To determine the average rate of change of the function [tex]\( f(x) = -2x^2 + 3x + 8 \)[/tex] from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex], we follow these steps:
1. Evaluate the function at [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = -2(3)^2 + 3(3) + 8
\][/tex]
Calculate the terms individually:
[tex]\[
-2(3)^2 = -2 \cdot 9 = -18
\][/tex]
[tex]\[
3(3) = 9
\][/tex]
[tex]\[
-18 + 9 + 8 = -1
\][/tex]
Thus, [tex]\( f(3) = -1 \)[/tex].
2. Evaluate the function at [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = -2(4)^2 + 3(4) + 8
\][/tex]
Calculate the terms individually:
[tex]\[
-2(4)^2 = -2 \cdot 16 = -32
\][/tex]
[tex]\[
3(4) = 12
\][/tex]
[tex]\[
-32 + 12 + 8 = -12
\][/tex]
Thus, [tex]\( f(4) = -12 \)[/tex].
3. Calculate the average rate of change:
The formula for the average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([x_1, x_2]\)[/tex] is given by:
[tex]\[
\frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Substitute [tex]\( x_1 = 3 \)[/tex], [tex]\( f(3) = -1 \)[/tex], [tex]\( x_2 = 4 \)[/tex], and [tex]\( f(4) = -12 \)[/tex]:
[tex]\[
\frac{f(4) - f(3)}{4 - 3} = \frac{-12 - (-1)}{4 - 3} = \frac{-12 + 1}{1} = \frac{-11}{1} = -11
\][/tex]
Thus, the average rate of change of [tex]\( f(x) \)[/tex] from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex] is [tex]\(-11\)[/tex].
