Chapter 4.6, Problem 75E

To determine

Find the damping factor and graph the given function and describe the behavior of the function as x increases without bound.

Answer to Problem 75E

The damping factor $2^{\frac{-x}{4}}$

Explanation of Solution

Given:

Period for $y=cosx$ is 2p.

The damping factor in $f(x)=2^{\frac{-x}{4}}\cos \pi x\text{is}{2}^{\frac{-x}{4}}$

Let's plot the function $f(x)=2^{\frac{-x}{4}}\cos \pi x$ and the damping factor $2^{\frac{-x}{4}}$ in the same viewing window from −10 to 10, using the command

  ${}_{plots[multiple]}\left({}_{plot,}\left[e^{\left(\frac{-x}{4}\right)}.\cos (\pi ·x),x=-2\pi \mathrm{..2}\pi \right],\left[2^{\left(\frac{-x}{4}\right)},x=-2\pi \mathrm{..2}\pi \right]\right)$

in the software Maple.

Thus, the graph is shown below.

  

The above graph represents the equation and the damping factor $2^{\frac{-x}{4}}$ .

Further, from the above graph, x increases and further approaches 0.

Therefore, $x\to \infty ,f(x)\to 0$