MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A chemical reactor converts reactant into product, and the concentration of product (in mol/L) measured t minutes after the start of the reaction is modeled by the differentiable and strictly decreasing function
C(t)=t+1050.
The function is invertible, and its inverse T=C−1 gives the time required to reach a specified concentration.
(a) Solve for t when C(t)=2 to find T(2).
(b) Determine C′(t).
(c) Find T′(2), the instantaneous rate of change (in minutes per mol/L) of the time with respect to concentration when the concentration is 2 mol/L. Interpret the sign and magnitude of your answer in the context of the reaction.