Chapter 1.4, Problem 18E

To determine

To Compute the values of the functions at the points indicated and Sketch its graph.

Answer to Problem 18E

The slope between the points r=0 and r=1 of the given function is 1.

Furthermore, the slope between the points r=1 and r=4 of the given function is 5.

Hence, the slope of the given function is not constant.

Explanation of Solution

Given:

The function

  $F(r)=r^2+5$

The given input points are as follows:

r=0

r=1

r=4

Consider the function

  

The given input points are as follows:

r=0

r=1

r=4

The slope between the input points r₁ and r₁ with output F(r₁) and F(r₂) , respectively is

  

Consider the value of function at the following points:

At r=0, we get

  $\begin{array}{l}F(0)=0^2+5\\ =0+5\\ =5\end{array}$

At r=1, we find that

  $\begin{array}{l}F(1)=1^2+5\\ =1+5\\ =6\end{array}$

At r=4, we have

  $\begin{array}{l}F(4)=4^2+5\\ =16+5\\ =21\end{array}$

Therefore, the value of the function at the points r=0, r=1, and r=4, respectively are

  

The slope between the points r=0 and r=1 of function (A) using formula (B) is shown below:

  $\begin{array}{l}\frac{F(1)-F(0)}{1-0}=\frac{6-5}{1}\\ =\frac{1}{1}\\ =1.\end{array}$

The slope between points r=1 and r=4 of function (A) using formula (B) is as follows:

  $\begin{array}{l}\frac{F(4)-F(1)}{4-1}=\frac{21-6}{3}\\ =\frac{15}{3}\\ =5\end{array}$

Therefore, the slope between the points r=0 and r=1 of the given function (A) is 1.

Furthermore, the slope between the points r=1 and r=4 of the given function (A) is 5.

By determining the slopes between the points r=0 , r=1, r=4, we get different values of slopes.

Hence, the slope of the given function is not contant.

The graph of the function is shown below: