MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
During the first minute of a rocket test firing, the rocket’s remaining fuel mass (in kilograms) is modeled by
m(t)=2.5×106−1.4×104t−120t2,0≤t≤60,
where t is measured in seconds.
(a) Calculate the average rate at which fuel mass changes over the interval [0,40]. Provide units and interpret the sign.
(b) Obtain a formula for the instantaneous rate of fuel consumption m′(t) and evaluate it at t=40 seconds.
(c) Verify that there exists at least one time k on 0<k<40 at which m′(k) equals the average rate found in part (a). State the theorem you use and the conditions that justify its application.
(d) Determine all critical times on 0≤t≤60 for m′(t) and classify each as a local maximum, local minimum, or neither for the rate of fuel consumption. Provide full justification.