Station #1 – Integrals
1.
Find y = f(x) if f ”(x) = 20x3, f ’(0) =-3 and f(0) = 8.
B) x5 – 3x + 8
A) 120x + 8
2. ∫ (8𝑥 5 + 2 − 4 ) 𝑑𝑥 =
𝑥 3 3√𝑥 2
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a. 40𝑥 4 − 𝑥 4 +
b.
c.
d.
3.
∫
4 6
𝑥
3
4 6
𝑥
3
4 6
𝑥
3
−
1 4
𝑥
2
2
+
−𝑥 +
−
1
𝑥2
8
3
3 √𝑥 5
12
3
3
D) x5 + 5
+𝑐
5 √𝑥 5
8
3 √𝑥
C) 5x4 – 3
+𝑐
+𝑐
3
− 12 √𝑥 + 𝑐
3𝑥 4 − 8𝑥 2 + 2
𝑑𝑥 =
𝑥2
2
a. 3𝑥 2 − 8 + 𝑥 2 + 𝑐
2
b. 𝑥 3 − 8𝑥 − 𝑥 + 𝑐
2
c. 𝑥 3 − 𝑥 + 𝑐
d. 𝑥 3 − 8𝑥 −
2
3𝑥 3
+𝑐
4. Given 𝑓(𝑥) = 4𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥𝑑𝑥,
a) Find the anti-derivative ∫ 𝑓(𝑥)𝑑𝑥
𝜋
b) Evaluate the definite integral ∫𝜋2 𝑓(𝑥)𝑑𝑥
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Station #2 – Calculator Inactive
1.
A particle moves along the x-axis so that its position at time t is given by 𝑥(𝑡) = 𝑡 2 − 6𝑡 + 5. For
what value of t is the velocity of the particle zero?
a) 1
b) 2
c) 3
d) 4
e) 5
2. A particle moves along the x-axis so that at time 𝑡 ≥ 0 its position is given by 𝑥(𝑡) = 2𝑡 3 − 21𝑡 2 +
72𝑡 − 53. At what time t is the particle at rest?
a) t = 1 only
b) t = 3 only
c) t = 3.5 only
d) t = 3 and t = 3.5
e) t = 3 and t = 4
3. The velocity v(t) of a particle is graphed below,
minute.
a) At what value of t does the particle change
direction?
b) What is the velocity of the particle at t = 12?
t = 25?
c) What is the particle’s acceleration at t = 12?
t = 25?
d) Is the speed of the particle increasing or
decreasing at t = 25? How do you know?
where v(t) is measured in meters per
4. A particle moves along the x-axis so that its velocity at time t, for 0 ≤ 𝑡 ≤ 6, is given by a
differentiable function v whose graph is shown below. The graph has horizontal tangents at t = 1
and t = 4. At time t = 0, the particle is at x = -2.
a) Where is the particle at rest?
b) When does the particle change directions?
c) On the interval 2 < t < 3, is the speed of the particle
increasing or decreasing? Give a reason for your answer.
d) During what time intervals, if any, is the acceleration
of the particle negative? Justify your answer.
5. Rocket A has a positive velocity v(t) after being launched upward from an initial height of 0 feet
at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the
interval 0 ≤ 𝑡 ≤ 80 seconds, as shown in the table below.
Find the average acceleration of Rocket A over the time interval 0 ≤ 𝑡 ≤ 80 seconds. Indicate
units of measure.
Station #3 – Calculator Active
1.
A particle moves along the x-axis so that at any time 𝑡 ≥ 0, its velocity is given by 𝑣(𝑡) = 3 +
4.1cos⁡(0.9𝑡). What is the acceleration of the particle at time t = 4?
a) -2.016
b) -0.677
c) 1.633
d) 1.814
e) 2.978
2. An object moves along the x-axis with initial position x(0) = 2. The velocity of the object at time
𝜋
𝑡 ≥ 0 is given by 𝑣(𝑡) = sin⁡( 3 𝑡).
a) What is the acceleration of the object at time t = 4?
b) Consider the following two statements. Evaluate their accuracy and explain your reasoning.
Statement I: For 3 < t < 4.5, the velocity of the object is decreasing.
Statement II: For 3 < t < 4.5, the speed of the object is increasing.
c) What is the displacement of the object from t = 0 to t = 4?
d) What is the position of the object at time t = 4?
3. A particle moves along the y-axis so that its velocity v, in cm/min at time 𝑡 ≥ 0 is given by 𝑣(𝑡) =
1 − tan−1 (𝑒 𝑡 ).
a) Find the acceleration of the particle at time t = 2.
b) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your
answer.
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6
c) Find ∫0 𝑣(𝑡) 𝑑𝑡. Using correct units, explain the meaning of ∫0 𝑣(𝑡) 𝑑𝑡.
4. A particle moves along the x-axis so that its velocity v at time t, for 0 ≤ 𝑡 ≤ 5, is given by 𝑣(𝑡) =
ln⁡(𝑡 2 − 3𝑡 + 3). The particle is at position x = 8 at time t = 0.
a) Find the acceleration of the particle at time t = 4.
b) Find all times t in the open interval 0 < t < 5 at which the particle changes direction. During
which time intervals, for 0 ≤ 𝑡 ≤ 5, does the particle travel to the left?
c) Find the position of the particle at time t = 2.
d) Find the average acceleration of the particle over the interval 0 ≤ 𝑡 ≤ 2.
5. A particle moves along the x-axis so that its velocity at time t is given by 𝑣(𝑡) = −(𝑡 + 1)𝑠𝑖𝑛 (𝑡 ).
2
a) Find the velocity of the particle at t = 3.
2
b) Find the speed of the particle at t = 3.
c) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t
= 2? Why or why not?
d) Find all times t on the open interval 0 < t < 3 when the particle changes direction. Justify your
answer.
6. A particle moves along the x-axis so that at any time 𝑡 ≥ 0, its velocity is given by 𝑣(𝑡) =
cos(2 − 𝑡 2 ). The position of the particle is 3 at time t = 0. What is the position of the particle
when its velocity is first equal to 0?
a) 0.411
b) 1.310
c) 2.816
d) 3,091
e) 3.411