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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 2: Q. 63 (page 91)

Find the average rate of change of $g (x) = x^{3} - 2 x + 1$
(a) From $- 3$ to $- 2$
(b) From $- 1$ to $1$
(c) From $1$ to $3$

Short Answer

Expert verified

(a) The average rate of change is $17 .$

(b) The average rate of change is $- 1 .$

(c) The average rate of change is $11 .$

Step by step solution

01

Step 1. Given information.

We have:

$g (x) = x^{3} - 2 x + 1$

02

Part a. Step 1. Find the average rate of change from -3 to -2.

The function is $g (x) = x^{3} - 2 x + 1$ .

Average rate of change is given by:

localid="1646219921983" $\frac{鈭� y}{鈭� x} = \frac{f (b) - f (a)}{b - a}$

Here, $a = - 3, b = - 2$ :

localid="1646220011891" $\frac{鈭� y}{鈭� x} = \frac{f (- 3) - f (- 2)}{- 3 - (- 2)} = \frac{{(- 3)}^{3} - 2 (- 3) + 1 - ({(- 2)}^{3} - 2 (- 2) + 1)}{- 3 + 2} = 17$

03

Part b. Step 1. Find the average rate of change from -1 to 1.

The function is $g (x) = x^{3} - 2 x + 1$ .

Average rate of change is given by:

$\frac{鈭� y}{鈭� x} = \frac{f (b) - f (a)}{b - a}$

Here, $a = - 1, b = 1$

role="math" localid="1646219347984" $\frac{鈭� y}{鈭� x} = \frac{f (1) - f (- 1)}{- 1 - 1} = \frac{{(1)}^{3} - 2 (1) + 1 - ({(- 1)}^{3} - 2 (- 1) + 1)}{- 3 + 2} = - 1$

04

Part c. Step 1. Find the average rate of change from 1 to 3.

The function is $g (x) = x^{3} - 2 x + 1$ .

Average rate of change is given by:

$\frac{鈭� y}{鈭� x} = \frac{f (b) - f (a)}{b - a}$

Here, $a = 1, b = 3$ :

$\frac{鈭� y}{鈭� x} = \frac{f (3) - f (1)}{3 - 1} = \frac{{(3)}^{3} - 2 (3) + 1 - ({(1)}^{3} - 2 (1) + 1)}{2} = 11$

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Most popular questions from this chapter

Graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. Verify your results using a graphing utility.

$h (x) = \sqrt{x + 1}$

Exploration Graph $y = x^{3}$ . Then on the same screen graph $y = {(x - 1)}^{3} + 2$ . Could you have predicted the result?

In Problems 7鈥�18, match each graph to one of the following functions:

$A . y = x^{2} + 2 B . y = - x^{2} + 2 C . y = |x| + 2 D . y = - |x| + 2 E . y = {(x - 2)}^{2} F . y = - {(x + 2)}^{2} G . y = |x - 2| H . y = - |x + 2| I . y = 2 x^{2} J . y = - 2 x^{2} K . y = 2 |x| L . y = - 2 |x|$

True or FalseThe cube root function is odd and is decreasing on the interval $(- 鈭�, 鈭�)$ .

Test the equation $y = 5 x^{2} - 1$ for symmetry with respect to the x-axis, the y-axis, and the origin.

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