MA 351, Spring 2017, Final Exam Preview:

Things to keep in mind as you take this practice test:

• The real test will not be this long. It will probably have around 16 problems.

• I tend to avoid harder problems on the test, but I don’t avoid them much on the practice.

• Some of the material should be memorized, it’s not as good if you have to look it up on yournotecard. Memorize what you can, and put the other stuff on the notecard.

• Since this test is for practice you should think about doing variations of some of the problems,especially the ones that you find difficult.

• Everything should be done algebraically without a computer or calculator.

• Don’t skip any steps and minimize how many steps you do in your head.

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1 Midterm 1 material

1. Let a = 2i− 4j + 4k, and b = 2j − k. Find the following:

(a) a + b,

(b) 2a + 3b,

(c) |a|,(d) |a− b|.

2. Let a = 〈2,−1, 4〉 and b =⟨12 , 2,

12

⟩. Find the following:

(a) a · b,

(b) a× b.

3. Find the vector and parametric equations for the line through (0, 14,−10) and parallel to theline x = −1 + 2t, y = 6− 3t, and z = 3 + 9t.

4. Determine if the following two planes are parallel, orthogonal, or neither. If neither, find theangle between them:

2z = 4y − x, 3x− 12y + 6z = 1.

5. Find the equation of the plane that is through the point (−1, 12 , 3) and normal to the vectori + 4j + k.

6. On the next two pages are shown surface graphs and traces of the surfaces defined by x2 −y2 − z = 0 and −4x2 − y2 + z2 = 1. Identify which equation goes with which set of graphs.

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Graph 1:

z = 1.5 trace

3

4

Graph 2:

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2 Midterm 2 material

1. Match the given space curves to their graphs

(a) r(t) = 〈(4 + sin(20t)) cos(t), (4 + sin(20t)) sin(t), cos(20t)〉(b) r(t) =

⟨t4, cos(2t), sin(2t)

⟩

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

x 104

−1

−0.5

0

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1

−1

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1

0

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−1

−0.5

0

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1

−1

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0

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1

0

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40

60

80

100

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

2

4

6

8

10

12

14

−3

−2

−1

0

1

2

3

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

−5

0

5

−5

0

5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2. Find parametric equations of the tangent line at the indicated point

r(t) =⟨e−t cos(t), e−t sin(t), e−t

⟩, 〈1, 0, 1〉 .

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3. Find the integral

∫r(t) dt where r(t) is defined below:

r(t) =⟨sec2(t), t(t2 + 1)3, t2 ln(t)

⟩.

4. Find the arc-length of the curve r(t) =⟨

2t3/2, cos(2t), sin(2t)⟩

, 0 ≤ t ≤ 1.

5. For the curve r(t) =⟨13 t

3, 12 t

2, t⟩, find

(a) the unit tangent vector,

(b) the unit normal vector, and

(c) the curvature.

6. Given r(t) =⟨et sin(t), cos(7t+ 11), t3

⟩, find v(t) and a(t).

7. Given a(t) =⟨1, sin(t), et

⟩, v(0) = 〈1, 1, 1〉 and r(0) = 〈2, 3, 4〉, find v(t) and r(t).

8. Match the following two functions with their graphs and contour map.

(a) z = sin(xy)

(b) z =x− y

1 + x2 + y2

7

−3−2

−10

12

3

−3

−2

−1

0

1

2

3

−4

−2

0

2

4

x

sin(x y)

y

−2

−1

0

1

−10−5

05

10−4

−3

−2

−1

0

1

2

3

4

x

exp(x) cos(y)

y

−5

0

5

−8−6−4−202468

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

(x−y)/(1+x2+y

2)

y

−5

0

5

−5

0

5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

sin(x−y)

y

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

−0.49985

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

0.49985

10 20 30 40 50 60 70 80 90 100

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0

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0

0

0

0

0

0 0

0

00

0

0

0

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0

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0

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0.5

0.5

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

−0.5

3012

−0.35341

−0.35341

−0.17671

−0.1

7671

−0.17671−0.1

7671

0

0

0

0

0.17671

0.1

7671

0.17671 0.17671

0.35341

0.35341

0.5

3012

10 20 30 40 50 60 70 80 90 100

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3 Midterm 3 material

1. Find all of the second partial derivatives of

z = tan −1

(x+ y

1− xy

)2. Find the equation of the tangent plane at (3, 1, 0) for the surface

z = ln(x− 2y)

3. Letw = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ.

Find∂w

∂r

∣∣∣r=2,θ=π/2

and∂w

∂θ

∣∣∣r=2,θ=π/2

.

4. Use the chain rule method to find the implicit derivatives∂z

∂xand

∂z

∂ywhere

yz + x ln(y) = z2

5. Find the directional derivative of g(p, q) = p4 − p2q3 at the point (2, 1) in the direction ofv = i + 3j.

6. Find the maximal directional derivative of f(x, y, z) =√x2 + y2 + z2 at the point (3, 6,−2),

and state the direction in which it occurs.

7. Find the critical points and identify them as local max/min/neither for the function

f(x, y) = y2 − 2y cos(x), −1 ≤ x ≤ 7.

8. Find the absolute max/min of the function on the indicated domain

f(x, y) = 4x− 6y − x2 − y2, D = {(x, y) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 5}.

9. Consider the function f(x, y) = y2 − x2 + x4. Find the critical points of f and classify themas max/min/saddle points.

10. Find the maximum of f(x, y) = xy restricted to the curve (x + 1)2 + y2 = 1. Give both thecoordinates of the point and the value of f .

11. Find the absolute maximum and minimum of f(x, y) = x2 + xy + y2 over the disk

D = {(x, y) |x2 + y2 ≤ 9}.

12. Find the absolute max and min of the function with the indicated constraint

f(x, y, z) = x2y2z2, x2 + y2 + z2 = 1

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13. Calculate the integral: ∫∫R

x sin(x+ y) dA, R = [0, π/6]× [0, π/3].

14. Find the volume of the solid under the paraboloid z = 3x2 +y2 and above the region boundedby y = x and x = y2 − y.

15. Evaluate

∫∫R

ex2+y2dA where R = {(x, y) | 16 ≤ x2 + y2 ≤ 25, x ≥ 0, y ≥ 0}.

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4 Material since midterm 3

1. Evaluate

∫∫∫D

πyz cos(π

2x5)dV where D = {(x, y, z) | 1 ≤ x ≤ 2, 0 ≤ y ≤ x, x ≤ z ≤ 2x}.

2. Find the Jacobian of the transformation x = eu−v, y = eu+v, z = eu+v+w.

3. Evaluate

∫∫R

(x2−xy+y2)dA, where R is the region bounded by the ellipse x2−xy+y2 = 2;

using the transformation x =√

2 u−√

2/3 v, y =√

2 u+√

2/3 v.

4. Find the gradient vector field of f(x, y, z) = sin(3x2 + 2y3) ln(5z + 7)

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5. Match each of the functions f with the plots of their gradient vector fields shown below.

(a) f(x, y) = x2 + y2

(b) f(x, y) = (x+ y)2

(c) f(x, y) = x(x+ y)

(d) f(x, y) = sin(√x2 + y2)

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6. Let F (x, y) = 〈y, x+ 2y〉. Find a function f such that F = ∇f .

7. Evaluate the integral

∫C

(x2 − 3y + z) ds where C is the path consisting of the line segment

from (0, 0, 0) and (8, 1, 0) followed by the curve r(t) = 〈3t+ 8, cos t, sin t〉 for 0 ≤ t ≤ 2π.

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