Math 260 Review questions for Multivariable ... - Gilles Cazelais

Math 260
Review questions for Multivariable Calculus
1. Find and classify the critical points of f(x, y) = x 3 + xy 2 − 3x 2 − 4y 2 + 4.
2. Show by substitution that φ(x, y, z) = e 3x+4y sin(5z) satisfy Laplace’s equation
∂ 2 φ
∂x 2 + ∂2 φ
∂y 2 + ∂2 φ
∂z 2 = 0.
3. Find an equation for the tangent plane at point (1, −1, 3) on the hyperboloid z 2 − 2x 2 − 2y 2 = 5.
4. Consider the function f(x, y) = x 2 + 3xy.
(a) Find the directional derivative of f at point (−1, 1) in the direction of vector v = 3i − 4j.
(b) Find the smallest directional derivative of f at point (−1, 1) and a unit vector pointing in
the direction in which it occurs.
5. Find Find ∇f(x 0 , y 0 ) given that
D u f(x 0 , y 0 ) = −1,
for u = − 4 5 i + 3 5 j
and
D v f(x 0 , y 0 ) = 2, for v = 3 5 i + 4 5 j.
6. Write the appropriate chain rule for ∂f
∂x
if f depends on u, v; both u and v depends on x, y, z;
and z depends on x and y.
7. Let f(x, y) = sin(x − y) where x = ln(t + s) and y = se 2t . Use the chain rule to find ∂f
∂s
8. The equation e xy + z 3 x − zy 2 = 0 defines z implicitly as a function of x and y. Find ∂z
∂x
and
∂f
∂t .
and
∂z
∂y .
9. Use differentials to estimate the value of f(3.1, 1.8) given that f(3, 2) = 5 and ∇f(3, 2) = 4i − j.
10. By approximately how much will the value of w = x2 y 3
by 2%, and z decreases by 3%
z
change if x increases by 1%, y increases
11. The centripetal acceleration of a particle moving in a circle is a = v2
r
where v is the velocity
and r is the radius of the circle. Use differentials to estimate the maximum relative error of the
acceleration if there are relative errors of ±2% in v and ±1% in r.
12. Find the absolute maximum and minimum values of f(x, y) = xy(3 − x − y) on the triangular
region: x ≥ 0, y ≥ 0, x + y ≤ 4. At what points are these extremum values attained
13. Suppose the temperature T on a disk x 2 + y 2 ≤ 1 is given by T = 2x 2 + y 2 − y. Find the hottest
and coldest spots on the disk.
14. Consider the curve x 2 y = 16.
(a) Find the vectors normal to the curve at points (4, 1) and (2, 4) on the curve.
(b) Use Lagrange multipliers to find the points on the curve that closest to the origin.
(c) Find the vectors normal to the curve at the points you found in (b).
15. Find the minimum value of f(x, y, z) = 3x + 2y + z on the paraboloid z = 9x 2 + 4y 2 . At what
point is the minimum value attained What about the maximum value
16. The curve 17x 2 +12xy+8y 2 = 100 is an ellipse centered at the origin. Use the method of Lagrange
multipliers to locate the points on the ellipse that are closest and farthest from the origin.

Math 260 Review questions for Multivariable Calculus Page 2 of 4
17. Evaluate ∫∫ R (x + y) dA where R is the region bounded by y = √ x and y = x 2 .
18. Locate the center of mass of the region bounded by the graphs of y = x 2 , x = 0, y = 1 with density
ρ(x, y) = xy.
19. Evaluate the following double integrals.
(a) I =
∫ 2 ∫ 4−x 2
0
0
xe 2y
∫ 2
4 − y dy dx (b) I =
0
∫ 1
y/2
sin(x 2 ) dx dy (c) I =
∫ ln 2 ∫ 2
0
e x 1
ln y
20. Evaluate ∫∫ R y2 dA where R is the region between the circles x 2 + y 2 = 1 and x 2 + y 2 = 4.
21. Set-up iterated double integrals in polar coordinates for the following quantities.
(a) The surface area of the part of the surface z = xy that lies in the region
x ≥ 0, 0 ≤ y ≤ x, x 2 + y 2 ≤ 9.
(b) The moment with respect to the y-axis, (i.e., ∫∫ R
x dA) of the region R defined by
1 ≤ x 2 + y 2 ≤ 2, y ≥ 0, 0 ≤ x ≤ √ 3 y.
dy dx.
(c) The moment of inertia I y of the first-quadrant region outside the circle r = 1 and inside the
cardioid r = 1 + cos θ. Assume density is constant.
(d) The volume of the part of the ball x 2 + y 2 + z 2 ≤ 4 that lies in the first octant and inside the
cylinder x 2 + y 2 = 2x.
22. Set-up an iterated triple integral for ∫∫∫ Q x2 dV where Q is the first-octant region under the plane
of equation 3x + 2y + z = 6.
23. Assume that p(x, y, z) is a function that measures the air pollutant particles per cubic meters of
air at any point inside BC Place stadium in Vancouver. What does
∫∫∫
p(x, y, z) dV
represent physically
(BC Place)
24. Set-up an iterated triple integral in cylinder coordinates to evaluate ∫∫∫ Q x2 dV where Q is the
region between the paraboloids z = 4 − x 2 − y 2 and z = x 2 + y 2 .
25. Set-up an iterated triple integral in spherical coordinates to evaluate I z = ∫∫∫ Q (x2 + y 2 ) dV where
Q is the region that lies between the spheres x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 9, and inside
the cone z = √ 4x 2 + 4y 2 .
26. Use spherical coordinates to evaluate
∫ 1 ∫ √ 1−x 2
−1
∫ √ 2−x 2 −y 2
− √ √
1−x 2 x 2 +y 2
√
x 2 + y 2 + z 2 dz dy dx.
27. Evaluate ∫∫∫ B (x2 + y 2 ) dV where B = {(x, y, z) | x 2 + y 2 + z 2 ≤ 1} is the unit ball in R 3 .
28. Let R be the region bounded by the curves xy = 1, xy = 4, y = 1, and y = 2. Evaluate ∫∫ R exy dA
by using an appropriate change of variables.
29.
∫∫
Let R be the parallelogram region with vertices at (0, 0), (3, −3), (5, −2), and (2, 1). Evaluate
(x + y) dA by using an appropriate change of variables.
R
30. Let R be the region in the first quadrant bounded by the curves xy = 1, xy = 2, y = x, and
y = 3x. Evaluate ∫∫ R y2 dA by using the change of variables u = xy and v = y/x.

Math 260 Review questions for Multivariable Calculus Page 3 of 4
Answers
1. (0, 0) → relative maximum, (2, 0) → saddle point.
2. We have φ xx = 9 φ, φ yy = 16 φ, and φ zz = −25 φ. Then, φ xx + φ yy + φ zz = 9φ + 16φ − 25φ = 0.
3. −2x + 2y + 3z = 5.
4. (a) 3 (b) Smallest D u f(−1, 1) is − √ 10 in direction u = −1 √
10
i + 3 √
10
j.
5. ∇f(x 0 , y 0 ) = 2i + j = 〈2, 1〉.
6.
7.
8.
∂f
∂x = ∂f ∂u
∂u ∂x + ∂f ∂u ∂z
∂u ∂z ∂x + ∂f ∂v
∂v ∂x + ∂f ∂v ∂z
∂v ∂z ∂x .
∂f
∂s
=
cos(x − y)
t + s
∂z
∂x = −(yexy + z 3 )
3z 2 x − y 2
9. f(3.1, 1.8) ≈ 5.6
− e 2t cos(x − y) and ∂f
∂t
and
10. w increases by about 11%.
∂z
∂y = −(xexy − 2zy)
3z 2 x − y 2 .
=
cos(x − y)
t + s
11. The maximum relative error of the acceleration is about 5%.
− 2se 2t cos(x − y).
12. Absolute maximum value is 1, attained at (1, 1). Absolute minimum value is −4, attained at
(2, 2).
( √ )
13. Hottest spots: ± 3
2 , − 1 2
where T = 9/4. Coldest spot: (0, 1 2
) where T = −1/4.
14. (a) A vector normal at (4, 1) is i + 2j. A vector normal at (2, 4) is 4i + j.
(b) The closest point to the origin are (±2 √ 2, 2).
(c) A vector normal at (−2 √ 2, 2) is − √ 2 i + j. A vector normal at (2 √ 2, 2) is √ 2 i + j.
15. The minimum value is − 1 2 and is attained at (x, y, z) = (− 1 6 , − 1 4 , 1 2
). The function does not have
a maximum value on the paraboloid.
16. Closest points: (−2, −1) and (2, 1). Farthest points: (2, −4) and (−2, 4).
17. 3/10
18. (x, y) = ( 4
7 , 3 )
4
19. (a) (e 8 − 1)/4, (b) 1 − cos(1), (c) 1.
20.
15π
4
21. (a) S =
22.
(c) I y =
∫ π/4 ∫ 3
0 0
∫ π/2 ∫ 1+cos θ
0 1
∫ 2 ∫ (6−3x)/2 ∫ 6−3x−2y
0
0
0
r √ 1 + r 2 dr dθ (b) M y =
r 3 cos 2 θ dr dθ (d) V =
x 2 dz dy dx
∫ π/2 ∫ √ 2
π/6 1
∫ π/2 ∫ 2 cos θ
23. It represents the total amount of pollution inside BC place.
0
0
r 2 cos θ dr dθ
r √ 4 − r 2 dr dθ

Math 260 Review questions for Multivariable Calculus Page 4 of 4
24.
25.
∫ 2π ∫ √ 2 ∫ 4−r 2
0 0 r 2
∫ 2π ∫ arctan(1/2) ∫ 3
0 0
1
r 3 cos 2 θ dz dr dθ
ρ 4 sin 3 φ dρ dφ dθ
26. π(2 − √ 2)
27. 8π/15
28. (e 4 − e) ln 2
29. We can use the change of variable u = x + y and v = x − 2y to get ∫∫ R
(x + y) dA = 27/2.
Note that if we use rectangular coordinates, we need to evaluate three double integrals.
30. 3/2
∫ 2 ∫ x/2
0
−x
(x + y) dy dx +
∫ 3 ∫ 3−x
2
−x
(x + y) dy dx +
∫ 5 ∫ 3−x
3
(x−9)/2
(x + y) dy dx = 27
2
Gilles Cazelais, March 13, 2008.