Question

Between x=2 and x=3, which function has a greater average rate of change than f(x)=2^x has?

Answer 1

Answer: f(3)

Step-by-step explanation:

First find the formula for the rate of change by taking the derivative of 2^x. Let f(x) equal some hypothetical y-value, then take the natural log of both sides.

$y=2^x\\\ln(y)=x \ln(2)$

Implicitly differentiate the left side and take the derivative of the right side

$\frac{y'}{y} =\ln(2)$

Multiply both sides by 'y' which was defined as 2^x

$y'=\ln(2)*2^x$

Plug in x = 2 and x = 3 to see which slope is larger

$y'=\ln(2)*2^2=4\ln(2)\\y'=\ln(2)*2^3=8\ln(2)$