MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let f(x)=ln(x2+1), and consider the point x=1.
(a) Calculate the average rate of change of f on the interval [1,2]. Express your answer exactly.
(b) Determine the instantaneous rate of change of f at x=1.
(c) Is f increasing faster at x=1 than it does on average over [1,2]? Justify your conclusion by comparing the results from parts (a) and (b) and interpreting them in terms of secant and tangent slopes.