Chapter 2.4, Problem 3E

(a)

To determine

To find: The average rate of change of the function $f\left(x\right)=e^x$ over the interval $\left[-2,0\right]$.

Answer to Problem 3E

The average rate of change of the function $f\left(x\right)=e^x$ over the interval $\left[-2,0\right]$ is $0.43$ approximately.

Explanation of Solution

Given information:

The function is $f\left(x\right)=e^x$ and the interval is $\left[-2,0\right]$.

Calculation:

The average rate of change of function $f\left(x\right)$ over the interval $\left[a,b\right]$ is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

The formula for the average rate of change of function $f\left(x\right)=e^x$ over the interval $\left[-2,0\right]$ is:

  $\frac{f\left(0\right)-f\left(-2\right)}{0-\left(-2\right)}=\frac{f\left(0\right)-f\left(-2\right)}{2}$

The function is given as $f\left(x\right)=e^x$.

Substitute $0$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=e^x\\ f\left(0\right)=e^0\\ f\left(0\right)=1\end{array}$

Substitute $-2$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=e^x\\ f\left(-2\right)=e^{-2}\\ f\left(-2\right)\approx 0.14\end{array}$

Substitute $1$ for $f\left(0\right)$ and $0.14$ for $f\left(-2\right)$ in the formula.

  $\begin{array}{c}\frac{f\left(0\right)-f\left(-2\right)}{2}=\frac{1-0.14}{2}\\ =\frac{0.86}{2}\\ =0.43\end{array}$

Therefore, the average rate of change of the function $f\left(x\right)=e^x$ over the interval $\left[-2,0\right]$ is $0.43$ approximately.

(b)

To determine

To find: The average rate of change of the function $f\left(x\right)=e^x$ over the interval $\left[1,3\right]$.

Answer to Problem 3E

The average rate of change of the function $f\left(x\right)=e^x$ over the interval $\left[1,3\right]$ is $8.68$.

Explanation of Solution

Given information:

The function is $f\left(x\right)=e^x$ and the interval is $\left[1,3\right]$.

Calculation:

The average rate of change of function $f\left(x\right)$ over the interval $\left[a,b\right]$ is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

The formula for the average rate of change of function $f\left(x\right)=e^x$ over the interval $\left[1,3\right]$ is:

  $\frac{f\left(3\right)-f\left(1\right)}{3-1}=\frac{f\left(3\right)-f\left(1\right)}{2}$

The function is given as $f\left(x\right)=e^x$ . Substitute $3$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=e^x\\ f\left(3\right)=e^3\\ f\left(3\right)\approx 20.08\end{array}$

Now substitute $1$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=e^x\\ f\left(1\right)=e^1\\ f\left(1\right)\approx 2.72\end{array}$

Substitute $20.08$ for $f\left(3\right)$ and $2.72$ for $f\left(1\right)$ in the formula.

  $\begin{array}{c}\frac{f\left(3\right)-f\left(1\right)}{2}=\frac{20.08-2.72}{2}\\ =\frac{17.36}{2}\\ =8.68\end{array}$

Therefore, the average rate of change of the function $f\left(x\right)=e^x$ over the interval $\left[1,3\right]$ is $8.68$ approximately.