MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the function y=x2. A fixed point P on its graph corresponds to x=3.
(i) Define the average rate of change of y with respect to x over the interval [3,3+h], where h=0.
(ii) Write the limit that gives the instantaneous rate of change of y at x=3 and state its numerical value.
(iii) Using the expressions obtained in (i) and (ii), explain why the average rate of change over [3,3+h] approaches the instantaneous rate of change as h tends to 0.
(iv) Interpret the conclusion of (iii) in terms of the secant line and the tangent line drawn to the curve at the point P.