Consider the natural logarithm function on the interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the natural logarithm function

p(x)=\ln x

on the interval

[1,\infty)

. (a) Explain why or why not the hypotheses of the Mean Value Theorem are satisfied for

p

on the interval

[1,\infty)

. Address both required conditions in your argument. (b) Let

b>1

be a finite real number. Determine all values of

b

for which the hypotheses of the Mean Value Theorem are satisfied on the interval

[1,b]

. Provide a clear justification. (c) For

b=3

, compute a value

c

(1,3)

such that

p'(c)

equals the average rate of change of

p

over

[1,3]

. Show all steps in your calculation and interpret the meaning of

c

in the context of the function.