Chapter 13.4, Problem 15WE

To determine

The equation of curve.

Answer to Problem 15WE

The equation of the curve is $y=\frac{1}{2}\cos \frac{1}{2}x$ .

Explanation of Solution

Given:

  

Calculations:

Suppose general form of a function,

  $y=c+a\sin bx$ ,

Here, a is the amplitude of the function.

And period is $\frac{2\pi }{b}$ .

Also,

The average of sum of maximum and minimum gives,

  $c=\frac{M+m}{2}$

The average of difference of maximum and minimum value gives amplitude,

  $a=\frac{M-m}{2}$

From the graph,

The value of curve at origin is 1/2, which is the maximum value.

  $\cos 0=1$

The given curve is a cosine curve.

The curve completes a cycle in interval of 4π . So, the period of the given cycle is 4π.

The maximum value of the curve is1/2 and the minimum value is -1/2.

Therefore,

  $\begin{array}{l}\frac{2\pi }{b}=4\pi \\ b=\frac{2\pi }{4\pi }\\ b=\frac{1}{2}\end{array}$

We know that,

The maximum value of the curve is 1 and the minimum value is -1.

Therefore,

  $\begin{array}{l}c=\frac{M+m}{2}=\frac{1/2+\left(-1/2\right)}{2}=0\\ a=\frac{M-m}{2}=\frac{1/2-\left(-1/2\right)}{2}=\frac{1}{2}\end{array}$

So, the equation of the graph is

  $\begin{array}{l}y=0+\left(1/2\right)\cos \left(1/2\right)x\\ y=\frac{1}{2}\cos \frac{1}{2}x\end{array}$