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Calculus AB · Unit 2 · 2.3
AP Calculus · Differentiation: Definition and Fundamental Properties
Estimating Derivatives at a Point
No formula means no limit. Take the best secant the data allows — and say which one you took.
key idea$f'(a)\approx\frac{f(b)-f(c)}{b-c}\ \text{with }a\text{ BETWEEN }b\text{ and }c$
The method
With no formula you cannot take a limit, so you take the best available secant slope. Which points you use decides how good the estimate is — and a symmetric pair beats a one-sided pair almost every time.
| $t$ (min) | $0$ | $2$ | $5$ | $9$ | $12$ |
|---|
| $P(t)$ (litres) | $40$ | $52$ | $76$ | $102$ | $113$ |
A tank's contents at five recorded times. Every estimate below comes from this table — there is no formula for $P$.
| Method | Formula for $f'(a)$ | Use when |
|---|
| Symmetric difference | $\dfrac{f(a+h)-f(a-h)}{2h}$ | data exists on both sides — most accurate |
| Forward difference | $\dfrac{f(a+h)-f(a)}{h}$ | $a$ is the first data point |
| Backward difference | $\dfrac{f(a)-f(a-h)}{h}$ | $a$ is the last data point |
| Nearest neighbours | $\dfrac{f(b)-f(c)}{b-c}$, $c\lt a\lt b$ | spacing is uneven — use the closest pair straddling $a$ |
Row 1 whenever you can. A symmetric estimate averages out the curvature error that one-sided estimates carry.
choosing the points
- Always pick points that straddle $a$ if the data allows it.
- Closer points give a better estimate — but only if the data is accurate.
- Divide by the actual gap, not by the number of rows.
- State the units: output units per input unit.
Worked examples
EX 1
Symmetric estimate
$P'(5)$Estimate $P'(5)$ from the table.
1
Find points straddling $t=5$
The neighbours are $t=2$ and $t=9$.
2
Apply the difference
$\dfrac{P(9)-P(2)}{9-2}=\dfrac{102-52}{7}$.
3
Compute
$=\dfrac{50}{7}\approx7.14$.
4
State with units
$P'(5)\approx7.14$ litres per minute.
5
Why straddle
The estimate uses information from both sides of $t=5$, so it captures the behaviour at $5$ rather than just after it.
EX 2
One-sided, because you must
$P'(0)$Estimate $P'(0)$.
1
Check for a left neighbour
There is none — $t=0$ is the first recorded time.
2
Use a forward difference
$\dfrac{P(2)-P(0)}{2-0}=\dfrac{52-40}{2}$.
3
Compute
$=\dfrac{12}{2}=6$ litres per minute.
4
Acknowledge the weakness
A forward difference measures the average rate over $[0,2]$, so it is biased toward behaviour after $t=0$.
5
Say so if asked
“An estimate using only data to the right of $t=0$” is the honest description.
EX 3
Which pair is better?
compareTwo students estimate $P'(9)$: one uses $[5,9]$, the other uses $[5,12]$. Whose is better?
1
First estimate
$\dfrac{P(9)-P(5)}{9-5}=\dfrac{102-76}{4}=6.5$ — a backward difference.
2
Second estimate
$\dfrac{P(12)-P(5)}{12-5}=\dfrac{113-76}{7}\approx5.29$ — a straddling difference.
3
Judge them
The second straddles $t=9$ and so reflects behaviour on both sides; the first only looks backward.
4
Note the tension
The first uses closer points, which usually helps. Straddling generally wins over slightly-closer-but-one-sided.
5
What you cannot do
Neither is right or wrong — both are estimates. Say which points you used, because that is what makes the answer checkable.
EX 4
Interpret the estimate
meaningUsing $P'(5)\approx7.14$ litres per minute, estimate $P(6)$ and comment.
1
Use a local-linear step
$P(6)\approx P(5)+P'(5)\cdot(6-5)$.
2
Substitute
$\approx76+7.14(1)=83.14$ litres.
3
Judge the reliability
One minute is a short step relative to the $3$–$4$ minute gaps in the data, so the estimate is reasonable.
4
Note the layered approximation
$P'(5)$ was itself an estimate, so $83.14$ carries two layers of error. Do not quote it to more decimals than that supports.
5
Better phrasing
“approximately $83$ litres” is defensible; “$83.14$ litres” overstates the precision.
How wrong can it be?
Symmetric beats one-sided
A straddling pair cancels much of the curvature error. Use a one-sided difference only when the data gives you no choice.
Divide by the real gap
In the table the gaps are $2$, $3$, $4$ and $3$ — unequal. Dividing by the number of steps instead of the elapsed time is a common slip.
Estimates inherit error
Using an estimated derivative to predict a value stacks two approximations. Round accordingly.
Say which points you used
Two defensible estimates of the same derivative can differ. Stating the pair is what makes your answer markable.