Question 1 (Essay Worth 30 points)
(08.01 HC)

A yo-yo is moving up and down a string so that its velocity at time t is given by v(t) = 3cos(t) for time t ≥ 0. The initial position of the yo-yo at time t = 0 is x = 3.

Part A: Find the average value of v(t) on the interval open bracket 0 comma pi over 2 close bracket. (10 points)

Part B: What is the displacement of the yo-yo from time t = 0 to time t = π? (10 points)

Part C: Find the total distance the yo-yo travels from time t = 0 to time t = π. (10 points)

Part A: The average value of v(t) on the interval [0, π/2] is 0.

Part B: The displacement of the yo-yo from time t = 0 to time t = π is 0.

Part C: The total distance the yo-yo travels from time t = 0 to time t = π is 6.

Part A: The average value of a function f(t) on the interval [a, b] is given by the following formula:

[tex]\frac{\int_a^b f(t) \, dt}{b - a}[/tex]

In this case, we have:

[tex]\frac{\int_0^{\pi/2} 3\cos(t) \, dt}{\pi/2 - 0} = \frac{3\sin(\pi/2) - 3\sin(0)}{\pi/2} = 0[/tex]

Part B: The displacement of a particle is given by the following integral:

[tex]x(t) = \int_0^t v(t) \, dt[/tex]

In this case, we have:

[tex]x(\pi) = \int_0^{\pi} 3\cos(t) \, dt = 3\sin(\pi) - 3\sin(0) = 0[/tex]

Part C: The total distance traveled by a particle is given by the following formula:

d = [tex]\int_0^t |v(t)| \, dt[/tex]

In this case, we have:

d = [tex]\int_0^{\pi} |3\cos(t)| , dt = \int_0^{\pi/2} 3\cos(t) , dt + \int_{\pi/2}^{\pi} 3\cos(t) , dt = 3\sin(\pi/2) + 3\sin(0) = 6[/tex]

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