Math 32A Exam 1 Solutions
Problem 1. Below, choose the answer which best answers each question. Write the answer
on the blank provided, and write the answer on the front page of your exam.
There is no partial credit on this problem.
(a) (1 point) What typ

Math 32A Quiz 5A Solution
Problem 1. Prove that aN =
av
v
. (Hint: start by decomposing a as aT T + aN N.)
Solution: Writing a = (aT T + aN N), we have
av
(aT T + aN N) v
=
v
v
aT T v + aN N v
=
v
aN N v
=
v
Note the last equality follows from the fact th

Math 32A Quiz 4B Solution
Name:
SID:
Problem 1. Find the length of the cycloid
r(t) = t sin(t), 1 cos(t)
on the interval 0 t 2. Recall the identity
sin2 (t/2) =
1 cos(t)
.
2
Solution: We compute
r (t) = 1 cos(t), sin(t)
r (t) =
(1 cos(t)2 + sin2 (t)
=
1 2

Math 32A Quiz 5A Solution
Problem 1. Prove that aN =
av
v
.
Solution: From Section 14.5 we have the formulas
a 2 aT
av
.
aT =
v
aN =
2
Because of the dot product formula |u v| = u v cos() (where is the angle between
the vectors), we have
aN =
a 2 aT 2
=
=

Math 32A Quiz 3B Solution
1
1
s
,
ds.
2 1 + s2
1+s
1
1
s
,
ds =
2 1 + s2
1+s
Problem 1. Evaluate
0
Solution:
We have
0
1
0
1
ds,
1 + s2
1
0
s
ds . Recall that
1 + s2
s
1
ds = arctan(s) + C. To evaluate
ds, we make the substitution u = 1 + s2
2
1+s
1 + s2

Math 32A Quiz 4A
Name:
SID:
Problem 1. Find the value of t in [0, 2] such that the speed of the cycloid
r(t) = ht sin(t), 1 cos(t)i
is at a maximum.
Solution:
We compute
r0 (t) = h1 cos(t), sin(t)i
q
0
kr (t)k = (1 cos(t)2 + sin2 (t)
q
= 1 2 cos(t) + cos2

Math 32A Quiz 2B Solution
Problem 1. Find parametric equations for the line through P0 = (3, 1, 1) perpendicular to
the plane 3x + 5y 7z = 29.
Solution: A normal vector to the plane is given by n = 3, 5, 7 , so our desired line goes
through (3, 1, 1) in t

Math 32A Quiz 3A Solution
Problem 1. Determine whether the space curves given by r 1 (t) = t, t2 , t + 1 and r 2 (s) =
s, s, s 1 intersect, and if they do, determine where.
s, s, s 1 to solve for s and t. Setting
Solution: Set r 1 (t) = t, t2 , t + 1 = r

Math 32A Quiz 2A Solution
Problem 1. Find the equation of the plane that contains the lines r 1 (t) = ht, 2t, 3ti and
r 2 (t) = h3t, t, 8ti.
Solution: Note that r 1 (0) = h0, 0, 0i = r 2 (0), so (0, 0, 0) lies on both lines. We then have
that v 1 = r 1 (1

Math 32A Quiz 1B Solution
1
2
2 along v = 0 .
Problem 1. Find the projection of u =
0
1
You must fully simplify your answer to receive full credit.
Solution: We use the formula
projv (u) =
uv
vv
2
4/9
2
0 =
0 .
v=
9
1
2/9
1

Math 32A Practice Exam 2
Problem 1. Compute the length of the curves over the given intervals
(a) r(t) = cos(t), sin(t), t3/2 on [0, 2]
(b) r(t) = t2 , 2t2 , t3 on [0, s]
Problem 2. Find an arc length parametrization of the curves
(a) r(t) = t2 , t3
(b) r

Math 32A Quiz 1A Solution
Problem 1. Find the intersection of the following lines in R2 :
1
2
r1 (t) =
+t
1
4
2
1
r2 (s) =
+s
.
1
6
You must fully simplify your answer to receive full credit.
Solution: We must find s, t R such that
1
2
2
1
+

Practice Exam 1 Answers
January 26, 2016
1) a) 4 3
b) 2
c) 9
2) a) r(t) = ht, 1 + t, 2 + ti
b) r(t) = ht, 1 + 2t, 2 ti
3) a) x 2y + z = 0
b) 2(x 1) + 2(y 2) + (z 4) = 0
4) a) h2, 1, 0i
b) h2t + 1, 3t, 5ti
5) 72
6) a) h 12 , 0, 12 i
b) h1, 1, 0i
c)
d)
3
7

Math 32A Practice Exam 1
Problem 1. Calculate the norms of the following vectors.
(a) 4(i j + k)
(b) h1, 1, 1, 1i R4
(c) h1, 2, 2i h2, 1, 2i
Problem 2. Find a parametrization of the line:
(a) passing through the points (0, 1, 2) and (1, 2, 3).
(b) passing

Math 32A Quiz 5B Solution
Problem 1. Let r(t) be given by r(t) = t, et , tet . Find the decomposition of a(t) = r (t)
into tangential and normal components at t = 0.
Solution: First we compute that r (t) = 1, et , (1 + t)et and r (t) = 0, et , (2 + t)et .