions to differential equations include motion along a line and exponential growth and decay. FUN-7.F.2 The model for exponential growth and decay that arises from the statement "The rate of change of a quantity is proportional to the size of the quantity" is $\dfrac{dy}{dt} = ky$.
	FUN-7.G Determine general and particular solutions for problems involving differential equations in context.	FUN-7.G.1 The exponential growth and decay model, $\dfrac{dy}{dt} = ky$, with initial condition $y = y_0$ when $t = 0$, has solutions of the form $y = y_0 e^{kt}$.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Exponential vs logistic growth

The equation $\dfrac{dy}{dt}=ky$ says the rate of change is proportional to the amount – giving exponential growth or decay 指数增长. Separating variables yields

$$y=y_0 e^{kt},$$

with $k>0$ for growth and $k<0$ for decay. This models unrestricted population growth, radioactive decay, and continuously compounded interest.

한국어

지수적 성장 vs 로지스틱 성장

방정식 $\dfrac{dy}{dt}=ky$은 변화율이 양에 비례함을 말하며, 이는 지수 성장 또는 감쇠를 의미합니다. 변수 분리를 통해 다음을 얻습니다

$$y=y_0 e^{kt},$$

여기서 $k>0$는 성장에, $k<0$는 감쇠에 해당합니다. 이는 무제한 인구 성장, 방사성 붕괴, 그리고 연속複利 이자를 모델링합니다.

Explore · ⁨탐색하기⁩

An exponential growth/decay model · ⁨지수 성장/감쇠 모델⁩

y = a·e^(bx) + c

The equation $dy/dt=ky$ has exponential solutions: quantity changes at a rate proportional to itself, giving unbounded growth ($k>0$) or decay to zero ($k<0$). · ⁨방정식 $dy/dt=ky$의 해는 지수 함수입니다: 양은自身에 비례하는 속도로 변화하므로 무한 성장($k>0$)이나 0으로 수렴하는 감쇠($k<0$)를 보입니다.⁩

7.9

Logistic Models with Differential Equations · ⁨미분 방정식을 이용한 로지스틱 모델⁩

Syllabus

English

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.H: Interpret the meaning of the logistic growth model in context. BC ONLY

FUN-7.H.1 The model for logistic growth that arises from the statement "The rate of change of a quantity is jointly proportional to the size of the quantity and the difference between the quantity and the carrying capacity" is $\dfrac{dy}{dt} = ky(a - y)$. BC ONLY
FUN-7.H.2 The logistic differential equation and initial conditions can be interpreted without solving the differential equation. BC ONLY
FUN-7.H.3 The limiting value (carrying capacity) of a logistic differential equation as the independent variable approaches infinity can be determined using the logistic growth model and initial conditions. BC ONLY
FUN-7.H.4 The value of the dependent variable in a logistic differential equation at the point when it is changing fastest can be determined using the logistic growth model and initial conditions. BC ONLY

한국어

지속적 이해(FUN-7): 미분 방정식을 풀면 함수를 결정하고 모델을 개발할 수 있다.

학습 목표 FUN-7.H: 로지스틱 성장 모델의 의미를 맥락 속에서 해석한다. BC ONLY

FUN-7.H.1 "량的变化率이 양의 크기와 그 양과 환경 수용력 사이의 차이에 비례한다"는 서술에서 도출되는 로지스틱 성장 모델은 $\dfrac{dy}{dt} = ky(a - y)$이다. BC ONLY
FUN-7.H.2 로지스틱 미분방정식과 초기 조건은 미분방정식을 풀지 않고도 해석할 수 있다. BC ONLY
FUN-7.H.3 로지스틱 미분방정식의 종속 변수가 무한대로 갈 때의 극한 값(환경 수용력)은 로지스틱 성장 모델과 초기 조건을 사용하여 결정할 수 있다. BC ONLY
FUN-7.H.4 로지스틱 미분방정식에서 종속 변수가 가장 빠르게 변화하는 지점에서의 값은 로지스틱 성장 모델과 초기 조건을 사용하여 결정할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

English

Real growth is limited by resources, so the logistic model 逻辑斯蒂模型 adds a carrying capacity 环境容纳量 $L$:

$$\frac{dP}{dt}=kP\!\left(1-\frac{P}{L}\right).$$

Growth is nearly exponential when $P$ is small, slows as $P$ approaches $L$, and stops at $P=L$. Key BC facts: the population levels off at $L$ ($\lim_{t\to\infty}P=L$), and it grows fastest when $P=\tfrac{L}{2}$ (the inflection point of the S-shaped curve). You are expected to read $L$ and the fastest-growth value directly from the equation.

Worked example. For $\dfrac{dP}{dt}=0.05\,P\!\left(1-\dfrac{P}{2000}\right)$, the carrying capacity is $L=2000$ (the population levels off there), and growth is fastest when $P=\dfrac{L}{2}=1000$ – both read straight off the equation, no solving needed.

한국어

실제 성장은 자원에 의해 제한되므로, 로지스틱 모델이 보유 능력 $L$을 추가합니다:

$$\frac{dP}{dt}=kP\!\left(1-\frac{P}{L}\right).$$

$P$가 작을 때 성장은 거의 지수함수적이며, $P$이 $L$에 가까워지면서 성장률이 둔화되고, $P=L$에서 정지합니다. 주요 BC 지식: 인구는 $L$에서 안정화됩니다 ($\lim_{t\to\infty}P=L$), 그리고 $P=\tfrac{L}{2}$일 때 가장 빠르게 성장합니다 (S자 곡선의 변곡점). $L$과 가장 빠른 성장 값을 식에서 직접 읽어야 합니다.

해설 예제. $\dfrac{dP}{dt}=0.05\,P\!\left(1-\dfrac{P}{2000}\right)$에 대해 환경 수용량은 $L=2000$(인구가 거기에 도달하여 안정화됨)이며, <<-$P=\dfrac{L}{2}=1000$일 때 성장률이 가장 빠릅니다. 두 값 모두 식에서 바로 읽을 수 있어 연산할 필요가 없습니다.

*로지스틱 모델은 P=L/2에서 가장 빠르게 성장하고 환경 수용량 L에서 안정화됩니다

7.9

Exam tips · ⁨시험 팁⁩

English

Solve a separable equation by getting all $y$ on one side and all $x$ on the other, then integrating both sides (add $+C$ once).
Use the initial condition to find $C$ (a particular solution).
Sketch or read a slope field: the little segments show $\tfrac{dy}{dx}$ at each point, and a solution curve follows them.
Recognise exponential models $\tfrac{dy}{dt}=ky\Rightarrow y=Ce^{kt}$ (growth/decay).
A differential equation gives the slope — you must integrate to recover the function.

한국어

분리 가능한 방정식을 풀려면 모든 $y$을 한 쪽에, 모든 $x$을 다른 쪽에 모은 후 양변을 적분합니다(상수 $+C$을 한 번 더 추가합니다).
초기 조건을 사용하여 $C$을 구합니다(특해).
**기울기 장(slope field)**을 그리거나 읽습니다: 작은 선분들은 각 점에서 $\tfrac{dy}{dx}$을 보여주며, 해 곡선은 이들을 따라갑니다.
지수 모델 $\tfrac{dy}{dt}=ky\Rightarrow y=Ce^{kt}$(성장/감쇠)을 인식합니다.
미분 방정식은 기울기를 제공하므로, 함수를 복원하려면 적분해야 합니다.

Applications of Integration · ⁨적분의 적용⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

8.1

Finding the Average Value of a Function on an Interval

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-4 Definite integrals allow us to solve problems involving the accumulation of change over an interval.	CHA-4.B Determine the average value of a function using definite integrals.	CHA-4.B.1 The average value of a continuous function $f$ over an interval $[a, b]$ is $\dfrac{1}{b-a}\int_a^b f(x)\,dx$.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The average value 平均值 of $f$ over $[a,b]$ is the integral divided by the width:

$$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

It is the constant height a rectangle would need to have the same area as the region under $f$. Do not confuse this with the average rate of change (which uses the derivative).

Worked example. The average value of $f(x)=x^2$ on $[0,3]$ is $\dfrac{1}{3}\displaystyle\int_0^3 x^2\,dx=\dfrac13\left[\dfrac{x^3}{3}\right]_0^3=\dfrac13(9)=3$.

Explore · ⁨탐색하기⁩

The average value of a function · ⁨함수의 평균값⁩

y = ax³ + bx² + cx + d

The average value of $f$ on $[a,b]$ is its integral divided by the width — the constant height whose rectangle has the same area as under the curve. · ⁨$f$의 $[a,b]$에서의 평균값은 적분을 너비로 나눈 값으로, 곡선 아랫면적과 같은 면적을 가진 사각형의 일정 높이에 해당합니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
average value/ˈævrɪdʒ ˈvæljuː/	평균값

8.2

Connecting Position, Velocity, and Acceleration Using Integrals

Syllabus

English

Enduring Understanding (CHA-4): Definite integrals allow us to solve problems involving the accumulation of change over an interval.

Learning Objective CHA-4.C: Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion.

CHA-4.C.1 For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle's displacement over the interval of time, and the definite integral of speed represents the particle's total distance traveled over the interval of time.

한국어

지속적 이해 (CHA-4): 정적분은 구간 내에서의 변화량 축적 문제를 해결할 수 있게 해준다.

학습 목표 CHA-4.C: 직선 운동이 포함된 문제에서 정적분을 사용하여 위치와 변화율의 값을 구할 수 있다.

CHA-4.C.1 시간 구간에서 직선 운동을 하는 입자에 대해, 속도의 정적분은 해당 시간 구간의 변위를 나타내며, 속력(steady speed)의 정적분은 해당 시간 구간의 이동 거리를 나타낸다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

For straight-line motion, integration reverses differentiation:

$$v(t)=\int a(t)\,dt,\qquad s(t)=\int v(t)\,dt.$$

Two key distinctions: displacement 位移 is $\int_a^b v\,dt$ (net change in position), while total distance 总路程 is $\int_a^b |v|\,dt$ (splitting where $v$ changes sign). Speed is $|v|$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
displacement/dɪˈspleɪsmənt/	변위
total distance/ˈtəʊtl ˈdɪstəns/	총 이동 거리

8.3

Using Accumulation Functions and Definite Integrals in Applied Contexts

Syllabus

English

Enduring Understanding (CHA-4): Definite integrals allow us to solve problems involving the accumulation of change over an interval.

Learning Objective CHA-4.D: Interpret the meaning of a definite integral in accumulation problems.

CHA-4.D.1 A function defined as an integral represents an accumulation of a rate of change.
CHA-4.D.2 The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.

Learning Objective CHA-4.E: Determine net change using definite integrals in applied contexts.

CHA-4.E.1 The definite integral can be used to express information about accumulation and net change in many applied contexts.

한국어

지속적 이해 (CHA-4): 정적분은 구간 내에서의 변화량 축적 문제를 해결할 수 있게 해준다.

학습 목표 CHA-4.D: 적분 함수와 정적분을 사용하여 변화율을 적분하는 의미를 해석할 수 있다.

CHA-4.D.1 적분으로 정의된 함수는 변화율의 적분(누적)을 나타낸다.
CHA-4.D.2 어떤 양의 변화율(steady rate)의 정적분은 그 양에 대한 순변화(net change)를 주어진 구간에 대해 제공한다.

학습 목표 CHA-4.E: 응용 맥락에서 정적분을 사용하여 순변화를 구할 수 있다.

CHA-4.E.1 정적분은 다양한 응용 맥락에서 누적과 순변화에 대한 정보를 표현하는 데 사용할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

When a rate is given (flow rate, sales per day), the definite integral gives the accumulated total, and $\int_a^b R(t)\,dt$ carries the units of $R$ times time. A common setup: initial amount $+\int(\text{rate in}-\text{rate out})\,dt$ gives the amount at a later time. Always interpret the answer in context, with units.

8.4

Finding the Area Between Curves Expressed as Functions of x

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.A: Calculate areas in the plane using the definite integral.

CHA-5.A.1 Areas of regions in the plane can be calculated with definite integrals.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.A: 정적분을 사용하여 평면상의 면적을 계산할 수 있다.

CHA-5.A.1 평면상의 영역의 면적은 정적분을 통해 계산할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A potter shaping a vessel: volumes of revolution rotate a plane region about an axis

The area between $y=f(x)$ (top) and $y=g(x)$ (bottom) from $a$ to $b$ is

$$\int_a^b\big(f(x)-g(x)\big)\,dx.$$

Find the intersection points for the limits, and always subtract top minus bottom.

The area between two curves is the integral of top minus bottom

Worked example. Between $y=x$ and $y=x^2$ (crossing at $x=0,1$, with $y=x$ on top), the area is $\displaystyle\int_0^1 (x-x^2)\,dx=\left[\dfrac{x^2}{2}-\dfrac{x^3}{3}\right]_0^1=\dfrac16$.

8.5

Finding the Area Between Curves Expressed as Functions of y

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.A: Calculate areas in the plane using the definite integral.

CHA-5.A.2 Areas of regions in the plane can be calculated using functions of either $x$ or $y$.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.A: 정적분을 사용하여 평면상의 면적을 계산할 수 있다.

CHA-5.A.2 평면상의 영역의 면적은 x-함수나 y-함수를 사용하여 계산할 수 있다. $x$ $y$

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

When curves are easier to describe as $x=f(y)$, integrate with respect to $y$ instead, using right minus left:

$$\int_c^d\big(f_{\text{right}}(y)-g_{\text{left}}(y)\big)\,dy.$$

Choosing to integrate in $y$ can avoid splitting the region into several pieces.

8.6

Finding the Area Between Curves That Intersect More Than Twice

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.A: Calculate areas in the plane using the definite integral.

CHA-5.A.3 Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.A: 정적분을 사용하여 평면상의 면적을 계산할 수 있다.

CHA-5.A.3 평면상의 특정 영역의 면적은 두 개 이상의 정적분의 합으로 계산하거나 두 함수의 차이의 절댓값의 정적분을 평가하여 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

If two curves cross several times, the top and bottom switch. Split the region at each intersection and integrate each piece with the correct top-minus-bottom (or use $\int|f-g|$), then add the pieces.

8.7

Volumes with Cross Sections: Squares and Rectangles

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.B: Calculate volumes of solids with known cross sections using definite integrals.

CHA-5.B.1 Volumes of solids with square and rectangular cross sections can be found using definite integrals and the area formulas for these shapes.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.B: 알려진 단면을 가진 입체의 부피를 정적분을 사용하여 계산할 수 있다.

CHA-5.B.1 정사각형 및 직사각형 단면을 가진 입체의 부피는 정적분과 해당 도형의 면적 공식으로 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

If a solid's cross sections 横截面 perpendicular to the $x$-axis are squares or rectangles, integrate their area. With side length equal to the distance between two curves, a square cross section gives

$$V=\int_a^b \big(f(x)-g(x)\big)^2\,dx.$$

The method is always "integrate the cross-sectional area."

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
cross sections/krɒs ˈsekʃnz/	단면

8.8

Volumes with Cross Sections: Triangles and Semicircles

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.B: Calculate volumes of solids with known cross sections using definite integrals.

CHA-5.B.2 Volumes of solids with triangular cross sections can be found using definite integrals and the area formulas for these shapes.
CHA-5.B.3 Volumes of solids with semicircular and other geometrically defined cross sections can be found using definite integrals and the area formulas for these shapes.
- Illustrative examples for CHA-5.B.3:
  - The volume of a funnel whose cross sections are circles can be found using the area formula for a circle and definite integrals (see 2016 AB Exam FRQ #5(b)).
  - The volume of a solid whose cross sectional area is defined using a function can be found using the known area function and a definite integral (see 2009 AB Exam FRQ #4(c)).

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.B: 알려진 단면을 가진 입체의 부피를 정적분을 사용하여 계산할 수 있다.

CHA-5.B.2 삼각형 단면을 가진 입체의 부피는 정적분과 해당 도형의 면적 공식으로 구할 수 있다.
CHA-5.B.3 반원 및 기타 기하학적으로 정의된 단면을 가진 입체의 부피는 정적분과 해당 도형의 면적 공식으로 구할 수 있다.
- *CHA-5.B.3 참고 예시:
  - 단면이 원인 깔대기의 부피는 원의 면적 공식과 정적분을 사용하여 구할 수 있다 (2016 AB Exam FRQ #5(b) 참조).
  - 단면적(steady area)이 함수로 정의된 입체의 부피는 알려진 면적(steady area) 함수와 정적분을 사용하여 구할 수 있다 (2009 AB Exam FRQ #4(c) 참조).

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Same idea, different area formula: for equilateral-triangle cross sections use $A=\tfrac{\sqrt3}{4}s^2$, and for semicircular ones $A=\tfrac{\pi}{8}s^2$ (with $s$ the distance between the curves). Substitute the area formula and integrate.

8.9

Volume with Disc Method: Revolving Around the x- or y-Axis

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.C: Calculate volumes of solids of revolution using definite integrals.

CHA-5.C.1 Volumes of solids of revolution around the $x$- or $y$-axis may be found by using definite integrals with the disc method.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.C: 정적분을 사용하여 회전체의 부피를 계산한다.

CHA-5.C.1 x축 또는 y축周围的人回전 입체의 부피는 원반법을 사용하여 정적분으로 구할 수 있다. $x$ $y$

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Solids of revolution: the disc method

Revolving a region around an axis makes a solid whose cross sections are discs. The disc method 圆盘法 integrates $\pi(\text{radius})^2$:

$$V=\pi\int_a^b \big(R(x)\big)^2\,dx,$$

where the radius $R$ is the distance from the curve to the axis. Use $dy$ when revolving around the $y$-axis.

The disc method: rotating y=f(x) about the axis sweeps out disks of radius f(x)

Worked example. Revolving the region under $y=\sqrt{x}$ from $0$ to $4$ about the $x$-axis gives discs of radius $\sqrt{x}$: $V=\pi\displaystyle\int_0^4 (\sqrt{x})^2\,dx=\pi\int_0^4 x\,dx=8\pi$.

Rotating a region about an axis sweeps out a solid of revolution

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
disc method/dɪsk ˈmeθəd/	원판 방법

8.10

Volume with Disc Method: Revolving Around Other Axes

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.C: Calculate volumes of solids of revolution using definite integrals.

CHA-5.C.2 Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.C: 정적분을 사용하여 회전체의 부피를 계산한다.

CHA-5.C.2 평면상의 모든 수평선 또는 수직선을 중심으로 회전하는 회전체의 부피는 원반법과 정적분을 사용하여 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

When the axis of revolution is a horizontal or vertical line like $y=k$ (not an axis), the radius adjusts: $R=|f(x)-k|$. Set up the radius as the distance from the curve to that line, then integrate $\pi R^2$ as before.

8.11

Volume with Washer Method: Revolving Around the x- or y-Axis

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.C: Calculate volumes of solids of revolution using definite integrals.

CHA-5.C.3 Volumes of solids of revolution around the $x$- or $y$-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.C: 정적분을 사용하여 회전체의 부피를 계산한다.

CHA-5.C.3 $x$축 또는 $y$축을 중심으로 회전하며 단면이 고리 모양인 회전체의 부피는 와셔법과 정적분을 사용하여 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Volume by the washer method

If the region does not touch the axis, revolving leaves a hole, so cross sections are washers (rings). The washer method 垫圈法 subtracts the inner disc:

$$V=\pi\int_a^b\Big(R_{\text{outer}}^2-R_{\text{inner}}^2\Big)\,dx.$$

Identify the outer and inner radii as distances from each curve to the axis.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
washer method/ˈwɒʃə ˈmeθəd/	와셔 방법

8.12

Volume with Washer Method: Revolving Around Other Axes

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.C: Calculate volumes of solids of revolution using definite integrals.

CHA-5.C.4 Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.C: 정적분을 사용하여 회전체의 부피를 계산한다.

CHA-5.C.4 평면상의 모든 수평선 또는 수직선을 중심으로 회전하며 단면이 고리 모양인 회전체의 부피는 와셔법과 정적분을 사용하여 구할 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

As with discs, revolving around a line $y=k$ or $x=k$ shifts both radii – each becomes the distance from its curve to that line. Sketch the region and the axis, label $R_{\text{outer}}$ and $R_{\text{inner}}$, then integrate the difference of squares.

8.13

The Arc Length of a Smooth Curve and Distance Traveled

Syllabus

English

Enduring Understanding (CHA-6): Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.

Learning Objective CHA-6.A: Determine the length of a curve in the plane defined by a function, using a definite integral. BC ONLY

CHA-6.A.1 The length of a planar curve defined by a function can be calculated using a definite integral. BC ONLY

한국어

지속적 이해 (CHA-6): 정적분을 이용하면 구간 내 길이의 누적 변화에 관한 문제를 해결할 수 있다.

학습 목표 CHA-6.A: 정적분을 사용하여 함수로 정의된 평면상의 곡선 길이를 구한다. BC ONLY

CHA-6.A.1 함수로 정의된 평면 곡선의 길이는 정적분을 사용하여 계산할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The arc length 弧长 of $y=f(x)$ from $a$ to $b$ is

$$L=\int_a^b\sqrt{1+\big(f'(x)\big)^2}\,dx.$$

This BC-only formula comes from summing tiny hypotenuses $\sqrt{dx^2+dy^2}$. The same idea gives the distance a particle travels along a curved path.

Worked example. Find the arc length of $y=\tfrac{2}{3}x^{3/2}$ from $x=0$ to $x=3$. Here $f'(x)=x^{1/2}$, so $1+(f')^2=1+x$ and

$$L=\int_0^3\sqrt{1+x}\,dx=\left[\tfrac{2}{3}(1+x)^{3/2}\right]_0^3=\tfrac{2}{3}(8-1)=\tfrac{14}{3}.$$

The Golden Gate Bridge and its sweeping main cables — The bridge's main cable hangs as a smooth curve; an integral gives its exact length

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
arc length/ɑːk leŋθ/	arc 길이를 가진다

8.13

Exam tips

Area between curves is $\int(\text{top}-\text{bottom})\,dx$ — find the intersection points for the limits and keep top minus bottom.
For a volume of revolution, add up disc/washer cross-sections of area $\pi r^2$ (or $\pi(R^2-r^2)$).
The average value of $f$ on $[a,b]$ is $\tfrac{1}{b-a}\int_a^b f\,dx$.
Accumulated change is $\int$ of a rate: total = initial value $+\int_a^b(\text{rate})\,dt$.
Integration means "adding up infinitely many tiny pieces" — set up the integrand as one thin slice.

Parametric Equations, Polar Coordinates, and Vector-Valued Functions · ⁨매개변수 방정식, 극좌표 및 벡터 함수⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

9.1

Defining and Differentiating Parametric Equations

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
CHA-3 Derivatives allow us to solve real-world problems involving rates of change.	CHA-3.G Calculate derivatives of parametric functions. BC ONLY	CHA-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to parametric functions. BC ONLY CHA-3.G.2 For a curve defined parametrically, the value of $\dfrac{dy}{dx}$ at a point on the curve is the slope of the line tangent to the curve at that point. $\dfrac{dy}{dx}$, the slope of the line tangent to a curve defined using parametric equations, can be determined by dividing $\dfrac{dy}{dt}$ by $\dfrac{dx}{dt}$, provided $\dfrac{dx}{dt}$ does not equal zero. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The ISS in orbit: parametric equations describe position as a function of time

Parametric equations 参数方程 give $x$ and $y$ each as functions of a parameter $t$ (often time): $x=x(t)$, $y=y(t)$. They trace a curve that need not be a function of $x$. The slope of the curve is found with the chain rule:

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\qquad(dx/dt\neq0).$$

Worked example. For $x=t^2$, $y=t^3-t$, find the slope at $t=2$. Here $\dfrac{dx}{dt}=2t$ and $\dfrac{dy}{dt}=3t^2-1$, so $\dfrac{dy}{dx}=\dfrac{3t^2-1}{2t}$; at $t=2$ this is $\dfrac{11}{4}$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Parametric equations/ˌpærəˈmetrɪk ɪˈkweɪʒnz/	매개변수 방정식

9.2

Second Derivatives of Parametric Equations

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.G: Calculate derivatives of parametric functions. BC ONLY

CHA-3.G.3 $\dfrac{d^2 y}{dx^2}$ can be calculated by dividing $\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)$ by $\dfrac{dx}{dt}$. BC ONLY

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.G: 매개변수 함수의 도함수를 계산한다. BC ONLY

CHA-3.G.3 $\dfrac{d^2 y}{dx^2}$은 $\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)$을 $\dfrac{dx}{dt}$로 나누어 계산할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The second derivative is not $\dfrac{d^2y/dt^2}{d^2x/dt^2}$. Instead, differentiate the first derivative with respect to $t$, then divide by $dx/dt$ again:

$$\frac{d^2y}{dx^2}=\frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{dx/dt}.$$

Use it to test concavity of a parametric curve.

9.3

Finding Arc Lengths of Curves Given by Parametric Equations

Syllabus

English

Enduring Understanding (CHA-6): Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.

Learning Objective CHA-6.B: Determine the length of a curve in the plane defined by parametric functions, using a definite integral. BC ONLY

CHA-6.B.1 The length of a parametrically defined curve can be calculated using a definite integral. BC ONLY

한국어

지속적 이해 (CHA-6): 정적분을 이용하면 구간 내 길이의 누적 변화에 관한 문제를 해결할 수 있다.

학습 목표 CHA-6.B: 정적분을 사용하여 매개변수 함수로 정의된 평면상의 곡선 길이를 구한다. BC ONLY

CHA-6.B.1 매개변수로 정의된 곡선의 길이는 정적분을 사용하여 계산할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The length of a parametric curve for $t$ from $a$ to $b$ is

$$L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt.$$

This is the parametric version of arc length – summing tiny hypotenuses $\sqrt{dx^2+dy^2}$ over the parameter.

9.4

Defining and Differentiating Vector-Valued Functions

Syllabus

English

Enduring Understanding (CHA-3): Derivatives allow us to solve real-world problems involving rates of change.

Learning Objective CHA-3.H: Calculate derivatives of vector-valued functions. BC ONLY

CHA-3.H.1 Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions. BC ONLY

한국어

지속적 이해 (CHA-3): 미분은 변화율과 관련된 실제 문제를 해결하게 해준다.

학습 목표 CHA-3.H: 벡터함수의 도함수를 계산한다. BC ONLY

CHA-3.H.1 실수함수의 도함수를 계산하는 방법은 벡터함수에도 적용할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A vector-valued function 向量值函数 $\vec{r}(t)=\langle x(t),\,y(t)\rangle$ gives a position vector for each $t$. Differentiate component-wise: the velocity is $\vec{v}(t)=\langle x'(t),\,y'(t)\rangle$ and the acceleration is $\vec{a}(t)=\langle x''(t),\,y''(t)\rangle$. The speed is the magnitude $|\vec{v}|=\sqrt{x'^2+y'^2}$.

A vector-valued line: start at a, then slide by t lots of the direction b

Explore · ⁨탐색하기⁩

Add vector-valued components · ⁨ 벡터 성분 추가하기⁩

A vector-valued function packs an $x(t)$ and $y(t)$ into one vector. Differentiating each component gives the velocity vector, tangent to the path. · ⁨벡터 값 함수는 ⟨$x(t)$, $y(t)$⟩를 하나의 벡터에 집약합니다. 각 성분을 미분하면 경로에 접하는 속도 벡터를 얻습니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
vector-valued function/ˈvektə ˈvæljuːd ˈfʌŋkʃn/	벡터 값 함수

9.5

Integrating Vector-Valued Functions

Syllabus

English

Enduring Understanding (FUN-8): Solving an initial value problem allows us to determine an expression for the position of a particle moving in the plane.

Learning Objective FUN-8.A: Determine a particular solution given a rate vector and initial conditions. BC ONLY

FUN-8.A.1 Methods for calculating integrals of real-valued functions can be extended to parametric or vector-valued functions. BC ONLY

한국어

지속적 이해 (FUN-8): 초기값 문제를 풀면 평면상에서 움직이는 입자의 위치를 나타내는 식을 구할 수 있다.

학습 목표 FUN-8.A: 속도 터와 초기 조건이 주어졌을 때 특해를 구한다. BC ONLY

FUN-8.A.1 실수함수의 적분을 계산하는 방법은 매개변수 함수나 벡터함수에도 적용할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Integrate a vector function component by component. Given acceleration or velocity plus an initial condition, integrate each component and use the condition to find the constants – recovering velocity from acceleration, or position from velocity.

9.6

Solving Motion Problems Using Parametric and Vector-Valued Functions

Syllabus

English

Enduring Understanding (FUN-8): Solving an initial value problem allows us to determine an expression for the position of a particle moving in the plane.

Learning Objective FUN-8.B: Determine values for positions and rates of change in problems involving planar motion. BC ONLY

FUN-8.B.1 Derivatives can be used to determine velocity, speed, and acceleration for a particle moving along a curve in the plane defined using parametric or vector-valued functions. BC ONLY
FUN-8.B.2 For a particle in planar motion over an interval of time, the definite integral of the velocity vector represents the particle's displacement (net change in position) over the interval of time, from which we might determine its position. The definite integral of speed represents the particle's total distance traveled over the interval of time. BC ONLY

한국어

지속적 이해 (FUN-8): 초기값 문제를 풀면 평면상에서 움직이는 입자의 위치를 나타내는 식을 구할 수 있다.

학습 목표 FUN-8.B: 평면 운동과 관련된 문제에서 위치와 변화율의 값을 구한다. BC ONLY

FUN-8.B.1 도함수를 이용하여 매개변수 또는 벡터함수로 정의된 평면상의 곡선을 따라 움직이는 입자의 속도, 속력, 가속도를 구할 수 있다. BC ONLY
FUN-8.B.2 시간 구간 동안 평면 운동을 하는 입자에 대해, 속도 벡터의 정적분은 해당 시간 구간 동안 입자의 변위(위치의 순변화)를 나타내며, 이를 통해 위치를 구할 수 있다. 속력의 정적분은 해당 시간 구간 동안 입자가 이동한 총 거리를 나타낸다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

For a particle moving in a plane: position is $\langle x(t),y(t)\rangle$, velocity and acceleration are its derivatives, speed is $|\vec v|$, and the distance traveled over $[a,b]$ is

$$\int_a^b\sqrt{x'(t)^2+y'(t)^2}\,dt.$$

Exam skill: these plane-motion problems appear on the BC free-response nearly every year – be fluent finding speed, the position at a later time (initial point plus the integral of velocity), and total distance.

Worked example. A particle has position $\langle t^2,\ t^3-t\rangle$. Its velocity is $\langle 2t,\ 3t^2-1\rangle$, so at $t=1$ the velocity is $\langle 2,\ 2\rangle$ and the speed is $\sqrt{2^2+2^2}=2\sqrt{2}$. Its position at $t=2$ is $\langle 4,\ 6\rangle$.

9.7

Defining Polar Coordinates and Differentiating in Polar Form

Syllabus

Enduring Understanding	Learning Objective	Essential Knowledge
FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.	FUN-3.G Calculate derivatives of functions written in polar coordinates. BC ONLY	FUN-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to functions in polar coordinates. BC ONLY FUN-3.G.2 For a curve given by a polar equation $r = f(\theta)$, derivatives of $r$, $x$, and $y$ with respect to $\theta$, and first and second derivatives of $y$ with respect to $x$ can provide information about the curve. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Tracing a polar curve

Polar coordinates 极坐标 locate a point by its distance $r$ from the origin and angle $\theta$: convert with $x=r\cos\theta$, $y=r\sin\theta$. A polar curve $r=f(\theta)$ is a parametric curve in $\theta$, so its slope is

$$\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta},\quad\text{with } x=r\cos\theta,\ y=r\sin\theta.$$

Polar coordinates give a point by its distance r and angle theta

Explore · ⁨탐색하기⁩

Plot a polar curve · ⁨극좌표 곡선 그리기⁩

In polar coordinates a point is a distance $r$ at angle $\theta$. Letting $r$ depend on $\theta$ traces curves like this cardioid. · ⁨극좌표에서 점은 거리 $r$와 각도 $\theta$로 정의됩니다. $r$를 $\theta$에 의존하게 하면 이 카르디오이드와 같은 곡선을 그리게 됩니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Polar coordinates/ˈpəʊlə kəʊˈɔːdɪnəts/	극좌표계

9.8

Finding the Area of a Polar Region

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.D: Calculate areas of regions defined by polar curves using definite integrals. BC ONLY

CHA-5.D.1 The concept of calculating areas in rectangular coordinates can be extended to polar coordinates. BC ONLY

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.D: 정적분을 사용하여 극곡선으로 정의된 영역의 면적을 계산한다. BC ONLY

CHA-5.D.1 직교 좌표계에서의 면적 계산 개념은 극좌표계로도 확장할 수 있다. BC ONLY

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The area swept out by a polar curve $r=f(\theta)$ from $\alpha$ to $\beta$ is

$$A=\frac12\int_\alpha^\beta \big(f(\theta)\big)^2\,d\theta.$$

The region is a "fan" of thin triangular sectors; choosing the correct $\theta$-limits (where the curve starts and finishes tracing the region) is the main challenge.

Worked example. Find the area of one petal of the rose $r=2\sin(2\theta)$ (traced for $\theta$ from $0$ to $\tfrac{\pi}{2}$). Using $\sin^2 u=\tfrac12(1-\cos 2u)$,

$$A=\frac12\int_0^{\pi/2}(2\sin 2\theta)^2\,d\theta=\int_0^{\pi/2}(1-\cos 4\theta)\,d\theta=\left[\theta-\tfrac{\sin 4\theta}{4}\right]_0^{\pi/2}=\frac{\pi}{2}.$$

The area of a polar region is swept out as one half the integral of r squared d theta — The shaded region is a fan of thin sectors swept from $\alpha$ to $\beta$; each has area $\tfrac12 r^2\,d\theta$, so the total is $\tfrac12\int_\alpha^\beta r^2\,d\theta$.

9.9

Finding the Area of the Region Bounded by Two Polar Curves

Syllabus

English

Enduring Understanding (CHA-5): Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Learning Objective CHA-5.D: Calculate areas of regions defined by polar curves using definite integrals. BC ONLY

CHA-5.D.2 Areas of regions bounded by polar curves can be calculated with definite integrals. BC ONLY

한국어

지속적 이해(CHA-5): 정적분은 구간에 따른 면적이나 부피의 누적 변화 관련 문제를 해결할 수 있게 한다.

학습 목표 CHA-5.D: 정적분을 사용하여 극곡선으로 정의된 영역의 면적을 계산한다. BC ONLY

CHA-5.D.2 극함수로 경계를 이루는 영역의 면적은 정적분을 사용하여 계산할 수 있습니다. BC만 해당

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

For the area between an outer curve $r_1$ and an inner curve $r_2$, subtract the sectors:

$$A=\frac12\int_\alpha^\beta\big(r_1^2-r_2^2\big)\,d\theta.$$

Find the intersection angles first (set $r_1=r_2$), and be careful which curve is outer over each interval – they can swap.

9.9

Exam tips

For parametric curves, $\tfrac{dy}{dx}=\tfrac{dy/dt}{dx/dt}$; speed is $\sqrt{(dx/dt)^2+(dy/dt)^2}$.
A vector-valued function carries the same information — differentiate/integrate it component by component.
In polar, convert with $x=r\cos\theta$, $y=r\sin\theta$; area swept is $\tfrac12\int r^2\,d\theta$.
Watch the direction of tracing (the sign of $dx/dt$) and set correct $\theta$-limits for polar area.
Eliminate the parameter to recover the ordinary $y$-vs-$x$ shape when it helps.

Infinite Sequences and Series · ⁨무한 수열 및 급수⁩

Watch lesson · ⁨수업 보기⁩

Open this topic on its own page · ⁨이 주제를 독립된 페이지로 열기⁩

10.1

Defining Convergent and Divergent Infinite Series

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.1 The $n$th partial sum is defined as the sum of the first $n$ terms of a series. BC ONLY
LIM-7.A.2 An infinite series of numbers converges to a real number $S$ (or has sum $S$), if and only if the limit of its sequence of partial sums exists and equals $S$. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.1 n번째 부분합(n-th partial sum)은 급수의 첫 n개 항의 합으로 정의된다. BC 전용 $n$ $n$
LIM-7.A.2 실수 S(또는 합이 S)로 수렴하는 무한급수는 오직 그 부분합의 극한이 존재하고 S와 같을 때이다. BC 전용 $S$ $S$ $S$

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Nested matryoshka dolls: an infinite series keeps adding terms — some converge, some diverge

An infinite series 无穷级数 adds infinitely many terms, $\sum_{n=1}^\infty a_n$. Its value is defined as the limit of the partial sums 部分和 $S_N=a_1+a_2+\cdots+a_N$. If $S_N$ approaches a finite number $L$, the series converges 收敛 to $L$; otherwise it diverges 发散. Every convergence question is really a question about the limit of the partial sums.

Partial sums of a convergent geometric series climb toward a over one minus r — For the geometric series with $a=1,\ r=\tfrac12$, the partial sums $S_1,S_2,S_3,\dots$ climb toward the limit $\dfrac{a}{1-r}=2$ – that limit is the series' value.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
infinite series/ˈɪnfɪnət ˈsɪəriːz/	무한 급수
partial sums/ˈpɑːʃl sʌmz/	부분합
converges/kənˈvɜːdʒɪz/	수렴함
diverges/daɪˈvɜːdʒɪz/	발산함

10.2

Working with Geometric Series

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.3 A geometric series is a series with a constant ratio between successive terms. BC ONLY
LIM-7.A.4 If $a$ is a real number and $r$ is a real number such that $|r| < 1$, then the geometric series $\sum_{n=0}^{\infty} ar^n = \dfrac{a}{1-r}$. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.3 기하급수는 연속된 항 사이의 상수비(공비)를 갖는 수열이다. BC 전공자만
LIM-7.A.4 $a$가 실수이고 $r$가 $|r| < 1$인 실수라면, 기하급수 $\sum_{n=0}^{\infty} ar^n = \dfrac{a}{1-r}$는 수렴한다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Geometric series & convergence

A geometric series 几何级数 $\sum ar^{n}$ has a constant ratio $r$ between terms. It converges exactly when $|r|<1$, and then

$$\sum_{n=0}^\infty ar^n=\frac{a}{1-r}.$$

This is the one series whose sum you can find exactly, and it underlies power series later in the unit.

Worked example. Sum $3+\tfrac32+\tfrac34+\tfrac38+\cdots$. Here $a=3$ and $r=\tfrac12$ (with $|r|<1$), so the sum is $\dfrac{a}{1-r}=\dfrac{3}{1-\tfrac12}=6$.

A geometric sequence multiplies by the same ratio at each step

Explore · ⁨탐색하기⁩

When a geometric series converges · ⁨기하 급수가 수렴하는 조건⁩

A geometric series $\sum ar^n$ converges only when $|r|<1$, summing to $\frac{a}{1-r}$. Change the ratio and watch the partial sums settle or blow up. · ⁨기하 급수 $\sum ar^n$는 $|r|<1$일 때만 수렴하며, 그 합은 $\frac{a}{1-r}$입니다. 공비를 변경하면 부분합이 수렴하거나 발산하는 것을 관찰할 수 있습니다.⁩

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
geometric series/ˌdʒiːəʊˈmetrɪk ˈsɪəriːz/	기하급수

10.3

The nth Term Test for Divergence

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.5 The $n$th term test is a test for divergence of a series. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.5 $n$항 판별법은 급수의 발산 여부를 확인하는 판별법입니다. BC 전용

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

If the terms do not shrink to zero, the sum cannot settle: if $\lim_{n\to\infty}a_n\neq0$, the series diverges. This is only a test for divergence – if the terms do go to zero, the test is inconclusive (the series may still diverge, like the harmonic series). Always check this quick test first.

10.4

Integral Test for Convergence

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.6 The integral test is a method to determine whether a series converges or diverges. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.6 적분 검정은 수열이 수렴하거나 발산하는지를 판단하는 방법이다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

If $a_n=f(n)$ for a positive, decreasing, continuous $f$, then $\sum a_n$ and $\int_1^\infty f(x)\,dx$ both converge or both diverge. The integral test 积分判别法 turns a series question into an improper-integral question, and it is what proves the p-series rule below.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
integral test/ˈɪntɪɡrəl test/	적분 검정

10.5

Harmonic Series and p-Series

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.7 In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and $p$-series. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.7 기하급수 외에도 일반적인 수열 급수로는 조화급수, 교차조화급수, 그리고 $p$-급수가 포함됩니다. BC 전용

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A p-series $\sum \dfrac{1}{n^p}$ converges if $p>1$ and diverges if $p\le1$. The special case $p=1$, $\sum\dfrac1n$, is the harmonic series 调和级数 – it diverges even though its terms go to zero (a famous, must-know fact). The p-series family is the standard yardstick for comparison tests.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
harmonic series/hɑːˈmɒnɪk ˈsɪəriːz/	조화 급수

10.6

Comparison Tests for Convergence

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.8 The comparison test is a method to determine whether a series converges or diverges. BC ONLY
LIM-7.A.9 The limit comparison test is a method to determine whether a series converges or diverges. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.8 비교 검정은 수열이 수렴하거나 발산하는지를 판단하는 방법이다. BC 전공자만
LIM-7.A.9 극한 비교 검정은 수열이 수렴하거나 발산하는지를 판단하는 방법이다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Compare an unfamiliar series to a known one (a p-series or geometric series):

Direct comparison 直接比较: if $0\le a_n\le b_n$ and $\sum b_n$ converges, so does $\sum a_n$; if $a_n\ge b_n\ge0$ and $\sum b_n$ diverges, so does $\sum a_n$.
Limit comparison 极限比较: if $\lim\dfrac{a_n}{b_n}$ is a finite positive number, the two series do the same thing. This is easier when the terms only behave like a known series.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Direct comparison/daɪˈrekt kəmˈpærɪsn/	직접 비교
Limit comparison/ˈlɪmɪt kəmˈpærɪsn/	비교 한계

10.7

Alternating Series Test for Convergence

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.10 The alternating series test is a method to determine whether an alternating series converges. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.10 교대 급수 검정은 교대 수열이 수렴하는지를 판단하는 방법이다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

An alternating series 交错级数 has terms that switch sign, $\sum(-1)^n b_n$. It converges if the $b_n$ are positive, decreasing, and $\lim b_n=0$. This lets series like $\sum\dfrac{(-1)^n}{n}$ converge even though the same terms without the signs (the harmonic series) diverge.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
alternating series/ˈɔːltəneɪtɪŋ ˈsɪəriːz/	교대 급수

10.8

Ratio Test for Convergence

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.11 The ratio test is a method to determine whether a series of numbers converges or diverges. BC ONLY
- Exclusion statement: The nth term test for divergence, and the integral test, comparison test, limit comparison test, alternating series test, and ratio test for convergence are assessed on the AP Calculus BC Exam. Other methods are not assessed on the exam. However, teachers may include additional methods in the course, if time permits.

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.11 비율 검정은 수열이 수렴하거나 발산하는지를 판단하는 방법이다. BC 전공자만
- 배제 문구: 발산의 n항 검정, 적분 검정, 비교 검정, 극한 비교 검정, 교대 급수 검정, 그리고 수렴을 위한 비율 검정은 AP Calculus BC 시험에서 평가된다. 기타 방법은 시험에서 평가되지 않는다. 단, 시간이 허용될 경우 교사들은 과정에 추가적인 방법을 포함시킬 수 있다.

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The ratio test 比值判别法 examines $L=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$:

$L<1$: the series converges absolutely;
$L>1$: it diverges;
$L=1$: inconclusive.

It is the go-to test for series with factorials or $n$th powers, and it is exactly how you find the radius of convergence of a power series.

Worked example. Test $\displaystyle\sum \frac{n}{2^n}$. The ratio is $\left|\dfrac{a_{n+1}}{a_n}\right|=\dfrac{n+1}{2^{n+1}}\cdot\dfrac{2^n}{n}=\dfrac{n+1}{2n}\to\dfrac12<1$, so the series converges.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
ratio test/ˈreɪʃɪəʊ test/	비율 검정

10.9

Determining Absolute or Conditional Convergence

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.A: Determine whether a series converges or diverges. BC ONLY

LIM-7.A.12 A series may be absolutely convergent, conditionally convergent, or divergent. BC ONLY
LIM-7.A.13 If a series converges absolutely, then it converges. BC ONLY
LIM-7.A.14 If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.A: 급수가 수렴하거나 발산하는지 판단할 수 있다. BC 전용

LIM-7.A.12 수열은 절대 수렴, 조건부 수렴, 또는 발산 중 하나일 수 있다. BC 전공자만
LIM-7.A.13 수열이 절대 수렴하면, 그 수열은 수렴한다. BC 전공자만
LIM-7.A.14 수열이 절대 수렴하면, 이를 통해 항을 재배열하거나 묶어 변경하여 얻은 모든 수열은 동일한 값을 가진다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A series converges absolutely 绝对收敛 if $\sum|a_n|$ converges. It converges conditionally 条件收敛 if $\sum a_n$ converges but $\sum|a_n|$ diverges (the classic example is $\sum\dfrac{(-1)^n}{n}$). Absolute convergence is the stronger property; conditional convergence relies on the cancellation of signs.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
converges absolutely/kənˈvɜːdʒɪz ˌæbsəˈluːtli/	절대 수렴
converges conditionally/kənˈvɜːdʒɪz kənˈdɪʃənəli/	조건부 수렴

10.10

Alternating Series Error Bound

Syllabus

English

Enduring Understanding (LIM-7): Applying limits may allow us to determine the finite sum of infinitely many terms.

Learning Objective LIM-7.B: Approximate the sum of a series. BC ONLY

LIM-7.B.1 If an alternating series converges by the alternating series test, then the alternating series error bound can be used to bound how far a partial sum is from the value of the infinite series. BC ONLY

한국어

지속적 이해(LIM-7): 극한을 적용하면 무한히 많은 항의 유한합을 결정할 수 있다.

학습 목표 LIM-7.B: 급수의 합을 근사할 수 있다. BC 전용

LIM-7.B.1 호환 급수 검정법(alternating series test)에 의해 수렴하는 호환 급수에 대해, 호환 급수 오차 한계를 사용하여 부분합이 무한급수의 값으로부터 얼마나 떨어지는지를界定(bound)할 수 있다. BC 전용

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

For a convergent alternating series, the error in stopping at the $N$th partial sum is no larger than the first omitted term:

$$|S-S_N|\le b_{N+1}.$$

This simple, powerful bound lets you say how many terms guarantee a desired accuracy.

10.11

Finding Taylor Polynomial Approximations of Functions

Syllabus

English

Enduring Understanding (LIM-8): Power series allow us to represent associated functions on an appropriate interval.

Learning Objective LIM-8.A: Represent a function at a point as a Taylor polynomial. BC ONLY

LIM-8.A.1 The coefficient of the $n$th degree term in a Taylor polynomial for a function $f$ centered at $x = a$ is $\dfrac{f^{(n)}(a)}{n!}$. BC ONLY
LIM-8.A.2 In many cases, as the degree of a Taylor polynomial increases, the $n$th degree polynomial will approach the original function over some interval. BC ONLY

Learning Objective LIM-8.B: Approximate function values using a Taylor polynomial. BC ONLY

LIM-8.B.1 Taylor polynomials for a function $f$ centered at $x = a$ can be used to approximate function values of $f$ near $x = a$. BC ONLY

한국어

지속적 이해(LIM-8): 멱급수(power series)는 적절한 구간에서 관련 함수를 표현할 수 있게 한다.

학습 목표 LIM-8.A: 특정 지점에서 함수를 테일러 다항식으로 표현할 수 있다. BC 전용

LIM-8.A.1 f(x)가 a에서 중심을 가진 테일러 다항식의 n차 항의 계수는 f^(n)(a)/n! 이다. BC 전용 $n$ $f$ $x = a$ $\dfrac{f^{(n)}(a)}{n!}$
LIM-8.A.2 많은 경우, 테일러 다항식의 차수가 증가함에 따라 n차 다항식이 특정 구간에서 원래 함수에 수렴한다. BC 전용 $n$

학습 목표 LIM-8.B: 테일러 다항식을 사용하여 함수의 값을 근사할 수 있다. BC 전용

LIM-8.B.1 a에서 중심을 가진 f(x)의 테일러 다항식은 a 근처의 x에 대한 함수값을 근사하는 데 사용될 수 있다. BC 전용 $f$ $x = a$ $f$ $x = a$

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Taylor series approximation

A Taylor polynomial 泰勒多项式 approximates a function near a center $x=a$ using its derivatives there:

$$P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k.$$

Each added term matches one more derivative, so the polynomial hugs the curve more closely near $a$. Centered at $a=0$ it is a Maclaurin polynomial.

Taylor polynomials of sin x hug the curve more closely as the degree grows — The Maclaurin polynomials of $\sin x$ – $T_1=x$, $T_3$, $T_5$ – each match one more derivative at $0$, so each hugs $\sin x$ over a wider interval before peeling away.

Exam skill: be able to build a Taylor polynomial from a table of derivative values and use it to estimate a function value.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Taylor polynomial/ˈteɪlə ˌpɒlɪˈnəʊmɪəl/	테일러 다항식

10.12

Lagrange Error Bound

Syllabus

English

Enduring Understanding (LIM-8): Power series allow us to represent associated functions on an appropriate interval.

Learning Objective LIM-8.C: Determine the error bound associated with a Taylor polynomial approximation. BC ONLY

LIM-8.C.1 The Lagrange error bound can be used to determine a maximum interval for the error of a Taylor polynomial approximation to a function. BC ONLY
LIM-8.C.2 In some situations, the alternating series error bound can be used to bound the error of a Taylor polynomial approximation to the value of a function. BC ONLY

한국어

지속적 이해(LIM-8): 멱급수(power series)는 적절한 구간에서 관련 함수를 표현할 수 있게 한다.

학습 목표 LIM-8.C: 테일러 다항식 근사와 관련된 오차 한계를 결정할 수 있다. BC 전용

LIM-8.C.1 라그랑주 오차 한계를 사용하여 테일러 다항식 근사의 오차가 최대가 되는 구간을 결정할 수 있다. BC 전용
LIM-8.C.2 어떤 상황에서는 호환 급수 오차 한계를 사용하여 함수 값에 대한 테일러 다항식 근사의 오차를界定(bound)할 수 있다. BC 전용

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

The Lagrange error bound 拉格朗日误差界 bounds how far a Taylor polynomial can be from the true value:

$$|R_n(x)|\le\frac{\max\big|f^{(n+1)}(z)\big|}{(n+1)!}\,|x-a|^{\,n+1}.$$

You bound the $(n+1)$th derivative on the interval, then compute – the standard way to prove a Taylor estimate is accurate enough.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
Lagrange error bound/ˈlæɡreɪndʒ ˈerə baʊnd/	라그랑주 오차 한계

10.13

Radius and Interval of Convergence of Power Series

Syllabus

English

Enduring Understanding (LIM-8): Power series allow us to represent associated functions on an appropriate interval.

Learning Objective LIM-8.D: Determine the radius of convergence and interval of convergence for a power series. BC ONLY

LIM-8.D.1 A power series is a series of the form $\sum_{n=0}^{\infty} a_n (x-r)^n$, where $n$ is a non-negative integer, $\{a_n\}$ is a sequence of real numbers, and $r$ is a real number. BC ONLY
LIM-8.D.2 If a power series converges, it either converges at a single point or has an interval of convergence. BC ONLY
LIM-8.D.3 The ratio test can be used to determine the radius of convergence of a power series. BC ONLY
LIM-8.D.4 The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence. BC ONLY
LIM-8.D.5 If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval. BC ONLY
LIM-8.D.6 The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series. BC ONLY

한국어

지속적 이해(LIM-8): 멱급수(power series)는 적절한 구간에서 관련 함수를 표현할 수 있게 한다.

학습 목표 LIM-8.D: 수열의 수렴 반경과 수렴 구간을 결정한다. BC 전공자만

LIM-8.D.1 멱급수는 $\sum_{n=0}^{\infty} a_n (x-r)^n$ 형태의 급수이며, 여기서 $n$은 음이 아닌 정수, $\{a_n\}$는 실수열, $r$은 실수입니다. BC 전용
LIM-8.D.2 수열이 수렴하면, 단일 점에서 수렴하거나 수렴 구간을 가진다. BC 전공자만
LIM-8.D.3 비율 검정(ratio test)을 사용하여 수열의 수렴 반경을 결정할 수 있다. BC 전공자만
LIM-8.D.4 수열의 수렴 반경을 사용하여 수열이 수렴하는 개구 구간을 식별할 수 있으나, 수렴 구간을 결정하기 위해서는 구간의 양 끝점을 모두 검정해야 한다. BC 전공자만
LIM-8.D.5 수열의 수렴 반경이 양수인 경우, 해당 수열은 개구 구간 내에서 수렴하는 함수의 테일러 급수이다. BC 전공자만
LIM-8.D.6 항별 미분 또는 항별 적분을 통해 얻은 수열의 수렴 반경은 원본 수열의 수렴 반경과 같다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

A power series 幂级数 $\sum c_n(x-a)^n$ converges for $x$ within a radius of convergence 收敛半径 $R$ of the center $a$. Find $R$ with the ratio test. Then test the two endpoints separately (the ratio test is inconclusive there) to state the full interval of convergence 收敛区间 – including or excluding each endpoint.

Worked example. Find the radius of convergence of $\displaystyle\sum \frac{x^n}{n}$. The ratio test gives $\left|\dfrac{x^{n+1}}{n+1}\cdot\dfrac{n}{x^n}\right|=|x|\dfrac{n}{n+1}\to|x|$, which is $<1$ when $|x|<1$, so $R=1$. Testing the endpoints, $x=-1$ gives the convergent alternating harmonic series and $x=1$ the divergent harmonic series, so the interval is $[-1,1)$.

Vocabulary · ⁨어휘⁩ Train · ⁨연습하기⁩

English	한국어
power series/ˈpaʊə ˈsɪəriːz/	power series(-power 급수)
radius of convergence/ˈreɪdɪəs ɒv kənˈvɜːdʒəns/	수렴 반경
interval of convergence/ˈɪntəvl ɒv kənˈvɜːdʒəns/	수렴 구간

10.14

Finding Taylor or Maclaurin Series for a Function

Syllabus

English

Enduring Understanding (LIM-8): Power series allow us to represent associated functions on an appropriate interval.

Learning Objective LIM-8.E: Represent a function as a Taylor series or a Maclaurin series. BC ONLY

LIM-8.E.1 A Taylor polynomial for $f(x)$ is a partial sum of the Taylor series for $f(x)$. BC ONLY

Learning Objective LIM-8.F: Interpret Taylor series and Maclaurin series. BC ONLY

LIM-8.F.1 The Maclaurin series for $\dfrac{1}{1-x}$ is a geometric series. BC ONLY
LIM-8.F.2 The Maclaurin series for $\sin x$, $\cos x$, and $e^x$ provides the foundation for constructing the Maclaurin series for other functions. BC ONLY

한국어

지속적 이해(LIM-8): 멱급수(power series)는 적절한 구간에서 관련 함수를 표현할 수 있게 한다.

학습 목표 LIM-8.E: 함수를 테일러 급수 또는 마클로린 급수로 표현한다. BC 전공자만

LIM-8.E.1 $f(x)$에 대한 테일러 다항식은 $f(x)$의 테일러 급수의 부분합이다. BC 전공자만

학습 목표 LIM-8.F: 테일러 급수와 마클로린 급수를 해석한다. BC 전공자만

LIM-8.F.1 $\dfrac{1}{1-x}$의 마클로린 급수는 기하급수이다. BC 전공자만
LIM-8.F.2 $\sin x$, $\cos x$, 및 $e^x$의 마클로린 급수는 다른 함수의 마클로린 급수를 구성하는 기초가 된다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Extending a Taylor polynomial to infinitely many terms gives a Taylor (or Maclaurin) series. Memorize the key Maclaurin series:

$$e^x=\sum\frac{x^n}{n!},\quad \sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!},\quad \cos x=\sum\frac{(-1)^n x^{2n}}{(2n)!},\quad \frac{1}{1-x}=\sum x^n.$$

New series come from manipulating these – substituting, differentiating, integrating, or multiplying.

Worked example. Find the Maclaurin series for $e^{x^2}$. Substitute $x^2$ for $x$ in $e^x=\sum\dfrac{x^n}{n!}$:

$$e^{x^2}=\sum_{n=0}^{\infty}\frac{(x^2)^n}{n!}=1+x^2+\frac{x^4}{2!}+\frac{x^6}{3!}+\cdots,$$

which converges for all $x$. This substitution trick is far faster than differentiating $e^{x^2}$ six times.

Each extra Maclaurin term hugs the function over a wider range

Explore · ⁨탐색하기⁩

The function a Taylor series approximates · ⁨테일러 급수가 근사하는 함수⁩

y = asin(bx + c) + d

A Taylor series builds a function from its derivatives at a point; more terms hug the curve (here $\sin x$) over a wider range. · ⁨테일러 급수는 특정 점에서의 도함수를 기반으로 함수를 구성합니다; 항이 많아질수록 더 넓은 범위에서 곡선(여기서 $\sin x$)에 밀착합니다.⁩

10.15

Representing Functions as Power Series

Syllabus

English

Enduring Understanding (LIM-8): Power series allow us to represent associated functions on an appropriate interval.

Learning Objective LIM-8.G: Represent a given function as a power series. BC ONLY

LIM-8.G.1 Using a known series, a power series for a given function can be derived using operations such as term-by-term differentiation or term-by-term integration, and by various methods (e.g., algebraic processes, substitutions, or using properties of geometric series). BC ONLY

한국어

지속적 이해(LIM-8): 멱급수(power series)는 적절한 구간에서 관련 함수를 표현할 수 있게 한다.

학습 목표 LIM-8.G: 주어진 함수를 수열로 표현한다. BC 전공자만

LIM-8.G.1 알려진 급수를 사용하여, 항별 미분이나 항별 적분, 대수적 과정, 치환, 또는 기하급수의 성질 활용 등 다양한 방법을 통해 주어진 함수의 수열을 유도할 수 있다. BC 전공자만

Source: College Board AP Course and Exam Description · ⁨출처: College Board AP Course and Exam Description⁩

Because a power series can be differentiated and integrated term by term (within its radius), you can build new series from known ones – e.g. integrate the geometric series for $\dfrac{1}{1-x}$ to get the series for $-\ln(1-x)=\sum_{n\ge 1}\dfrac{x^n}{n}$, or substitute $-x^2$ to get the series for $\dfrac{1}{1+x^2}$. Representing a function as a power series lets you approximate values and integrals that have no elementary antiderivative.

Exam skill: the BC series free-response usually asks you to derive a new Maclaurin series from a known one, find its interval of convergence, and use the alternating-series or Lagrange bound to estimate the error – the capstone skills of the course.

10.15

Exam tips

Test a series for convergence with the right tool: geometric ($|r|<1$, sum $\tfrac{a}{1-r}$), $n$th-term, ratio, integral, comparison, or alternating-series test.
A geometric infinite sum converges only when $|r|<1$; otherwise it diverges.
Build a Taylor/Maclaurin series to approximate a function; more terms give a better fit near the centre.
Know the standard Maclaurin series for $e^x$, $\sin x$, $\cos x$, and $\tfrac{1}{1-x}$.
Find the radius/interval of convergence with the ratio test, then check the endpoints separately.