Question

compare the average rates of change for ( f(x) = sqrt3{x} + c ) and ( g(x) = ksqrt3{x} + c ) over the interval (a, b). express the average rate of change for ( g(x) ) in terms of the rate of change for ( f(x) ) over (a, b). the average rate of change for ( f(x) ) is (square) and the average rate of change of ( g(x) ) is (square). (simplify your answer. type exact answers, using radicals as needed.)

Explanation:

Step1: Recall the average rate of change formula

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

Step2: Calculate the average rate of change for \( f(x)=\sqrt[3]{x}+c \)

Substitute \( f(x) \) into the formula:
\[
\frac{f(b)-f(a)}{b - a}=\frac{(\sqrt[3]{b}+c)-(\sqrt[3]{a}+c)}{b - a}
\]
Simplify the numerator: \( (\sqrt[3]{b}+c)-(\sqrt[3]{a}+c)=\sqrt[3]{b}-\sqrt[3]{a} \)
So, the average rate of change for \( f(x) \) is \(\frac{\sqrt[3]{b}-\sqrt[3]{a}}{b - a}\).

Step3: Calculate the average rate of change for \( g(x)=k\sqrt[3]{x}+c \)

Substitute \( g(x) \) into the formula:
\[
\frac{g(b)-g(a)}{b - a}=\frac{(k\sqrt[3]{b}+c)-(k\sqrt[3]{a}+c)}{b - a}
\]
Simplify the numerator: \( (k\sqrt[3]{b}+c)-(k\sqrt[3]{a}+c)=k(\sqrt[3]{b}-\sqrt[3]{a}) \)
So, the average rate of change for \( g(x) \) is \(\frac{k(\sqrt[3]{b}-\sqrt[3]{a})}{b - a}=k\cdot\frac{\sqrt[3]{b}-\sqrt[3]{a}}{b - a}\).

Answer:

The average rate of change for \( f(x) \) is \(\boldsymbol{\frac{\sqrt[3]{b}-\sqrt[3]{a}}{b - a}}\), and the average rate of change of \( g(x) \) is \(\boldsymbol{k\cdot\frac{\sqrt[3]{b}-\sqrt[3]{a}}{b - a}}\) (or equivalently, the average rate of change of \( g(x) \) is \( k \) times the average rate of change of \( f(x) \)).