Chapter 5, Problem 22P

To determine

To graph: $f(x)=\sin (e^x)$ and use the graph to estimate the value of t such that ${\displaystyle \underset{t}{\overset{t+1}{\int }}f(x)dx}$ is a maximum and then find the exact value of t that maximizes this integral.

Answer to Problem 22P

  $\begin{array}{l}t\in [-0.2,0]\\ t\approx -0.169.\end{array}$

Explanation of Solution

Given information: Given function $f(x)=\sin (e^x)$ .

Calculation:

Graph the function $f(x)=\sin (e^x)$ is shown below:

  

As the integral that have to maximize is the area of a rectangle with width t +1- t =1, from the graph it seems that the corresponding t so that the graph of the function is above the x −axis and the enclosed area is as great as possible is somewhere between [-0.2, 0].

  $\begin{array}{c}LetI(t)={\displaystyle \underset{t}{\overset{t+1}{\int }}f(x)dx}={\displaystyle \underset{t}{\overset{t+1}{\int }}\sin (e^x)dx}\\ I\text{'}(t)=\sin (e^{t+1})-\sin (e^t)=0.\end{array}$

Solve the equation I’(t) = 0 : graph the function I’(t):

  

From the graph the value of t so that I(t) is maximum is $\approx -0.169$ .