Question

find the average rate of change of the function over the given intervals. f(x)=11x^3 + 11; a) 2,4, b) -5,5 a) the average rate of change of the function f(x)=11x^3 + 11 over the interval 2,4 is (simplify your answer.)

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Calculate $f(4)$ and $f(2)$ for part a

For $f(x)=11x^{3}+11$, when $x = 4$, $f(4)=11\times4^{3}+11=11\times64 + 11=704+11 = 715$. When $x = 2$, $f(2)=11\times2^{3}+11=11\times8 + 11=88 + 11=99$.

Step3: Compute average rate of change for part a

Using the formula $\frac{f(4)-f(2)}{4 - 2}=\frac{715 - 99}{2}=\frac{616}{2}=308$.

Step4: Calculate $f(5)$ and $f(-5)$ for part b

When $x = 5$, $f(5)=11\times5^{3}+11=11\times125+11 = 1375+11=1386$. When $x=-5$, $f(-5)=11\times(-5)^{3}+11=11\times(-125)+11=-1375 + 11=-1364$.

Step5: Compute average rate of change for part b

Using the formula $\frac{f(5)-f(-5)}{5-(-5)}=\frac{1386-(-1364)}{10}=\frac{1386 + 1364}{10}=\frac{2750}{10}=275$.

Answer:

a) 308
b) 275