find the average rate of change of ( f(x)=x^{5} ) over the interval ( 0,2 ).\nwrite your answer as an…

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)