On the interval define

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

On the interval I=(−π2,π2)I=(-\tfrac{\pi}{2},\tfrac{\pi}{2}) define s(x)=xcos⁡x+2xs(x)=x\cos x+2x. The function ss is differentiable and strictly increasing on II, so an inverse function s−1s^{-1} exists with domain s(I)s(I). (a) Justify that ss is strictly increasing on II by analyzing s′(x)s'(x). (b) Compute s′(0)s'(0) and then find s−1 ′(0)\displaystyle s^{-1}\,'(0). (c) Using part (b), write the equation of the line that best approximates s−1s^{-1} near x=0x=0. (d) Use your linear approximation from part (c) to estimate the value of s−1(0.20)s^{-1}(0.20). Comment on the reasonableness of your estimate with respect to the behavior of ss.

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