Spring 2007 - Takeda's Class - Practice Exam 1 - Math 20E...

Math 20E, Practice Midterm 1
April 24, 2007
Name:
Section:
This exam consists of 7 pages including this front page.
Ground Rules
1. No calculator is allowed.
2. Show your work for every problem. A correct answer without any justification
will receive no credit.
3. You may use one 4-by-6 index card, both sides.
4. Show your ID on your desk.
Score
1 10
2 10
3 10
4 10
5 10
6 10
Total 60
1

1. Let a = 2 i + k and b = - i + 2 j - 3 k be vectors in R 3 .
(a) Compute the orthogonal projection of a on b .
Answer :
The orthogonal projection p is
p =
b · a
|| b || 2
b
=
- 2 - 3
1 + 4 + 9
b
=
- 5
14
b
=
5
14
i -
5
7
j +
15
14
k .
(b) Compute a × b .
Answer :
a × b =
i j k
2 0 1
- 1 2 - 3
=
0 1
2 - 3
i -
2 1
- 1 - 3
j +
2 0
- 1 2
= - 2 i + 5 j + 4 k .
2

2. (a) Let f : R 3 → R given by f ( x, y, z ) = x 2 + y 2 + z 2 - 2 z . Describe level
surfaces by words.
Answer :
f ( x, y, z ) = x
2
+ y
2
+ z
2
- 2 z
x
2
+ y
2
+ ( z - 1)
2
- 1 .
For each c , the level surface is
x
2
+ y
2
+ ( z - 1)
2
- 1 = c,
i.e.
x
2
+ y
2
+ ( z - 1)
2
= c + 1 .
Hence the level surfaces are the spheres with center (0 , 0 , 1).
(b) Let c : R → R 3 be a path in space defined by c ( t ) = ( e t - 1 , t, t 2 ). Find