MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A company models the daily price P, in dollars, of one share of its stock by
P(t)=50+10e0.05tsin(0.2t),
where t is the number of days since the stock was first offered to the public.
(a) Compute P′(t).
(b) Evaluate P′(0) and interpret this value in the context of the stock’s price.
(c) Write the equation of the normal line to the graph of P at t=0.
(d) Use the tangent line at t=0 to estimate the stock price on day t=1. Then, using the second derivative of P, explain whether this linear estimate is an overestimate or an underestimate of the actual price at t=1. Support your conclusion with appropriate reasoning.