1. Given two non-parallel vectors u and v, which of the following are perpendicular to u × v?
B) (u · v)(u × v)
D) (u + v) × v
F) u · v
A) 2u + 4v
C) v × u
E) (u × v) × u
2. Find a vector perpendicular to the two lines
< x, y, z >=< 2, −2, 1 > +t < 1, 0, 3 >
< x, y, z >=< 0, 2, 7 > +t < 4, −2, 7 > .
and
A) < −16, −14, 4 >
C) < 6, 5, −2 >
E) < 6, 15, 2 >
B) < 3, 12, 14 >
D) < 6, −5, −2 >
3. What is the trace of the surface 6z = x2 + 4y 2 − 1 in the plane 2y − z − 1 = 0?
A) Ellipse
C) Hyperbola
B) Circle
D) Parabola
4. Compute the partial derivative of the function f (x, y, z) = e1−x cos y + ze−1/(1+y
to x at the point (1, 0, π).
A) −1
C) 0
E) π
2)
with respect
B) −1/e
D) π/e
5. Considering the following contour map of f (x, y), which are the most accurate estimates for
f (2, 4), fx (2, 4) and fy (2, 4)?
A) f (2, 4) = 6, fx (2, 4) = −3, fy (2, 4) = 3/2
B) f (2, 4) = 6, fx (2, 4) = 3/2, fy (2, 4) = −3
C) f (2, 4) = −3, fx (2, 4) = 3/2, fy (2, 4) = −3
D) f (2, 4) = −3, fx (2, 4) = −3, fy (2, 4) = 3/2
6. The tangent plane to z = x2 y + 1/(1 + y 2 ) at the point (1, 1, 3/2) contains the point (2, 2, t)
for which value of t?
√
√
A) 1 + 7/4 2
B) 2
C) 4
D) 5
E) none of the above
7. Which of these quantities is closest to 1.01e0.03 ?
A) 0.03
C) 1.04
B) 1.01
D) 1.13
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8. Find a vector normal to the curve x2 y + ln y − 2x = 0 at the point (2, 1).
B) < 5, −2 >
D) < 2, 1 >
A) < 2, 5 >
C) < −2/5, 1 >
E) none of the above
9. Given z = f (x, y) with ∇f (2, −5) =< 3, −2 >, which vector points along a level curve of f ?
B) < −2, 5 >
D) < 2, 3 >
A) < 5, 2 >
C) < 3, 2 >
E) < −3, 2 >
10. If (a, b) is a critical point of z = f (x, y), which of the following statement must be true?
A) All directional derivatives of f at (a, b) are zero.
B) Either fx (a, b) = 0 or fx (a, b) = 0.
C) The point (a, b) is either a local maximum or a local minimum.
D) fxx (a, b) = 0 = fyy (a, b)
11. Which of the following statements is NOT true?
A) If the value of a function at a saddle point is zero, then near the saddle point this function
takes both positive and negative values.
B) The only possible candidates for local extrema of a differentiable function are those points
at which both first-order partial derivatives are zero.
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C) If at a critical point fxx fyy < fxy
, then it is a saddle point.
D) If at a critical point all the second partial derivatives are equal to zero, then there is a
local minimum at that point.
12. Several contour lines for a certain function f (x, y) are shown below. What is the maximum
value of the function f (x, y) subject to the constraint x + 2y = 10?
A) 10
C) 40
E) 60
B) 25
D) 50
13. Consider the region under z = ax2 + 3y 2 , a > 0 and over the triangle with vertices (0, 0, 0),
(0, 2, 0) and (2, 2, 0). For what value of a does the volume of this region equal to 20?
A) 4
C) 6
D) 12
B) 5
D) 9
E) none of the above
2
RR
14. Given the contour map from Question 5, estimate the value of
[0, 2] × [0, 2].
A) 9
C) 24
f (x, y)dA over the region
B) 18
D) not close to any of the above
15. For the last 4 questions,
p we will be looking at the ice cream cone, that is the solid bound
below by the cone z = x2 + y 2 and above by x2 + y 2 + z 2 = 2.
Which of the following integrals represents its volume?
A)
R 1 R √1−x2
√
− 1−x2
−1
dydx
B)
R 1 R √1−x2 R √1−x2 −y2
C) −1 −√1−x2 √ 2 2 dydx
R 1 R √1−x2
−1
√
− 1−x2
dxdy
R 1 R √1−x2 R √1−x2 −y2
D) −1 −√1−x2 √ 2 2 dxdy
x +y
x +y
16. How about the same volume but in polar coordinates?
A)
R 1 R 2π R √1−r2
C)
R 1 R 2π √
( 1 − r2 − r)drdθ
0 0
0
0
r
rdrdθ
B)
R 1 R 2π √
( 1 − r2 − r)rdrdθ
−1 0
D) none of the above
17. How about the same volume but in cylindrical coordinates?
R 2π R 1 R √1−x2 −y2
R 2π R 1 R √1−r2 √
( 1 − r2 − r)rdzdrdθ
A) 0 0 √ 2 2 rdzdrdθ
B) 0 0 r
x +y
C)
R 2π R 1 R √1−r2
0
0
r
rdzdrdθ
D) none of the above
18. How about the same volume but in spherical coordinates?
A)
R 2π R π/4 R √2
C)
R 2π R π/4 R 2
0
0
−π/4
0
0
0
ρ2 sin φdρdφdθ
B)
ρ2 sin φdρdφdθ
R 2π R π/4 R √2
0
0
0
ρ2 sin φdρdφdθ
D) none of the above
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