Chapter 1.8, Problem 49E

(a)

To determine

To plot: The given function on one graph.

Explanation of Solution

Given information:

Consider the following functions,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};f_7(x)=\frac{cos\left(7x-\frac{\pi }{2}\right)}{7}\end{array}$

Graph:

The graph of $f_1(x),f_3(x),f_5(x),f_7(x)$ is shown below as,

  

Interpretation:

The graph of the function is getting decreased with respect to $\pi$ and $f(x)$ .

(b)

To determine

To sketch:The graph by plotting the sum of $f_1(x)\text{and}{f}_3(x)$ .

Explanation of Solution

Given information:

Consider the following functions,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ \end{array}$

Graph:

The given function is, $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ \end{array}$

On adding $f_1(x),f_3(x)$ we get,

  $\begin{array}{l}=f_1(x)+f_3(x)\\ =\cos \left(x-\frac{\pi }{2}\right)+\frac{\cos \left(3x-\frac{\pi }{2}\right)}{3}\end{array}$

The graph of $f_1(x)+f_3(x)$ is shown below as,

  

Interpretation:

The graph of the function on adding $f_1(x),f_3(x)$ will have same amplitude $f(x)$ and time period $\pi$ on both the positive and negative axis. There is a monotonically increase and decrease in the values.

(c)

To determine

To Sketch:The graph by plotting the sum of $f_1(x),f_3(x)\text{and}{f}_5(x)$ .

Explanation of Solution

Given information:

Consider the following functions,

  $f_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};$

Graph:

The given function is, $f_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};$

On adding $f_1(x),f_3(x),f_5(x)$ we get,

  $\begin{array}{l}=f_1(x)+f_3(x)+f_5(x)\\ =\cos \left(x-\frac{\pi }{2}\right)+\frac{\cos \left(3x-\frac{\pi }{2}\right)}{3}+\frac{\cos \left(5x-\frac{\pi }{2}\right)}{5}\end{array}$

The graph of $f_1(x)+f_3(x)+f_5(x)$ is shown below as,

  

Interpretation:

The graph of the function on adding $f_1(x),f_3(x),f_5(x)$ will have same amplitude $f(x)$ and time period $\pi$ on both the positive and negative axis. There is a monotonically increase and decrease in the values with the spikes.

(d)

To determine

To sketch:The graph by plotting the sum of $f_1(x),f_3(x),f_5(x)\text{and}{f}_7(x).$ .

Explanation of Solution

Given information:

Consider the following functions,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};f_7(x)=\frac{cos\left(7x-\frac{\pi }{2}\right)}{7}\end{array}$

Graph:

The given function is,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};f_7(x)=\frac{cos\left(7x-\frac{\pi }{2}\right)}{7}\end{array}$

On adding $f_1(x),f_3(x),f_5(x),f_7(x)$ we get,

  $\begin{array}{l}=f_1(x)+f_3(x)+f_5(x)+f_7(x)\\ =\cos \left(x-\frac{\pi }{2}\right)+\frac{\cos \left(3x-\frac{\pi }{2}\right)}{3}+\frac{\cos \left(5x-\frac{\pi }{2}\right)}{5}+\frac{\cos \left(7x-\frac{\pi }{2}\right)}{7}\end{array}$

The graph of $f_1(x)+f_3(x)+f_5(x)+f_7(x)$ is shown below as,

  

Interpretation:

The graph of the function on adding $f_1(x),f_3(x),f_5(x),f_7(x)$ will have same amplitude $f(x)$ and time period $\pi$ on both the positive and negative axis. There is a monotonically increase and decrease in the values with the spikes. The entire value lies within $\pi$ only.

(e)

To determine

To Explain:How does it look on plotting the sum of $f_1(x),f_3(x),f_5(x)\text{and}{f}_7(x).$

Answer to Problem 49E

The sum of $f_1(x)+f_3(x)+f_5(x)+f_7(x)$ looks like the function which switches between 1 and   -1.

Explanation of Solution

Given information:

Consider the following functions,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};f_7(x)=\frac{cos\left(7x-\frac{\pi }{2}\right)}{7}\end{array}$

Calculation:

The sum of $f_1(x)+f_3(x)+f_5(x)+f_7(x)$ looks like the function which switches between 1 and

-1 on adding more and more number of cosine terms.

(f)

To determine

To analyze:The pattern on adding $f_9(x),f_{11}(x)$ to $f_1(x),f_3(x),f_5(x)\text{and}{f}_7(x).$

Explanation of Solution

Given information:

Consider the following functions,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};f_7(x)=\frac{cos\left(7x-\frac{\pi }{2}\right)}{7}\end{array}$

Graph:

The given function is,

  $\begin{array}{l}{f}_1(x)=\cos \left(x-\frac{\pi }{2}\right);f_3(x)=\frac{cos\left(3x-\frac{\pi }{2}\right)}{3};\\ f_5(x)=\frac{cos\left(5x-\frac{\pi }{2}\right)}{5};f_7(x)=\frac{cos\left(7x-\frac{\pi }{2}\right)}{7}\\ f_9(x)=\frac{cos\left(9x-\frac{\pi }{2}\right)}{9};f_{11}(x)=\frac{cos\left(11x-\frac{\pi }{2}\right)}{11}\end{array}$

On adding $f_1(x),f_3(x),f_5(x),f_7(x),f_9(x),f_{11}(x)$ we get,

  $\begin{array}{l}=f_1(x)+f_3(x)+f_5(x)+f_7(x)+f_9(x)+f_{11}(x)\\ =\cos \left(x-\frac{\pi }{2}\right)+\frac{\cos \left(3x-\frac{\pi }{2}\right)}{3}+\frac{\cos \left(5x-\frac{\pi }{2}\right)}{5}+\frac{\cos \left(7x-\frac{\pi }{2}\right)}{7}+\frac{\cos \left(9x-\frac{\pi }{2}\right)}{9}+\frac{\cos \left(11x-\frac{\pi }{2}\right)}{11}\end{array}$

The graph of $f_1(x)+f_3(x)+f_5(x)+f_7(x)+f_9(x)+f_{11}(x)$ is shown below as,

  

Interpretation:

On adding more number of cosine terms the graph becomes more stable at 1 and -1.