Chapter 11.4, Problem 57E

Solution

a.

To calculate: The value of $A\left(x\right)$ by using the calculator NINT method.

The value explained in explanation.

Given Information:

The function is $f\left(t\right)=3t^2$ .

Calculation:

Consider the given function,

  $f\left(t\right)=3t^2$

On a TI-83 Plus 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 $2X$ , the variable is $X$, the lower limit is 0 with different values of upper limit as given:

  

And,

  

And,

  

And,

  

And,

  

And,

  

And,

  

Therefore, all the obtained are mentioned in the window each value of x .

b.

To graph: The table of pairs $\left(x,A\left(x\right)\right)$ for the values of x and plot the graph paper.

Tables and graph shown in figure.

Given Information:

The function is $f\left(t\right)=3t^2$ .

Explanation:

Consider the given information,

  $f\left(t\right)=3t^2$

Using the results from part (a), we construct the table as shown on the left. Plot the ordered pairs and draw a smooth curve through them as shown on the right:

  

And,

  

Therefore, the obtained table and graph shown above.

c.

To determine: The best fit to model the data in part (b) ad overlay its graph on a scatter plot of the data.

The cubic model is $A\left(x\right)=x^3$ .

Given Information:

The function is $f\left(t\right)=3t^2$ .

Explanation:

Consider the given information,

  $f\left(t\right)=3t^2$

Using a graphing calculator, enter the x -values in L1 and the areas in L2. Then use the

feature and graph it together with the scatter plot of the data. The quadratic regression equation is defined as:

  

And,

  

Therefore, the required best model is $A\left(x\right)=x^3$ .

d.

To determine: The conjecture about the exact value of $A\left(x\right)$ for any x greater than zero.

The function is $A\left(x\right)=x^3$ .

Given Information:

The function is $f\left(t\right)=3t^2$ .

Explanation:

Consider the given information,

  $f\left(t\right)=3t^2$

Using the result from part (c), the exact value of $A\left(x\right)$ for any $x$ greater than zero is the square of $x$ .

  $A\left(x\right)=x^3$

Therefore, the required function is $A\left(x\right)=x^3$ .

e.

To determine: The derivative of the function $A\left(x\right)=x^3$ .

The derivative is $A\text{'}\left(x\right)=3x^2$ .

Given Information:

The function is $A\left(x\right)=x^3$ .

Explanation:

Consider the given information,

  $A\left(x\right)=x^3$

Find the derivative above function with respect to x .

  $\begin{array}{c}\frac{d}{dx}A\left(x\right)=\frac{d}{dx}{x}^3\\ A^{\prime }\left(x\right)=3x^2\end{array}$

Therefore, the derivative is $A\text{'}\left(x\right)=3x^2$ .