Question

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

Answer 1

Answer:

The function touches the damping factor

at x=$\frac{(4n-3)\pi}{2}$ and x=$\frac{(4n-1)\pi}{2}$

The x-intercept of f(x) is

at x=$n\pi$

Step-by-step 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=$\frac{(4n-3)\pi}{2}$

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=$\frac{(4n-1)\pi}{2}$

Therefore, The function touches the damping factor x=$\frac{(4n-3)\pi}{2}$ and x=$\frac{(4n-1)\pi}{2}$

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=$n\pi$