Let be a nonzero real constant and consider the exponential function (p(x)=e^ kx )

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let

k

be a nonzero real constant and consider the exponential function

p(x)=e^{kx}

. The goal of this problem is to obtain a general formula for its instantaneous rate of change at an arbitrary point

x=a

. (a) Starting from the difference quotient

\displaystyle \frac{p(a+h)-p(a)}{h}

, evaluate the limit as

h\to 0

in terms of

k

and

e^{ka}

. Show each algebraic step and justify any limit property you use. (b) Simplify the result from part (a) to an expression of the form

C\,e^{ka}

, identifying the constant

C

. Describe how the value of

k

influences the magnitude of the instantaneous rate of change at any point. (c) Compute the instantaneous rate of change of

p

x=0

when

k=-0.4

. State both the exact value and its decimal approximation to three decimal places. (d) For

k<0

, explain why

p

is decreasing on its entire domain, making explicit reference to the sign of the quantity you found in part (a).

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