Question

find the average rate of change for the function between the given values.

1. f(x) = 4x^3 - 8x^2 - 1; from -4 to 1
2. f(x) = x^2 + 3x; from 1 to 8

Explanation:

Step1: Recall the average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve problem 17

First, find $f(-4)$ and $f(1)$ for $f(x)=4x^{3}-8x^{2}-1$.

• Calculate $f(-4)$:

$f(-4)=4(-4)^{3}-8(-4)^{2}-1=4\times(-64)-8\times16 - 1=-256-128 - 1=-385$.

• Calculate $f(1)$:

$f(1)=4\times1^{3}-8\times1^{2}-1=4 - 8 - 1=-5$.

• Then, find the average rate of change:

$\frac{f(1)-f(-4)}{1-(-4)}=\frac{-5-(-385)}{1 + 4}=\frac{-5 + 385}{5}=\frac{380}{5}=76$.

Step3: Solve problem 18

First, find $f(1)$ and $f(8)$ for $f(x)=x^{2}+3x$.

• Calculate $f(1)$:

$f(1)=1^{2}+3\times1=1 + 3=4$.

• Calculate $f(8)$:

$f(8)=8^{2}+3\times8=64+24=88$.

• Then, find the average rate of change:

$\frac{f(8)-f(1)}{8 - 1}=\frac{88 - 4}{7}=\frac{84}{7}=12$.

Answer:

1. 76
2. 12