MATH 32A/3: Practice Final Exam
Dr. Frederick Park
Note: This Exam is Slightly Longer than the Actual Final
1. Find the equation of the tangent plane and the normal line to the surface given
by
xy
+
yz
+
zx
= 3 at the point (1
,
1
,
1).
2. Find the points on the sphere
x
2
+
y
2
+
z
2
= 1 where the tangent line is parallel
to the plane 2
x
+
y

3
z
= 2.
3. Use the chain rule to find
∂w/∂t
where
w
=
√
x
+ (
y
2
/z
),
y
=
t
3
+4
t
,
x
=
e
2
t
,
and
z
=
t
2

4.
4. Use a tree diagram to write out the chain rule for the case where
w
=
f
(
t, u, v
),
t
=
t
(
p, q, r, s
),
u
=
u
(
p, q, r, s
), and
v
=
v
(
p, q, r, s
) are all differentiable
functions.
5. The length
x
of a side of a triangle is increasing at a rate of 3 in/s, the length
y of another side is decreasing at a rate of 2 in/s, and the contained angle
θ
is increasing at a rate of 0.05 radians/s. How fast is the area of the triangle
changing when
x
= 40 in,
y
= 50 in, and
θ
=
π/
6?
6. If
yz
4
+
x
2
z
3
=
e
xyz
, find
∂z/∂x
and
∂z/∂y
.
7. Directional Derivative. Let
f
=
f
(
x, y, z
) be a differentiable function:
(a) When is the directional derivative of f a maximum?
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 Spring '09
 Park
 Math, Calculus, Multivariable Calculus, Optimization, Mathematical analysis

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