A function is given by for in the closed interval .

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

Problem

A function pp is given by p(x)=∣x−4∣p(x)=|x-4| for xx in the closed interval [4,9][4,9]. (a) Explain why only one one–sided limit is relevant when evaluating p′(4)p'(4). (b) Evaluate lim⁡x→4+p(x)−p(4)x−4\displaystyle\lim_{x\to 4^+}\frac{p(x)-p(4)}{x-4} and state whether p′(4)p'(4) exists. (c) Interpret the meaning of your result in part (b) in terms of the geometry of the graph of pp on [4,9][4,9]. Discuss the slope of the appropriate tangent line segment.

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