The function is defined for all real numbers by p(x)= cases x, & x 0

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

Problem

The function pp is defined for all real numbers by p(x)={sin⁡x,x≤0,excos⁡x,x>0.p(x)= \begin{cases} \sin x, & x\le 0,\\ e^{x}\cos x, & x>0. \end{cases} Numerical work should be shown to at least three decimal places. (a) Approximate p′(0−)p'(0^-) and p′(0+)p'(0^+), the left-hand and right-hand derivatives of pp at x=0x=0. Present the numerical values and describe your computational approach. (b) Using the results from part (a), determine whether pp is differentiable at x=0x=0. Justify your conclusion by referencing the formal definition of differentiability. (c) Compute an accurate numerical approximation for p′(1)p'(1). Then write the equation of the line tangent to the graph of pp at x=1x=1.

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