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 $- 1 .$

Step by step solution

Step 1. Given information.

We have:

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

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" $\frac{∆ y}{∆ x} = \frac{f (b) - f (a)}{b - a}$

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

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

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:

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

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

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

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:

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

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

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

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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

Step by step solution

Step 1. Given information.

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

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

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

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Find the average rate of change of gx=x3-2x+1(a) From -3to-2(b) From-1to1(c) From1to3

Short Answer

Step by step solution

Step 1. Given information.

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

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

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 $3$