Chapter 2: Q. 62 (page 90)

Find the average rate of change of $f (x) = - x^{3} + 1$
(a) From $0$ to $2$
(b) From $1$ to $3$
(c) From $- 1$ to $1$

Short Answer

Expert verified

(a) The average rate of change is $- 4$ .

(b) The average rate of change is $- 13 .$

Step by step solution

Step 1. Given information.

We have:

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

Part a. Step 1. Find the average rate of change from 0 to 2.

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

Average rate of change is given by:

localid="1646218831507" $\frac{∆ y}{∆ x} = \frac{f (b) - f (a)}{b - a}$

Here, $a = 0$ and $b = 2 :$

localid="1646217782079" $\frac{∆ y}{∆ x} = \frac{f (2) - f (0)}{2 - 0} \frac{∆ y}{∆ x} = \frac{- {(2)}^{3} + 1 - (- {(0)}^{3} + 1)}{2} = - 4$

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

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

Average rate of change is given by:

localid="1646217897240" $\frac{∆ y}{∆ x} = \frac{f (b) - f (a)}{b - a}$

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

localid="1646217942520" $\frac{∆ y}{∆ x} = \frac{f (3) - f (1)}{3 - 1} \frac{∆ y}{∆ x} = \frac{- {(3)}^{3} + 1 - (- {(3)}^{3} + 1)}{2} = - 13$

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

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

Average rate of change is given by:

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

Here, $a = - 1$ and $a = 1$ :

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

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Find the average rate of change of $f (x) = - x^{3} + 1$
(a) From $0$ to $2$
(b) From $1$ to $3$
(c) From $- 1$ to $1$

Short Answer

Step by step solution

Step 1. Given information.

Part a. Step 1. Find the average rate of change from 0 to 2.

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

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

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Find the average rate of change of f(x)=-x3+1(a) From0to2(b) From1to3(c) From-1to1

Short Answer

Step by step solution

Step 1. Given information.

Part a. Step 1. Find the average rate of change from 0 to 2.

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Study anywhere. Anytime. Across all devices.

Find the average rate of change of $f (x) = - x^{3} + 1$
(a) From $0$ to $2$
(b) From $1$ to $3$
(c) From $- 1$ to $1$