MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
For small values of t (measured in hours), the temperature change in a laboratory experiment is modeled by T(t)=ln(1+0.02t) degrees Celsius. A technician uses only the first-degree Maclaurin approximation to estimate the change after 12 hours.
(a) Write the first-degree Maclaurin polynomial L1(t) for T(t) and use it to approximate the temperature change after 12 hours.
(b) Establish a numerical upper bound for the magnitude of the error incurred by replacing T(12) with L1(12).
(c) The technician claims that the linear approximation is accurate to within 0.1∘C for a 12-hour period. Determine whether this claim is justified and explain your reasoning.