Question

Find the relative rate of change of f(x)=12+2e^-2x

Answer 1

The relative rate of change of a function f(x) is the ratio of the derivative f'(x) to the function f(x). Since f'(x) = -4e^(-2x) the relative rate of change r = [-4e^(-2x)]/[12 + 2e^(-2x)]. This can be simplified by factoring a value of two resulting in: r = [-2e^(-2x)]/[6 + e^(-2x)]. This can be further re-arranged if desired into: r = -1/3 e^(-2x) - 2

Answer 2

The rate of change of a given function is given by f'(x)=dy/dx
given that the function is:
f(x)=12+2^(-2x)
f'(x)=2*(-2)e^(-2x)
f'(x)=-4e^(-2x)
the relative rate of change will be the ratio of the derivative to the original function:
f'(x)/f(x)
=(12+2^(-2x))/(-4e^(-2x))