FrQ AP CALCULUS

Question

A particle moves along a straight line. For 0 ≤ t ≤ 5, the position of the particle is given by s(t) = 3t + 4t² - t³.

a) What is the average velocity of the particle over the interval 0 ≤ t ≤ 4?

b) Find a general expression for the velocity of the particle as a function of time, v(t), and determine the value of v at t = 2. Is the particle traveling to the left or the right at this point? Justify your answer.

c) Find all times t during the interval 0 ≤ t ≤ 5 that the particle changes direction. Justify your answer.

d) At t = 4, is the speed of the particle increasing or decreasing? Justify your answer.

Answer 1

(a) Differentiate s(t) to get v(t). Then calculate ( v(4) - v(0)) / 4

(b) You have the general expression from (a). Find v(2). The sign of v(2) gives the answer : positive moving to the right, negative to the left

(c) It will change direction at time t when (a) v(t) = 0, (b) v(t) changes sign moving through the point . So you can plot v(t) on Desmos say and spot from the graph valid t values

(d) a(t) is dv/dt. Find that, then find a(4). The sign of this gives your answer (positive increasing, negative decreasing).