Question

find the average rate of change of ( f(x)=x^{5} ) over the interval ( 0,2 ).
write your answer as an integer, fraction, or decimal rounded to the nearest tenth.
simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 0\), \(b=2\), and \(f(x)=x^{5}\).

Step2: Calculate \(f(a)\) and \(f(b)\)

When \(x = 0\), \(f(0)=0^{5}=0\). When \(x = 2\), \(f(2)=2^{5}=32\).

Step3: Substitute into the formula

Substitute \(f(0) = 0\), \(f(2)=32\), \(a = 0\), and \(b = 2\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{32-0}{2 - 0}=\frac{32}{2}\).

Step4: Simplify the fraction

\(\frac{32}{2}=16\).

Answer:

\(16\)