2.10 Approximating a Derivative
If a function $f$ is defined by a table of values, then the approximation values of its derivatives at $b$ can be obtained from the average rate of change using values that are close to $b$.
For $a<b<c$,
$
\begin{aligned}
& f^{\prime}(b) \approx \frac{f(c)-f(b)}{c-b} \text { or } \\
& f^{\prime}(b) \approx \frac{f(b)-f(a)}{b-a} \text { or } \\
& f^{\prime}(b) \approx \frac{f(c)-f(a)}{c-a}
\end{aligned}
$
Example 1
- The temperature of the water in a coffee cup is a differentiable function $F$ of time $t$. The table below shows the temperature of coffee in a cup as recorded every 3 minutes over 12 minute period.
(a) Use data from the table to find an approximation for $F^{\prime}(6)$ ?
(b) The rate at which the water temperature decrease for $0 \leq t \leq 12$ is modeled by $F(t)=120+85 e^{-0.03 t}$ degrees per minute. Find $F^{\prime}(6)$ using the given model.
▶️Answer/Explanation
Solution
$
\text { (a) } \begin{aligned}
& F^{\prime}(6) \approx \frac{F(6)-F(3)}{6-3}=\frac{192-197}{3}=-\frac{5}{3}{ }^{\circ} F / \mathrm{min} \text { or } \\
& F^{\prime}(6) \approx \frac{F(9)-F(6)}{9-6}=\frac{186-192}{3}=-2{ }^{\circ} \mathrm{F} / \mathrm{min} \text { or } \\
& F^{\prime}(6) \approx \frac{F(9)-F(3)}{9-3}=\frac{186-197}{6}=-\frac{11}{6}{ }^{\circ} \mathrm{F} / \mathrm{min}
\end{aligned}
$
$
\begin{aligned}
(b) & F^{\prime}(t)=0+85 e^{-0.03 t} \frac{d}{d x}(-0.03 t) \\
& =85 e^{-0.03 t}(-0.03)=-2.55 e^{-0.03 t} \\
& F^{\prime}(6)=-2.55 e^{-0.03(6)}=-2.55 e^{-0.18} \approx-2.129^{\circ} \mathrm{F} / \mathrm{min}
\end{aligned}
$
Exercises – Approximating a Derivative
Multiple Choice Questions
1. Some values of differentiable function $f$ are shown in the table below. What is the approximation value of $f^{\prime}(3.5)$ ?
(A) 8
(B) 10
(C) 13
(D) 16
▶️Answer/Explanation
Ans:C
Free Response Questions
- 2. The normal daily maximum temperature $F$ for a certain city is shown in the table above.
(a) Use data in the table to find the average rate of change in temperature from $t=1$ to $t=6$.
(b) Use data in the table to estimate the rate of change in maximum temperature at $t=4$.
(c) The rate at which the maximum temperature changes for $1 \leq t \leq 6$ is modeled by $F(t)=40-52 \sin \left(\frac{\pi t}{6}-5\right)$ degrees per minute. Find $F^{\prime}(4)$ using the given model.
▶️Answer/Explanation
Ans:
2. (a) $19.2^{\circ} \mathrm{F} / \mathrm{mon}$
(b) $23.5^{\circ} \mathrm{F} / \mathrm{mon}$
(c) $26.472^{\circ} \mathrm{F} / \mathrm{mon}$