Let and on . Which expression gives the value of in where the instantaneous rate of…

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

k\ne 0

and

f(x)=e^{kx}

[0,1]

. Which expression gives the value of

c

(0,1)

where the instantaneous rate of change of

f

equals its average rate of change over the interval? (a)

\dfrac{1}{k}\,(e^{k}-1)

(b)

\ln\!\left(\dfrac{e^{k}-1}{k}\right)

(c)

\dfrac{1}{k}\,\ln\!\left(\dfrac{e^{k}-1}{k}\right)

(d)

\dfrac{1}{2}