Chapter 12.3, Problem 73E

a.

To determine

Find a cubic model for the data.

Answer to Problem 73E

  $\boxed{y=159.9917x^3-4040.411x^2+34145.10x-88063.6}$

Explanation of Solution

Given information: The table shows the revenues (in millions of dollars) for eBay, Inc. from 2006 through 2011.

Use the regression feature of a graphing utility to find a cubic model for the data. Let $x$ represent the time in years, with $x=6$ corresponding to 2006.

Calculation:

Using as number of years since 2000, the given table can be replaced as

  $\begin{array}{cc}\text{Number of years }x\text{ since 2000}& \text{Revenue }(\text{in millions of dollars)}\\ 6& 5969.7\\ 7& 7672.3\\ 8& 8541.3\\ 9& 8727.4\\ 10& 9156.3\\ 11& 11652.2\end{array}$

Enter the data from given table into your graphing utility calculator.

To enter the data first press $STAT$ the key and then press $1$ to select $Edit$ .

  

  

Now input all the data given sets followed by $ENTER$ key.

  

Before you use regression feature, make sure to enter the DignosticOn command stored in $CATALOG$ menu.

The calculator does not show the value of regression coefficient unless the DignosticOn command has been entered.

  

  

Press the $STAT$ key and then switch to $CALC$ menu.

Move the cursor to the cubic regression function.

  

  

Press $ENTER$ key or $6$ key to select the option $6$ : CubicReg and then write the argument $Y_1$ .

Regression equation is stored in $Y_1$ for further use.

  

Now press $ENTER$ key to find the Cubic Regression equation.

  

Hence, from the above display, write the Cubic Regression equation $y\approx a{x}^3+b{x}^2+cx+d$ as,

  $\boxed{y=159.9917x^3-4040.411x^2+34145.10x-88063.6}$ .

b.

To determine

Estimate the slope of the graph when $x=10$ and give an interpretation of the result.

Answer to Problem 73E

  $\boxed{1334.4}$ .

The rate of change in revenue in 2010 is $\$1334.4$ million.

Explanation of Solution

Given information: The table shows the revenues (in millions of dollars) for eBay, Inc. from 2006 through 2011.

Use the graphing utility to graph the model found in part (a). Estimate the slope of the graph when $x=10$ and give an interpretation of the result.

Calculation:

The regression equation $y=159.9917x^3-4040.411x^2+34145.10x-88063.6$ is already stored in $Y_1$ .

Press the $WINDOW$ key and then set up the viewing window as $[6,11,X_{scl}=1]$ by $[5500,12000,Y_{scl}=500]$ :

  

Press the $GRAPH$ key to obtain the graph of $Y_1$ in viewing window $[6,11]$ by $[5500,12000]$ as shown below:

  

To estimate slope of $Y_1$ at $x=10$ using graphic calculator, input the command $nDeriv\left(Y_1,X,10\right)$

To copy the function $nDeriv$ ,press the $MATH$ key and then scroll down to select the option $8$ : $nDeriv$ .

Hence, from the above display, slope of the graph when $x=10$ is about $\boxed{1334.4}$ which represents the rate of change in revenue in 2010 is $\$1334.4$ million.

c.

To determine

Graph the tangent line to the model.

Compare the slope.

Answer to Problem 73E

  

  $same$ as estimated slope.

Explanation of Solution

Given information: The table shows the revenues (in millions of dollars) for eBay, Inc. from 2006 through 2011.

(c) Use the graphing utility to graph the tangent line to the model when $x=10$ .Compare the slope given by the graphing utility with the estimate in part (b)

Calculation:

To graph the tangent line to the $Y_1$ at $x=10$ ,press the key and then select the option $5$ : Tangent.

  

  

Input the value $x=10$ and then press the $ENTER$ key to draw the tangent line. The graphing calculator also gives the equation of tangent line in the slope-intercept form $y=mx+b$ , where coefficient of $x$ is slope $m$ .

  

From the above display, at $x=10$ the coefficient of $x$ is about $1334.4$ , which is slope of the function $Y_1$ .

Hence, the slope obtained above $1334.4$ is $same$ as estimated slope in part b.