Chapter 4.6, Problem 74E

To determine

To sketch: the graph of the given function, to describe the behavior of the function as $x$ increases without bound.

Answer to Problem 74E

The intersection points are $x=3.87\times {10}^{-7},3.14$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=e^{-x}\cos x$

Calculation:

To plot the graph on the same viewing window, proceed as below:

Press the button $Y=$ and type:

  $\begin{array}{l}{Y}_1=e^{-x}\cos x\\ Y_2=e^{-x}\\ Y_3=-e^{-x}\end{array}$

  

Press the button WINDOW and adjust the window as shown:

  

Press the button TRACE or GRAPH to obtain the graph:

  

Now to find the intersection point, press the button 2^nd CALC select option “5”

  

Move cursor on $Y_1$ curve and ENTER

  

Move cursor on $Y_2$ curve and ENTER

  

Move cursor near point of intersection of graphs and press ENTER

  

It will show following coordinates.

  

Now to find the second intersection, press the button 2^nd CALC select option “5”

  

Move cursor on $Y_1$ curve and ENTER

  

Move cursor on $Y_3$ curve and ENTER

  

Move cursor near point of intersection of graphs and press ENTER

  

It will show following coordinates.

  

Hence the intersection points are $x=3.87\times {10}^{-7},3.14$ . The above graph show that as $x$ increases to infinity the function $f\left(x\right)=e^{-x}\cos x$ approaches to zero.