Question

1. lesson quiz for which function is the average rate of change over the interval ( 1 < x < 5 ) greater than the average rate of change over the same interval for the function ( g(x) = 1.8x^2 )?

a. ( f(x) = x^2 )
b. ( g(x) = 1.2x^2 )
c. ( h(x) = 1.5x^2 )
d. ( k(x) = 2x^2 )

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $f(x)$ over $a

Step2: Calculate rate for $g(x)=1.8x^2$

First find $g(5)=1.8(5)^2=1.8\times25=45$, $g(1)=1.8(1)^2=1.8$.
Rate: $\frac{45-1.8}{5-1}=\frac{43.2}{4}=10.8$

Step3: Calculate rate for $f(x)=x^2$

$f(5)=25$, $f(1)=1$.
Rate: $\frac{25-1}{5-1}=\frac{24}{4}=6$

Step4: Calculate rate for $g(x)=1.2x^2$

$g(5)=1.2\times25=30$, $g(1)=1.2$.
Rate: $\frac{30-1.2}{4}=\frac{28.8}{4}=7.2$

Step5: Calculate rate for $h(x)=1.5x^2$

$h(5)=1.5\times25=37.5$, $h(1)=1.5$.
Rate: $\frac{37.5-1.5}{4}=\frac{36}{4}=9$

Step6: Calculate rate for $k(x)=2x^2$

$k(5)=2\times25=50$, $k(1)=2$.
Rate: $\frac{50-2}{4}=\frac{48}{4}=12$

Step7: Compare rates

Only 12 > 10.8, which is the rate for $k(x)=2x^2$.

Answer:

D. $k(x)=2x^2$