Certainly! To find the average rate of change of the function [tex]\( f(x) = \sqrt[3]{x+5} \)[/tex] on the interval [tex]\([-4, 3]\)[/tex], we follow these steps:
1. Define the function: The function given is [tex]\( f(x) = \sqrt[3]{x+5} \)[/tex].
2. Identify the interval: We are provided with the interval [tex]\([a, b]\)[/tex] which is [tex]\([-4, 3]\)[/tex].
3. Evaluate the function at the endpoints:
- Calculate [tex]\( f(a) \)[/tex] where [tex]\( a = -4 \)[/tex]:
[tex]\[
f(-4) = \sqrt[3]{-4 + 5} = \sqrt[3]{1} = 1
\][/tex]
- Calculate [tex]\( f(b) \)[/tex] where [tex]\( b = 3 \)[/tex]:
[tex]\[
f(3) = \sqrt[3]{3 + 5} = \sqrt[3]{8} = 2
\][/tex]
4. Find the average rate of change: The average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
Substituting the values we found:
[tex]\[
\text{Average rate of change} = \frac{f(3) - f(-4)}{3 - (-4)} = \frac{2 - 1}{3 - (-4)} = \frac{2 - 1}{3 + 4} = \frac{1}{7}
\][/tex]
So, the average rate of change of the function [tex]\( f(x) = \sqrt[3]{x+5} \)[/tex] over the interval [tex]\([-4, 3]\)[/tex] is:
[tex]\[
0.14285714285714285
\][/tex]
The result is [tex]\( 0.14285714285714285 \)[/tex] (or approximately [tex]\( \frac{1}{7} \)[/tex]).
