To determine the average rate of change of the function [tex]\( f(z) = 1 - 3z^2 \)[/tex] between [tex]\( z = -2 \)[/tex] and [tex]\( z = 0 \)[/tex], we can follow these steps:
1. Calculate the function value at [tex]\( z = -2 \)[/tex]: [tex]\[
f(-2) = 1 - 3(-2)^2
\][/tex] Simplify the expression inside the parentheses: [tex]\[
f(-2) = 1 - 3(4)
\][/tex] [tex]\[
f(-2) = 1 - 12
\][/tex] [tex]\[
f(-2) = -11
\][/tex]
2. Calculate the function value at [tex]\( z = 0 \)[/tex]: [tex]\[
f(0) = 1 - 3(0)^2
\][/tex] Simplify the expression inside the parentheses: [tex]\[
f(0) = 1 - 3(0)
\][/tex] [tex]\[
f(0) = 1
\][/tex]
3. Determine the change in the function values ([tex]\( \Delta f \)[/tex]): [tex]\[
\Delta f = f(0) - f(-2)
\][/tex] Substituting the calculated values: [tex]\[
\Delta f = 1 - (-11)
\][/tex] [tex]\[
\Delta f = 1 + 11
\][/tex] [tex]\[
\Delta f = 12
\][/tex]
4. Determine the change in the input values ([tex]\( \Delta z \)[/tex]): [tex]\[
\Delta z = 0 - (-2)
\][/tex] [tex]\[
\Delta z = 0 + 2
\][/tex] [tex]\[
\Delta z = 2
\][/tex]
5. Calculate the average rate of change: [tex]\[
\text{Average Rate of Change} = \frac{\Delta f}{\Delta z}
\][/tex] Substituting the values: [tex]\[
\text{Average Rate of Change} = \frac{12}{2}
\][/tex] [tex]\[
\text{Average Rate of Change} = 6.0
\][/tex]
So, - The function value at [tex]\( z = -2 \)[/tex] is [tex]\( -11 \)[/tex]. - The function value at [tex]\( z = 0 \)[/tex] is [tex]\( 1 \)[/tex]. - The average rate of change of the function [tex]\( f(z) = 1 - 3z^2 \)[/tex] between [tex]\( z = -2 \)[/tex] and [tex]\( z = 0 \)[/tex] is [tex]\( 6.0 \)[/tex].
