Chapter 4.4, Problem 71E

Solution

a.

To graph: The functions $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$ for x in the interval $\left[-1,8\right]$ . State the points of intersection that appear to be.

Given information:

The given functions are: $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$

Graph:

Consider the given functions:

  $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$

Now draw the graphs of the given functions $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$

:

  

Now observe the graph carefully and observe that there are two intersecting points for x in the graph in the interval $\left[-1,8\right]$ .

Interpretation: The graphs of $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$ are intersecting at two points.

b.

To state: The coordinates of each point of intersection.

The points are $\left(1,0\right)\text{ and }\left(2\pi ,{1.3}^{-2\pi }\right)$ .

Given information:

The given functions are: $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$

Explanation:

Consider the given functions: $y={1.3}^{-x}\text{ and }y={1.3}^{-x}\cos x$

The value of both the functions should be same at the point of intersection. So,

  $\begin{array}{l}{1.3}^{-x}={1.3}^{-x}\cos x\\ 1=\cos x\\ x=0,2\pi \end{array}$

Thus, the x- coordinates are $0\text{ and }2\pi$ .

Now find the x- coordinates for $x=0\text{ and }x=2\pi$ .

  $\begin{array}{c}y={1.3}^{-x}\\ ={1.3}^{-0}\\ =1\\ y={1.3}^{-x}\\ ={1.3}^{-2\pi }\end{array}$

Thus, the y- coordinates are $\text{1 and 1}{\text{.3}}^{-2\pi }$ .

Therefore, the resultant coordinates of points of intersection are $\left(1,0\right)\text{ and }\left(2\pi ,{1.3}^{-2\pi }\right)$ .