PRACTICE FINAL EXAM
Math 1320 sec 004
December XXX, 2013
Name: |
{z
by writing my name I swear by the honor code
}
Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I may take off points if
I cannot see how you arrived at your answer (even if your final answer is correct).
• No calculators or books are allowed for this test.
• You may use a flowchart you have created that describes series convergence tests. You
cannot use one you have not created or has more information than series convergence tests.
• No other notes are allowed for this test.
• Please keep your written answers brief; be clear and to the point. You may lose points for
incorrect or irrelevant statements.
• Please turn off cellphones, turn hats backwards, take out headphones and sit every other
seat, if possible.
• You have 120 minutes to work on this exam.
• Good luck!
Page (value)
1 (20 points)
2 (15 points)
3 (15 points)
4 (15 points)
6 (15 points)
7 (15 points)
Total (110 points)
Score
1.
(20 points) Short Answer. For True or False you must write ‘True’ or ‘False’ and justify
your answer to receive credit.
a. (4 pts)
Consider working in spherical coordinates (ρ, θ, φ). Describe the surface with
equation φ = π/3.
b. (4 pts)
A force F~ = 4x̂ + 9ŷ N moves a mass in a straight line from point (0,0) to
(4,1) in meters. What is the work done by this force?
c. (4 pts)
Consider a 3-d curve ~r(t). Write down a formula that calculates arc length,
s, as a function of a time-like parameter t.
d. (4 pts)
Does
∞
P
n=1
e. (4 pts)
2n
n2013
converge?
What is the gradient of f (x) = x sin(xy)?
2.
(15 points)
a. (5 pts) Calculate the average value of f (x) = sin x cos x over the interval 0 ≤ x ≤ π/4.
b. (4 pts) Set up an integral that represents the volume of the solid created by rotating
the region enclosed by the curves y = x2 and y = x4 about the x-axis.
c. (6 pts) A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building
120 ft high. How much work is required to pull the rope to the top of the building?
3.
(15 points)
a. (5 pts) A parachute falling to the ground has a veloctiy modeled by m dv
dt = −mg + kv
where m is its mass, g is gravity and kv represents the force due to air resistance. Are there any
equilibrium solutions to this system, and if so, what are they?
b. (10 pts) Solve the differential equation
dy
dt
= √te
y
t
1+y 2
.
4.
(15 points)
a. (8 pts) Suppose a ball is dropped from 2 meters height and bounces to 1.5 meters on
the second bounce, and 9/8ths of a meter on the second bounce, and so on indefinitely. What
is the total vertical distance traveled by the ball?
b. (7 pts)
2
Find the third degree Taylor polynomial of f (x) = e−x about x = 0.
5.
(15 points)
a. (7 pts)
A 30 cm wrench lies along the y-axis and grips a bolt at the origin. A force
is applied in the direction givn by vector 0x̂ + 3/5ŷ − 4/5ẑ. Find the magnitude of the force
required to supply 100 Nm of torque.
b. (8 pts) Find the equation for the plane that passes through the line of intersection of
the planes x − z = 1 and y + 2z = 3 and is perpendicular to the plane x + y − 2z = 1. (hint:
the point (1,3,0) is on the line of intersection)
6.
(15 points)

a. (7 pts)

t
Find the curvature of the twisted cubic curve ~r(t) =  t2  at the point
t3
(0, 0, 0).
b. (8 pts)  What is the force required to have a particle of mass m have the position
cos t

te−t .
function ~r(t) =
t ln t
7.
(15 points)
a. (5 pts)
Find
∂z
∂x
if a surface z = f (x, y) is given implicitly by yz = ln(x + z).
b. (10 pts) Find the absolute maximum of f (x, y) = x2 − 2xy + 2y on the closed set given
by R = {(x, y)|0 ≤ x ≤ 3, 0 ≤ y ≤ 2}.