Question

d) compare the rate of increase of the function ( f(x)=3^{x} ) and ( g(x)=3 x ) from ( x = 0 ) to ( x = 5 ) by calculating the average rate of change for each function from ( x = 0 ) to ( x = 5 ). determine which function grows faster, on average, in the given interval.
a. ( g(x)=3 x ) increases at a faster rate, on average, than does ( f(x)=3^{x} ). in this interval, ( g(x) ) is increasing about 10 times faster.
b. ( f(x)=3^{x} ) increases at a faster rate, on average, than does ( g(x)=3 x ). in this interval, ( f(x) ) is increasing about 10 times faster.
c. ( f(x)=3^{x} ) increases at a slower rate, on average, than does ( g(x)=3 x ). in this interval, ( f(x) ) is increasing about 17 times slower.
d. both ( f(x)=3^{x} ) and ( g(x)=3 x ) increase at the same rate, on average.

Explanation:

Step1: Calculate average rate of change formula

The average rate of change of a function \( y = h(x) \) from \( x=a \) to \( x = b \) is given by \( \frac{h(b)-h(a)}{b - a} \).
For \( g(x)=3x \), \( a = 0 \), \( b = 5 \). Then \( g(0)=3\times0 = 0 \), \( g(5)=3\times5=15 \). The average rate of change of \( g(x) \) is \( \frac{g(5)-g(0)}{5 - 0}=\frac{15 - 0}{5}=3 \).

Step2: Calculate average rate of change for \( f(x)=3^{x} \)

For \( f(x)=3^{x} \), \( a = 0 \), \( b = 5 \). Then \( f(0)=3^{0}=1 \), \( f(5)=3^{5}=243 \). The average rate of change of \( f(x) \) is \( \frac{f(5)-f(0)}{5 - 0}=\frac{243 - 1}{5}=\frac{242}{5}=48.4 \).

Step3: Compare the two average - rate - of - change values

Divide the average rate of change of \( f(x) \) by the average rate of change of \( g(x) \): \( \frac{48.4}{3}\approx16.13\approx16 \). So \( f(x)=3^{x} \) has a much faster average rate of change than \( g(x)=3x \) over the interval \( [0,5] \).

Answer:

B. \( f(x)=3^{x} \) increases at a faster rate, on average, than does \( g(x) = 3x \). In this interval, \( f(x) \) is increasing about 10 times faster.