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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A school administrator believes that, in his school, the GPA of
seniors in their second semester is less than the GPA of seniors in
their first semester. The class size is large, so a census is not
practical because of time constraints. To test his belief, he selects a
SRS of senior GPA's from the first semester (n1 - 50) and an
independent SRS of senior GPA's from the second semester (n2-
50). He finds that x1 bar-3.1 and s1-0.8, and x2 bar.-2.8 and $2
-0.5. He defines p1 as the average GPA of seniors in semester 1
and u2 as the average GPA of seniors in semester 2.
The test of his belief for
statistical significance will be
a:
A. two-tailed test.
B. one-tailed left test.
C. one-tailed right test.