Rates of Change
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| rate of change/reɪt ɒv tʃeɪndʒ/ | tốc độ thay đổi |
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | tốc độ thay đổi trung bình |
| interval/ˈɪntəvl/ | khoảng |
| slope/sləʊp/ | hệ số góc |
| secant line/ˈsiːkənt laɪn/ | đường cắt |
| constant/ˈkɒnstənt/ | hằng số |
How fast, exactly?
- A car's speedometer answers a question the odometer cannot: not how far, but how fast right now.
- "How fast is the output changing?" is the central question of this whole course.
- The answer is a number called a rate of change 变化率.
- This lesson measures it two ways: over a stretch, and at a single instant.
Average rate of change
- Over an interval 区间 from $x = a$ to $x = b$, compare how much the output changed to how much the input changed.
- The average rate of change 平均变化率 is $\dfrac{f(b) - f(a)}{b - a}$ — change in output over change in input.
- It is a ratio, so it always carries units: metres per second, dollars per year, °C per minute.
- Example in words: if temperature rose 12°C over 4 hours, the average rate was 3°C per hour.
The average rate of change of $f$ over the interval $[a, b]$ equals…
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
It is the change in output divided by the change in input — a ratio, so it carries units of output per unit of input.
It is the slope of a secant line
- Draw the straight line joining the two endpoints $(a, f(a))$ and $(b, f(b))$: that is the secant line 割线.
- Its slope 斜率 — rise over run — is exactly the average rate of change.
- Steeper secant → faster average change; a downhill secant → a negative rate.
A runner's distance is $f(t)$ metres. If $f(2) = 10$ and $f(5) = 40$, what is the average speed over $[2, 5]$?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Average speed $= \dfrac{40 - 10}{5 - 2} = \dfrac{30}{3} = 10$ m/s — the secant slope over that interval.
On a graph, the average rate of change over $[a, b]$ is the slope of the…
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The secant line joins $(a, f(a))$ and $(b, f(b))$; its slope is exactly the average rate of change.
The rate at a single point
- A secant needs two points. To get the rate at just one point, shrink the interval.
- Take average rates over smaller and smaller intervals around the point.
- As the interval closes toward zero, the secant tips over into a tangent — its slope is the rate at that exact point.
- This "rate at an instant" is the idea that grows into calculus next year.
Shrink the interval until the secant becomes a tangent
f(x) = 0.5x²
Drag the point to read the slope at a single spot — this is the limit of the average rate of change as the interval shrinks to zero.
To estimate the rate of change at a single point, average rates are taken over very ____ intervals around it.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
As the interval shrinks toward zero, the secant slope approaches the slope at that exact point.
Sign, size, and units
- A positive rate means the output is rising; a negative rate means it is falling; zero means no change.
- A bigger number (ignoring sign) means faster change.
- For a straight line the rate is the same everywhere — it is constant 常数.
- Always state the units: a rate of change with no units is only half an answer.
A negative average rate of change means the output increased over the interval.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
False. A negative ratio means the output ended lower than it started, so the function decreased on average over the interval.
Select all true statements about average rate of change.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
It can be positive, negative, or zero. Everything else is true: it is a slope, it carries units, and it approximates a point rate over small intervals.
Average rate of change is about the two endpoints only — it ignores everything in between. A hiker who climbs then descends back to the same height has an average rate of zero, even though they were moving the whole time.
A car travels 150 km between 1:00 and 3:00.
- Average speed $= \dfrac{150 \text{ km}}{2 \text{ h}} = 75$ km/h.
- That is the secant slope of the distance–time graph over those two hours.
- The car's actual speed changed constantly — 75 km/h is just the average.
The average rate of change over $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$, the slope of the secant line through the endpoints. Shrink the interval toward zero and it becomes the rate at a single point. Always report the units.
Write from an average rate of 5
- f of 1 is 3 and f of 4 is 18.
- Find the average rate of change.
A runner's distance is f of t. f of 1 is 3 and f of 4 is 18. What is the average rate of change?
15 divided by 3 is 5.
What is the average rate of change?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Output change over input change is 5.