HL Unit 3, Day 3: Infinite Sums D=
Rb a
79
|v(t)| dt
Just like area, total distance traveled must be positive. In a graph of velocity, the definite integral yields the displacement, i.e., change in position. The area yields the total distance traveled. Example 3.3.10 (adapted from AB 1997) A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 ≤ t ≤ 8, is given by the function whose graph is shown in Figure 3.9 Figure 3.9: Vertical velocity of a bug
(a) At value of t does the bug change direction? (b) What is the total distance the bug traveled from t = 0 to t = 8? (c) What is the net displacement of the bug between t = 0 and t = 8? (d) What is the total distance traveled downward by the bug? upward? (e) What is the bug’s velocity at t = 5? The acceleration at t = 5? (f) What other questions could we ask about the bug? Example 3.3.11 (AB ’03) A particle moves along the x-axis so that its velocity at time t is given by 2 t v(t) = − (t + 1) sin 2 At time t = 0, the particle is at position x = 1. (a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2? Why or why not? Mr. Budd, compiled September 29, 2010