A particle moves along a horizontal track so that its velocity (in meters per second)…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle moves along a horizontal track so that its velocity (in meters per second) at time t seconds is given by
v(t)=sint−0.2t, 0≤t≤10.
(1) Part 1: (a) Show that v(t) has at least one zero in the interval (0,π). Provide an interval of length 0.5 seconds that contains such a zero.
(2) Part 2: (b) Using an appropriate numerical method or technology, approximate the smallest positive time t1 at which the particle is at rest. Report t1 to the nearest hundredth of a second.
(3) Part 3: (c) Determine every time in 0≤t≤10 at which the particle changes direction. For each sub-interval determined by these times, indicate the direction of motion and justify your conclusions.
(4) Part 4: (d) Evaluate the particle’s displacement on 0≤t≤10. Give an exact expression and a decimal approximation correct to three decimal places.
(5) Part 5: (e) Compute the total distance traveled by the particle on 0≤t≤10, giving your answer to three decimal places.