MATH 200
—Calculus III (December 1999)
Closed book examination.
Time:
2.5 hours.
Special Instructions:
Calculators are NOT required but may be used.
Books or notes are NOT permitted.
Show your work in the spaces provided.
1) [10 marks]
Answer each of the following with either true [T] or false [F].
Reasons need
not be given.
No credit for ambiguous or wrong answers.
a) Let
a
and
b
be vectors in
R
3
.
If
c
a
b
a
⋅
=
⋅
then
b
=
c
.
b) If
a
and
b
are parallel vectors in
R
3
, then
a x b = 0
.
c)
.
1
)
sin(
lim
)
0
,
0
(
)
,
(
=
→
x
xy
y
x
d) The equation
x
2
+
y
2
= 0 describes a line in
xyz
space.
e) The space curve parametrized by
k
j
i
r
t
t
t
t
+
+
=
)
(cos
)
(sin
)
(
lies on the cylinder
1
2
2
=
+
y
x
.
f) If a function
f
= (
x
,
y
,
z
) has directional derivative
0
)
,
,
(
0
0
0
=
z
y
x
f
D
u
for all unit
vectors
u
, then
)
,
,
(
0
0
0
z
y
x
is a critical point of
f
.
g) Assuming
f
=
f
(
t
) is a differentiable function and
g
(
x
,
y
) =
f
(
xy
), then
)
(
'
xy
yf
x
g
=
∂
∂
.
h) The function
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 Spring '03
 Unknown
 Math, Calculus, Derivative, Multivariable Calculus, dzdydx, directional derivative Du, spherical integral hemisphere

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