Download Spring 15

Math 170
Common Final Exam
Spring 2015
Part 1: Short Answer
• Pages 3 to 8 are short answer.
• Most problems do not require you to show work.
• Partial credit will be rare.
1. (14 points) Find the following time derivatives. f and g are functions of time while a and b are
constants.
d at (a)
fe
dt
d
a(b − g 2 )3
(b)
dt
2. (10 points) Suppose the derivative of a function is
df
1
= cos(at + b) − 3
dt
t
where a and b are constants. Give a possible function f .
3. (12 points) Suppose that the pressure in a cylinder, P in kilopascals (kPa), is a function of time t,
in minutes. Write a short sentence that describes the physical meaning of each of the following:
(a) ∆P on [1, 3]
∆P
on [3, 7]
(b)
∆t
dP (c)
dt t=1
d2 P (d)
dt2 t=3
4. (12 points) The pressure in a cylinder, P in kilopascals (kPa), is measured every 2 minutes and
recorded in the following table.
t (min)
0
2
P (kPa) 340
(a) Find ∆P on the interval [2, 8].
dP
(b) Estimate
at the instant t = 6.
dt
1
4
6
8
141 113 99 89
5. (12 points) Consider the definite integral
Z
5
√
x 3x − 5dx
0
Use the substitution u = 3x − 5 to transform this integral into one using the variable u. Do not
calculate the integral.
Z
6. (10 points) The potential of a circuit, V in volts, is a function of time. The initial potential and rate
of change in the potential of this circuit are given by
• V (0) = 20 Volts
dV
= −12.60e−0.63t V/min
•
dt
Write an integral formula that gives the voltage V at the time t = 2 minutes. Do not calculate
the integral.
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7. (18 points) The height of a moving object is graphed in each figure below. For each quantity on
the left sketch the secant or tangent line (whichever is appropriate) that corresponds to the quantity
using the graph on the right.
Velocity at time t = 5 seconds.
∆h
on [1, 3]
∆t
Average velocity on the interval [3, 7]
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8. (18 points) The force on an object is a function of time. Suppose the force starts out at 100 Newtons
(N) at time t = 0 and decreases to 0 Newtons at time t = 5 minutes. For each description to the
left, sketch a graph of a possible force function on the right.
The force decreases at a constant rate.
The rate of change in the force is increasing.
The rate of change in the force is decreasing.
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Part 2: Long Answer
• The rest of the exam requires you show all work.
• Unsupported answers will not receive full credit.
• Present your work clearly with correct notation. Neatness counts.
• If you use your calculator beyond basic arithmetic you must
– State what calculator tool/function you used.
– Write what you entered into the calculator in calculator notation.
– State what information you got from the calculator.
– If you used a graph, recreate the graph with proper labeling.
• All answers must include correct units.
• If you solve an equation you must show how you solved the equation.
• If you take a derivative you must show how you took the derivative.
• If you calculate an integral, you must show how you calculate the integral.
• Most problems require you answer the question with a sentence. Pay attention
to the instructions on each problem.
1. (18 points) The height, h, of a mass attached to a spring is given by
h(t) = 20e−0.2t cos(1.2t) + 50
Where h is in inches and t in seconds.
Find the velocity of this mass after 1 second. Write your answer in a sentence in the context of this
problem.
2. (18 points) The distance, s, a skydiver has fallen is a function of time given by
s(t) = 270e−0.2t + 54t − 270
Where s is in meters and t is in seconds.
How far has the skydiver fallen the instant the skydiver is moving at a rate of 36 m/s? Write your
answer in a sentence in the context of this problem.
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3. (18 points) An object moves up and down so that its height, h, in inches is a function of time, t, in
seconds. The initial height and rate of change in the height are
• h(0) = 50 in.
dh
•
= −20 sin
dt
π
t
2
in/s.
Find ∆h on the time interval 0 ≤ t ≤ 3. Write your answer in a sentence in the context of this
problem.
4. (20 points) The following graph gives the velocity in meters of an object shot into the air as a function
of time in seconds.
(a) Shade in the area on the above graph that represents the change in the height of this object on
the time interval 1 ≤ t ≤ 3.
(b) Find the change in the height of this object on the time interval 1 ≤ t ≤ 3.
(c) If the object reaches a maximum height of 100 meters, what was the initial height of the object
(t = 0)?
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5. (20 points) Solve only one of the following two problems. Circle the one you want graded. Use
the back page for your work.
(a) Becky sails in a ship heading due north of a port
at a speed of 10 miles per hour. Sometime later
Ryan sails in a ship heading due east of the same
port at a speed of 15 miles per hour. What is the
rate of change in the distance between the two
ships at the instant Becky is 6 miles north and
Ryan is 8 miles east of the port?
Write your answer in a sentence in the context of
this problem.
(b) A rancher wishes to build three corrals, such that
the total enclosed area is A = 5000 m2 . To save
on fencing he builds the corrals side by side as
shown in the diagram. What is the minimum
fence length, F , needed? Be sure to state the objective function and constraint equation in your
work.
Write your answer in a sentence in the context of
this problem.
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