Let be the differentiable function defined by

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let ff be the differentiable function defined by f(x)=x3−4xf(x) = x^3 - 4x. Consider the point a=1a = 1. Respond to the prompts below. (a) Write an expression for the average rate of change of ff over the interval [1,1+h][1, 1+h] where h≠0h \neq 0. (b) Simplify the expression from part (a) so that it is a rational function of hh. (c) Evaluate lim⁡h→0\lim_{h \to 0} of your simplified expression to obtain f′(1)f'(1). Show the algebraic steps leading to the limit value. (d) Without using the power rule, explain why the limit you found in part (c) represents the slope of the tangent line to the graph of ff at x=1x = 1.

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