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Midterm Exam # 1
September 24, 2007
MthSc 206 – 013: Calculus of Several Variables
Instruction:
PRINT your name clearly
in
capitals
and underline your last name
.
NAME:
EXAM SOLUTION
1.
(3pts). Show that the following equation represents a sphere and ﬁnd its center and radius:
x
2
+
y
2
+
z
2
+ 2
x

6
y
+ 4
z
+ 10 = 0
.
Solution:
x
2
+ 2
x
+
y
2

6
y
+
z
2
+ 4
z
=

10
x
2
+ 2
x
+ 1 +
y
2

6
y
+ 9 +
z
2
+ 4
z
+ 4 =

10 + 1 + 9 + 4
(
x
+ 1)
2
+ (
y

3)
2
+ (
z
+ 2)
2
= 4
It is an equation for a sphere. The center is
c
(

1
,
3
,

2)
and its radius is
2
.
2.
(8pts) Two vectors
a
=
h
1
,
√
2
,
√
3
i
and
b
=
h
0
,
1
,
0
i
are given.
(a) (3pts) Find the scalar projection (scalar component) of
b
onto
a
.
Solution:
Comp
a
b
=
a
◦
b

a

=
h
1
,
√
2
,
√
3
i ◦ h
0
,
1
,
0
i
q
(1)
2
+ (
√
2)
2
+ (
√
3)
2
=
√
2
√
6
=
√
3
3
∼
= 0
.
577
.
(b) (3pts) Find the vector projection of
b
onto
a
.
Solution:
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This note was uploaded on 04/03/2008 for the course MTHSC 206 taught by Professor Chung during the Spring '07 term at Clemson.
 Spring '07
 Chung
 Math, Calculus

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