A runner’s position along a straight track is modeled (in meters) by s(t)=10…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A runner’s position along a straight track is modeled (in meters) by
s(t)=10sin(0.5t)+0.2t2,0≤t≤8,
where t is measured in seconds.
(1) Write an algebraic expression for the average velocity of the runner over the time interval [t,t+h] with h>0. Do not yet take any limits.
(2) Algebraically simplify the expression from part (a) so that common factors in the numerator are clear.
(3) Compute the limit of the simplified expression as h→0 and verify that the result equals s′(t). State the explicit formula for the instantaneous velocity.
(4) Using your work, approximate the average velocity over the interval [5,5.1] and compare it to the instantaneous velocity at t=5. Comment on the closeness of the two values in the context of motion.