Question

how do the average rates of change for the pair of functions compare over the given interval?
f(x) = 0.1x²
g(x) = 0.4x²
1 ≤ x ≤ 6
the average rate of change of f(x) over 1 ≤ x ≤ 6 is the average rate of change of g(x) over 1 ≤ x ≤ 6 is the average rate of change of g(x) is times that of f(x).
(simplify your answers. type integers or decimals.)

Explanation:

Step1: Recall the formula for average rate of change

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

Step2: Calculate the average rate of change for \( f(x)=0.1x^{2} \) over \([1, 6]\)

First, find \( f(6) \) and \( f(1) \):

• \( f(6)=0.1\times(6)^{2}=0.1\times36 = 3.6 \)
• \( f(1)=0.1\times(1)^{2}=0.1\times1 = 0.1 \)

Now, use the average rate of change formula:
\(\frac{f(6)-f(1)}{6 - 1}=\frac{3.6-0.1}{5}=\frac{3.5}{5}=0.7\)

Step3: Calculate the average rate of change for \( g(x)=0.4x^{2} \) over \([1, 6]\)

First, find \( g(6) \) and \( g(1) \):

• \( g(6)=0.4\times(6)^{2}=0.4\times36 = 14.4 \)
• \( g(1)=0.4\times(1)^{2}=0.4\times1 = 0.4 \)

Now, use the average rate of change formula:
\(\frac{g(6)-g(1)}{6 - 1}=\frac{14.4 - 0.4}{5}=\frac{14}{5}=2.8\)

Step4: Find how many times the average rate of change of \( g(x) \) is that of \( f(x) \)

Divide the average rate of change of \( g(x) \) by that of \( f(x) \):
\(\frac{2.8}{0.7}=4\)

Answer:

The average rate of change of \( f(x) \) over \( 1\leq x\leq6 \) is \( 0.7 \). The average rate of change of \( g(x) \) over \( 1\leq x\leq6 \) is \( 2.8 \). The average rate of change of \( g(x) \) is \( 4 \) times that of \( f(x) \).