MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the family of functions f(x)=xn, where n is a positive integer. The point P(a,an) with a>0 lies on the graph of f.
(a) Write an expression, in terms of h, for the slope of the secant line that passes through P and the point Q(a+h,(a+h)n) on the graph.
(b) Evaluate the limit of the expression from part (a) as h→0 and state the resulting slope of the tangent line to the graph at P in terms of a and n.
(c) Describe geometrically how the secant line PQ changes as h approaches 0 and explain why the limit in part (b) represents the slope of the tangent line at P. Distinguish clearly between the roles of the secant line and the tangent line.