Chapter 5.3, Problem 56E

a.

To determine

To graph: The given functions in the given interval.

Explanation of Solution

Given information:

The following functions:

  $\begin{array}{l}y={\cos }^{2}x\\ y=e^{-x}+x-1\end{array}$

The following interval:

  $[0,2\pi )$

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{Y}}_1={\cos }^{2}x$ and ${\text{Y}}_2=e^{-x}+x-1$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

The graphs of the functions intersect at a point.

b.

To determine

To state: An equation whose solutions are the points of intersection of the graph.

Answer to Problem 56E

An equation whose solutions are the points of intersection of the graph is:

  ${\cos }^{2}x=e^{-x}+x-1$

Explanation of Solution

Given information:

The following functions:

  $\begin{array}{l}y={\cos }^{2}x\\ y=e^{-x}+x-1\end{array}$

The following interval:

  $[0,2\pi )$

Suppose $f(x)$ and $g(x)$ are two functions that take a real number as input and give a real number as output. The intersection points of $f(x)$ and $g(x)$ are given by the points for which $f(x)=g(x)$ . The values of those points can be found out by solving $f(x)=g(x)$ algebraically.

Hence, the equation whose solutions are the points of intersection of the graph is:

  ${\cos }^{2}x=e^{-x}+x-1$

c.

To determine

To state: The points of intersection on the graph (to four decimal places).

Answer to Problem 56E

The intersection point is:

  $A(0.951,0.3374)$

Explanation of Solution

Given information:

The following functions:

  $\begin{array}{l}y={\cos }^{2}x\\ y=e^{-x}+x-1\end{array}$

The following interval:

  $[0,2\pi )$

The graph of the function is obtained as:

  

Using $\boxed{\text{INTERSECT}}$ feature on the graphing utility, the point of intersection is obtained as:

  $A(0.951,0.3374)$