MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The natural logarithm function h(x)=lnx is defined for x>0. Examine the behavior of h on two separate intervals that share an endpoint.
(a) Compute the average rate of change of h on [1,e] and on [e,e4]. Express each answer exactly.
(b) For each interval in part (a), find a point c in its interior where the instantaneous rate of change of h matches the corresponding average rate of change. Express your answers exactly and verify that each lies within the appropriate open interval.
(c) Compare the two values of c obtained in part (b). Which is larger? Give a reason for your conclusion that references the concavity of h on (1,e4).
(d) Determine whether either value of c coincides with a point of inflection of h. Support your answer with appropriate reasoning.