To find the average rate of change of the function [tex]\( f(x) = 3x^2 + 2 \)[/tex] over given intervals, you use the formula for the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Let's apply this formula to each of the intervals provided:
(a) From -2 to 0
1. Find [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 3(-2)^2 + 2 = 3 \cdot 4 + 2 = 12 + 2 = 14
\][/tex]
2. Find [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3(0)^2 + 2 = 0 + 2 = 2
\][/tex]
3. Calculate the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{2 - 14}{0 + 2} = \frac{-12}{2} = -6
\][/tex]
The average rate of change from -2 to 0 is [tex]\( -6.0 \)[/tex].
(b) From 1 to 3
1. Find [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3(1)^2 + 2 = 3 \cdot 1 + 2 = 3 + 2 = 5
\][/tex]
2. Find [tex]\( f(x) \)[/tex] at [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 3(3)^2 + 2 = 3 \cdot 9 + 2 = 27 + 2 = 29
\][/tex]
3. Calculate the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(3) - f(1)}{3 - 1} = \frac{29 - 5}{3 - 1} = \frac{24}{2} = 12
\][/tex]
The average rate of change from 1 to 3 is [tex]\( 12.0 \)[/tex].
(c) From 1 to 4
1. Find [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3(1)^2 + 2 = 3 \cdot 1 + 2 = 3 + 2 = 5
\][/tex]
2. Find [tex]\( f(x) \)[/tex] at [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 3(4)^2 + 2 = 3 \cdot 16 + 2 = 48 + 2 = 50
\][/tex]
3. Calculate the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(4) - f(1)}{4 - 1} = \frac{50 - 5}{4 - 1} = \frac{45}{3} = 15
\][/tex]
The average rate of change from 1 to 4 is [tex]\( 15.0 \)[/tex].
The final results are:
(a) The average rate of change from -2 to 0 is [tex]\( -6.0 \)[/tex].
(b) The average rate of change from 1 to 3 is [tex]\( 12.0 \)[/tex].
(c) The average rate of change from 1 to 4 is [tex]\( 15.0 \)[/tex].
