Giải tích hàm biến số và ứng dụng
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| continuity/kɒntɪˈnjuːɪti/ | Tính liên tục |
| chain rule/tʃeɪn ruːl/ | The rule for differentiating a composite function by multiplying the appropriate derivatives. |
| Riemann sum/ˈriːmən sʌm/ | A sum of function values multiplied by subinterval widths, used to approximate a definite integral. |
A corner with no slope
- The graph of $f(x) = |x|$ has no gap at $0$, yet a tangent line cannot be drawn there: the slope jumps from $-1$ to $+1$.
- GRE single-variable calculus keeps asking which hypothesis a statement needs: limit, continuity 连续性, differentiability or integrability.
Limits and continuity
- $\lim_{x \to a} f(x)$ describes values near $a$; $f(a)$ need not exist. Continuity at $a$ adds: $f(a)$ exists and equals the limit.
- The two one-sided limits must agree. $\dfrac{x^2 - 1}{x - 1}$ is undefined at $1$, but cancelling (valid for $x \neq 1$) shows the limit is $2$: a removable discontinuity.
- $0/0$ is an indeterminate form to be resolved, never an answer.
Derivatives and the chain rule 链式法则
- $f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$. Differentiable implies continuous; the converse fails ($|x|$, $\sqrt[3]{x}$ at $0$).
- Chain rule: $\dfrac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)$. Differentiating $\sin(x^2)$ twice gives $2\cos(x^2) - 4x^2 \sin(x^2)$: the product rule produces two terms.
What happens at x = 0?
Separate continuity from differentiability at a point.
At $x = 0$, the function $f(x) = |x|$ is
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The one-sided derivatives are −1 and 1.
Find the derivative of $\ln(x^2 + 1)$ at $x = 1$.
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$\frac{2x}{x^2+1}$ at $x = 1$ is $1$.
The fundamental theorem with moving bounds
- If $f$ is continuous, $\dfrac{d}{dx}\displaystyle\int_a^{x} f(t)\,dt = f(x)$. With upper bound $h(x)$, multiply by $h'(x)$; a variable lower bound contributes with a minus sign.
$F(x) = \displaystyle\int_0^{x^2} e^t\,dt$. Then $F'(x) = e^{x^2} \cdot 2x$, so $F'(1) = 2e \approx 5.44$. Forgetting the factor $2x$ gives $e$, the most common wrong answer.
If $F(x) = \int_0^{x^2} e^t\,dt$, find $F'(1)$ to two decimal places.
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$F'(x) = 2x e^{x^2}$, so $F'(1) = 2e$.
A Riemann sum 黎曼和 or a series?
- A Riemann sum has a partition that refines as $n \to \infty$. Rewrite $\displaystyle\sum_{k=1}^{n} \frac{n}{n^2 + k^2} = \frac{1}{n}\sum_{k=1}^{n} \frac{1}{1 + (k/n)^2} \to \int_0^1 \frac{dt}{1+t^2} = \frac{\pi}{4}$.
- A series adds ever more terms of a fixed sequence. Terms tending to $0$ is necessary, not sufficient: $\sum 1/k$ diverges.
Do not treat $n$-dependent factors as constants. Pull out exactly $1/n$ and read the rest as $f(k/n)$.
Evaluate $\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n} \left(\frac{k}{n}\right)^2$.
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It is $\int_0^1 t^2\,dt = 1/3$.
Although $1/k \to 0$, the series $\sum_{k=1}^{\infty} 1/k$ diverges.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Terms tending to 0 is necessary but not sufficient; the harmonic series diverges (compare with $\int_1^\infty dx/x$ or group terms).
Recap
- Continuity: the limit exists and equals $f(a)$. Differentiability is stronger.
- Chain rule and moving-bound FTC need the inner derivative.
- Rewrite sums as $\frac{1}{n}\sum f(k/n)$ before naming the integral; the term test only rules convergence out.