MATH 2551 PRACTICE EXAM 1
COVERS MATERIAL UP TO SECTION 14.1
Name
Section
1
2
3
4
5
6
Total
Please read all instructions carefully before beginning.
• You have 50 minutes to complete this exam.
• There are no aids of any kind (notes, text, etc.) allowed.
• Please show your work.
• Good luck!
This is a practice exam. It is similar in format, length, and difficulty to
the real exam. It is not meant as a comprehensive list of study problems. I recommend completing the practice exam in 50 minutes, without notes or distractions.
1. True or false. Clearly indicate your answer. If the statement is ever false, answer false.
There is no partial credit, and you do not need to justify your answer.
a) The volume of the parallelepiped determined by vectors u, v, and w is
|(u · v) × w|.
TRUE
FALSE
b) If ⟨A1 , B1 , C1 ⟩ and ⟨A2 , B2 , C2 ⟩ are perpendicular nonzero vectors, then the planes
given by A1 x + B1 y + C1 z = 1 and A2 x + B2 y + C2 z = 3 are perpendicular.
TRUE
FALSE
c) The domain of f (x, y) = ln(x 2 + y 2 − 1) is a closed set.
Z1
d)
⟨1, 0, 2t⟩ d t = 2.
TRUE
TRUE
FALSE
FALSE
0
2.
a) A curve and its unit tangent vector at a point P are given below. Draw the principal
unit normal vector for the curve at P.
b) Identify, and roughly sketch, the surface given by x 2 + z 2 − y 2 = 1.
3. Find an equation for the plane determined by the intersecting lines `1 and `2 given below
(in other words, find the plane that contains both lines).
Write the plane in the form Ax + B y + Cz = D.
`1 :
`2 :
x = 1 + 2t,
y = 3 − 2t,
z = 5 − 4t
(−∞ < t < ∞)
x = 3 − s,
y = 5 − 3s,
z =2+s
(−∞ < s < ∞)
4.
a) Let r(t) = ⟨ln(t), t 3 − t⟩ for t > 0. Find the angle between v(1) and a(1).
(as usual, v(t) is velocity and a(t) is acceleration)
¬
¶
b) For the curve r(t) = cos(2t), sin(2t), t − 1 , find κ(t).
5.
p
a) Let f (x, y) = (x 2 − 4)( y + 1). Graph the domain of f on the axes provided below.
Show your work for obtaining the graph. Your work must be clear in order to receive
full credit.
y
4
3
2
1
-4 -3 -2 -1
1
-1
-2
-3
-4
b) At the point (−3, −5), an infinitely-differentiable curve r(t) satisfies
¬4 3¶
¬3 4¶
1
,−
N=
,
κ= .
T=
5 5
5 5
5
Find the equation for the circle of curvature for r at (−3, −5).
2
3
4
x
6.
¬
¶
a) Consider the curve r(t) = 3 sin(t), 5 cos(t), −4 sin(t) for 0 ≤ t ≤ π. Find the arc
p
length along r from ( 32 , 5 2 3 , −2) to (3, 0, −4). Fully simplify your final answer.
b) Let u = ⟨−4, 2, 3⟩.
(i) Suppose v is a unit vector. What is the maximum possible value of |u × v|?
Justify your answer.
(ii) Find a unit vector v which finds the maximum value from part (i). Justify your
answer, either with a computation or a mathematical justification.
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