MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let g(x)=ln(x2+1) be defined on the closed interval [−2,3].
(a) Calculate the average rate of change of g over [−2,3]. Express your answer in exact form.
(b) Justify that there exists at least one number c in (−2,3) for which g′(c) equals the average rate of change found in part (a). Clearly reference any required hypotheses in your reasoning.
(c) Describe geometrically what your conclusion in part (b) implies about the graph of y=g(x) on the interval [−2,3]. Discuss the relationship between a secant line and at least one tangent line.