week 5 homework problem set: average rate of change\ncalculate the average rate of change of the following…

week 5 homework problem set: average rate of change\ncalculate the average rate of change of the following functions with the provided intervals.\n1) $f(x)=sqrt{x - 4}+5$ on the interval $4,29$\n2) $g(x)=sqrt3{2x}-3$ on the interval $-4,4$\n3) $h(x)=-2sqrt{x + 5}$ on the interval $-1,11$

week 5 homework problem set: average rate of change\ncalculate the average rate of change of the following functions with the provided intervals.\n1) $f(x)=sqrt{x - 4}+5$ on the interval $4,29$\n2) $g(x)=sqrt3{2x}-3$ on the interval $-4,4$\n3) $h(x)=-2sqrt{x + 5}$ on the interval $-1,11$

Answer

Explanation:

Step1: Recall average rate - of - change formula

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

Step2: Solve for $f(x)=\sqrt{x - 4}+5$ on $[4,29]$

First, find $f(29)$ and $f(4)$. $f(29)=\sqrt{29 - 4}+5=\sqrt{25}+5=5 + 5=10$. $f(4)=\sqrt{4 - 4}+5=5$. Then, $\frac{f(29)-f(4)}{29 - 4}=\frac{10 - 5}{25}=\frac{5}{25}=\frac{1}{5}$.

Step3: Solve for $g(x)=\sqrt[3]{2x}-3$ on $[-4,4]$

Find $g(4)$ and $g(-4)$. $g(4)=\sqrt[3]{2\times4}-3=\sqrt[3]{8}-3=2 - 3=-1$. $g(-4)=\sqrt[3]{2\times(-4)}-3=\sqrt[3]{-8}-3=-2 - 3=-5$. Then, $\frac{g(4)-g(-4)}{4-(-4)}=\frac{-1-(-5)}{8}=\frac{-1 + 5}{8}=\frac{4}{8}=\frac{1}{2}$.

Step4: Solve for $h(x)=-2\sqrt{x + 5}$ on $[-1,11]$

Find $h(11)$ and $h(-1)$. $h(11)=-2\sqrt{11 + 5}=-2\sqrt{16}=-2\times4=-8$. $h(-1)=-2\sqrt{-1 + 5}=-2\sqrt{4}=-2\times2=-4$. Then, $\frac{h(11)-h(-1)}{11-(-1)}=\frac{-8-(-4)}{12}=\frac{-8 + 4}{12}=\frac{-4}{12}=-\frac{1}{3}$.

Answer:

  1. $\frac{1}{5}$
  2. $\frac{1}{2}$
  3. $-\frac{1}{3}$