MATH 126 D & E
Exam I
April 25, 2013
Name
Student ID #
Section
HONOR STATEMENT
“I affirm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized assistance
on this exam.”
SIGNATURE:
1
2
3
4
5
Total
10
12
12
8
8
50
• Your exam should consist of this cover sheet, followed by 5 problems. Check that you have
a complete exam.
• Pace yourself. You have 50 minutes to complete the exam and there are 5 problems. Try
not to spend more than 5 minutes on each page.
• Unless otherwise indicated, show all your work and justify your answers.
• Unless otherwise indicated, your answers should be exact values rather than decimal approximations. (For example, π4 is an exact answer and is preferable to its decimal approximation
0.7854.)
• You may use a scientific calculator and one 8.5×11-inch sheet of handwritten notes. All
other electronic devices (including graphing calculators) are forbidden.
• The use of headphones or earbuds during the exam is not permitted.
• There are multiple versions of the exam, you have signed an honor statement, and cheating
is a hassle for everyone involved. DO NOT CHEAT.
• Turn your cell phone OFF and put it AWAY for the duration of the exam.
GOOD LUCK!
Math 126 — Spring 2013
1. (10 points) Let a = �0, 3, 2�, and b = �−1, −1, 4�.
(a) Find two unit vectors orthogonal to both a and b.
(b) Find a vector v, parallel to the y-axis, such that compa v = 5.
1
Math 126 — Spring 2013
2
2. (12 points) Consider two lines
�1 : x = 4 + 2t, y = 3 − t, z = 1 + 6t
1
�2 : x = 3 + 3s, y = s, z = 20 − 2s.
4
(a) Find the point at which these two lines intersect.
(b) Find an equation of the form Ax + By + Cz = D for the plane that contains these two
lines.
Math 126 — Spring 2013
3. (12 points) Let r(t) =
3
�
1
t + cos t, t2 − 4t, 2 sin t
π
�
(a) Find the curvature at t = π2 .
(b) Let � be the line tangent to r(t) at t = π. Find the point at which the line � intersects
the xz-plane.
Math 126 — Spring 2013
4
4. (8 points) Let X be the surface in R3 determined by the equation z 2 = x2 − 2y 2 . You are
not required to show any work for the following questions.
(a) Identify the traces of X in the indicated plane.
i. The trace in the plane x = 0 is a(n):
circle
hyperbola
point
ellipse
parabola
pair of lines
ii. The trace in the plane x = k (k �= 0) is a(n):
circle
hyperbola
point
ellipse
parabola
pair of lines
iii. The trace in the plane y = 0 is a(n):
circle
hyperbola
point
ellipse
parabola
pair of lines
iv. The trace in the plane y = k (k �= 0) is a(n):
circle
hyperbola
point
ellipse
parabola
pair of lines
v. The trace in the plane z = 0 is a(n):
circle
hyperbola
point
ellipse
parabola
pair of lines
vi. The trace in the plane z = k (k �= 0) is a(n):
circle
hyperbola
point
ellipse
parabola
pair of lines
(b) Identify the surface X.
cone
elliptic paraboloid
hyperboloid of one sheet
ellipsoid
hyperbolic paraboloid
hyperboloid of two sheets
(c) True or False?
The path described by the vector function r(t) =
�
1 1
t, t, √ t
2
2
�
lies on X.
Math 126 — Spring 2013
5
5. (8 points) Match the graph of r = f (θ) with its corresponding polar curve. You are not
required to show any work.
r
r
2
2
1
1
!
A.
2!
!
!
B.
r
r
1
1
!
2!
!
!
-1
-1
C.
D.
Place the letter of the graph from above in the appropriate box below.
i.
ii.
iii.
iv.
2!
2!
!
!