A differentiable

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

Problem

A differentiable, invertible function y=f(x)y=f(x) satisfies f(2)=5f(2)=5 and f′(2)=−3f'(2)=-3. A particle’s position along a beam is recorded by the coordinate x(t)x(t), and the temperature at that position is given by T(t)=f(x(t))T(t)=f(x(t)). (a) Calculate (f−1)′(5)\displaystyle \left(f^{-1}\right)'(5). Show your work and include units. (b) At the instant when x=2x=2, the coordinate is increasing at dxdt=0.02 units/s\displaystyle \frac{dx}{dt}=0.02\text{ units/s}. Determine dTdt\displaystyle \frac{dT}{dt} at that instant. (c) Instead, suppose that when T=5 unitsT=5\text{ units} the temperature is rising at dTdt=0.06 units/s\displaystyle \frac{dT}{dt}=0.06\text{ units/s}. Determine the instantaneous rate of change of xx with respect to time at that moment.

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