Question

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window.

f(x) = e−3x sin(x)

Analyze the graph of the function using the method in Example 6. (Enter your answers as a comma-separated list. Use n to represent an arbitrary integer if necessary.)
y = ±e−3x touches y = e−3x sin(x) at:

y = e−3x sin(x) has x-intercepts at:

Answer 1

The function touches the damping factor

at x=
`((4n-3)\pi)/(2)` and x=
`((4n-1)\pi)/(2)`

The x-intercept of f(x) is

at x=
`n\pi`

Explanation:

Given function is f(x)=
`e^(-3x) sin(x)` and damping factor as y=
`e^(-3x)` and y=
`(-1)e^(-3x)`

To find when function touches the damping factor:

For f(x)=
`e^(-3x) sin(x)` and y=
`e^(-3x)`

Equating the both the equation,

`e^(-3x) sin(x)=e^(-3x)`

`sin(x)=1`

x=
`((4n-3)\pi)/(2)`

For f(x)=
`e^(-3x) sin(x)` and y=
`(-1)e^(-3x)`

Equating the both the equation,

`e^(-3x) sin(x)=(-1)e^(-3x)`

`sin(x)=(-1)`

x=
`((4n-1)\pi)/(2)`

Therefore, The function touches the damping factor x=
`((4n-3)\pi)/(2)` and x=
`((4n-1)\pi)/(2)`

To find x-intercept of f(x):

For x-intercept, y=0

f(x)=
`e^(-3x) sin(x)`

y=
`e^(-3x) sin(x)`

`e^(-3x) sin(x)=0`

Hence,
`e^(-3x)` is always greater than zero.

Therefore,
`sin(x)=0`

x=
`n\pi`

Thus,

The x-intercept of f(x) is at x=
`n\pi`