Introducing Calculus: Can Change Occur at an Instant?
Introduced| English | Meaning |
|---|---|
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | The change in a function's output divided by the change in its input across an interval. |
| slope/sləʊp/ | The change in a line's vertical coordinate divided by the corresponding change in its horizontal coordinate. |
| secant line/ˈsiːkənt laɪn/ | A straight line passing through two distinct points on a curve. |
| tangent line/ˈtændʒənt laɪn/ | A straight line through a point on a curve with the same gradient as the curve at that point. |
| instantaneous rate of change/ˌɪnstənˈteɪnɪəs reɪt ɒv tʃeɪndʒ/ | The rate of change at a particular input value, given by the derivative where it exists. |
| limit/ˈlɪmɪt/ | A value that a function or sequence approaches as its input or index approaches a specified value. |
| derivative/dɪˈrɪvətɪv/ | The instantaneous rate of change of a function with respect to its input. |
How fast, right now?
- Your car's speedometer reads $60\ \tfrac{\text{km}}{\text{h}}$ at a single instant.
- But speed is distance ÷ time — and at an instant, no time passes and no distance is covered.
- So how can you have a speed at a single moment?
- Answering this question is where calculus begins.
Average rate of change
- Over an interval, change is easy: it's the total change divided by the time taken.
- This is the average rate of change 平均变化率 of a function.
- On a graph it is the slope 斜率 of the straight line joining two points — a secant line 割线.
- Example: from $x=1$ to $x=3$, the average rate is $\dfrac{f(3)-f(1)}{3-1}$.
The slope of a secant line between two points on a curve gives the...
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A secant joins two points, so its slope is the average rate of change over that interval.
Squeeze the two points together
- Now slide the second point closer and closer to the first.
- The secant line pivots and gets closer to just touching the curve at one point.
- That touching line is the tangent line 切线.
As the second point slides toward the first, the secant line approaches the...
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The secant pivots until it just touches the curve — it approaches the tangent line.
The instantaneous rate of change
- The slope of the tangent line is the instantaneous rate of change 瞬时变化率 at that point.
- That is the speedometer reading — the rate at one exact instant.
- For a position-time graph, it is the instantaneous velocity.
Average rate or instantaneous rate?
Decide whether each quantity is measured over an interval or at a single instant.
The slope of the tangent line is the ____ rate of change.
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The tangent line touches at one point, so its slope is the instantaneous rate of change there.
You cannot find the instantaneous rate by plugging in one point — that gives $\tfrac{0}{0}$, which is undefined. You have to watch what the average rate approaches as the interval shrinks.
You can find the instantaneous rate of change by plugging a single point into the average-rate formula.
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That gives $\tfrac{0}{0}$, which is undefined. You must take the limit as the interval shrinks to zero.
This "approaches" is a limit
- We never actually divide by zero; we ask what value the slope approaches.
- That value is a limit 极限 — the central idea of calculus.
- The limit of the average rate of change, as the interval shrinks to zero, is called the derivative 导数.
- The whole first unit of the course is about making this idea precise.
Select all true statements about the derivative at a point.
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The derivative is the limit of average rates as the interval shrinks — the tangent slope, i.e. the instantaneous rate.
A ball's height is $h(t) = 5t^2$ metres. Estimate its speed at $t=2$ s.
- Average rate from $t=2$ to $t=2.1$: $\dfrac{h(2.1)-h(2)}{0.1} = \dfrac{22.05-20}{0.1} = 20.5\ \tfrac{\text{m}}{\text{s}}$.
- From $t=2$ to $t=2.01$: $\dfrac{20.2005-20}{0.01} = 20.05\ \tfrac{\text{m}}{\text{s}}$.
- The values close in on $20\ \tfrac{\text{m}}{\text{s}}$ — that limit is the instantaneous speed.
A ball's height is $h(t)=5t^2$ m. Estimate its instantaneous speed at $t=2$ s (in m/s).
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Shrinking the interval, the average rate closes in on $20\ \tfrac{\text{m}}{\text{s}}$ — that limit is the instantaneous speed.
Average rate of change = slope of a secant line over an interval. Shrink the interval to zero and the secant approaches the tangent line, whose slope is the instantaneous rate of change. That limiting value is the derivative.