Define for all real numbers.

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

Problem

Define p(x)=x2sin⁡x+x+13p(x)=x^2\sin x+\sqrt[3]{x+1} for all real numbers. (a) Verify that pp is continuous for all real xx and, in particular, at x=−1x=-1. (b) Derive an explicit expression for p′(x)p'(x) for x≠−1x\neq -1. (c) Evaluate lim⁡x→−1p′(x)\displaystyle \lim_{x\to -1}p'(x) by carefully justifying each step. Address any potential indeterminate forms that arise. (d) Decide whether pp is differentiable at x=−1x=-1. If not, describe the nature of the non-differentiability and give a geometric interpretation involving the tangent line (or lack thereof). (e) Discuss how adding the term x2sin⁡xx^2\sin x alters, or does not alter, the differentiability properties inherited from x+13\sqrt[3]{x+1} at x=−1x=-1. Provide supporting mathematical evidence.

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