Chapter 11, Problem 40RE

Solution

a.

To graph: The scatter plot of the pairs $\left(x,A\left(x\right)\right)$ for $x=0,0.4,0.,0.12,\cdots ,6.0,6.4$ .

Given Information:

The sequence is defined as $x=0,0.4,0.,0.12,\cdots ,6.0,6.4$ and $A\left(x\right)=\text{NINT}\left(\cos t,t,0,x\right)$ .

Explanation:

Consider the given table,

On a TI-83 Plus graphing calculator press MATH then select fnInt. Enter the function, the variable, the lower limit, and the upper limit of the integral. In this case, the function we input is $\cos x$ , the variable is $x$ , the lower limit is 0 with different values of upper limit as given. The completed table and its scatter plot is shown below,

  

Now, use graphing calculator to make the scatter plot and set the given window.

  

Therefore, the above graph is the required graph.

b.

To determine: The model of function part (a) and overlay its scatter plot of data.

The is the basic sine function graph

Given Information:

The sequence is defined as $x=0,0.4,0.,0.12,\cdots ,6.0,6.4$ and $A\left(x\right)=\text{NINT}\left(\cos t,t,0,x\right)$ .

Explanation:

Consider the given information,

  

The scatter plot appears to be the basic sine function, $A\left(x\right)=\sin x$. Graphing it together with the scatter plot, it appears to be a very good fit:

Therefore, the obtained function is $A\left(x\right)=\sin x$ .

c.

To determine: The basic function appears to be the derivative of $A\left(x\right)$ .

The obtained derivative is $A\text{'}\left(x\right)=\cos x$ .

Given Information:

The best fit function from previous part (b) is $A\left(x\right)=\sin x$ .

Explanation:

Consider the given information,

  $A\left(x\right)=\sin x$

Find the derivative of it.

  $\begin{array}{c}{A}^{\prime }\left(x\right)=\frac{d}{dx}\left(\sin x\right)\\ =\cos x\end{array}$

Therefore, the obtained derivative is $A\text{'}\left(x\right)=\cos x$ .