d) compare the rate of increase of the function ( f(x)=3^{x} ) and ( g(x)=3 x ) from ( x = 0 ) to ( x = 5 )…

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.\na. ( 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.\nb. ( 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.\nc. ( 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.\nd. both ( f(x)=3^{x} ) and ( g(x)=3 x ) increase at the same rate, on average.

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.\na. ( 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.\nb. ( 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.\nc. ( 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.\nd. both ( f(x)=3^{x} ) and ( g(x)=3 x ) increase at the same rate, on average.

Answer

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.

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] ).