PRACTICE FINAL
General Exam Information
The final exam covers everything we did in class in Chapters 11-14 in the book. For material
from chapters 11-13 you can go back and look at the review materials and exams on those
chapters. I have included a couple more problems here to look at, but be aware that anything
on any of the previous practice examsn is fair for the final as well!
You may make a reference sheet for yourself of both sides of an 8.5 x 11 paper. You may
put anything you’d like on the sheet.
The following is a list of things from Chapter 14 which you will not need to know.
Everything else covered in those sections can be asked about on the final.
- None of the proofs of anything from Ch. 14 will be required to know. It is basically
knowing formulas and how to apply them (and how to calculate the resulting double/triple
integrals)
- 14.3 - Conservation of Energy. This section explained the meaning behind the name
conservative vector field. Students in physics should read this section for your own benefit,
but it will not be tested over.
- 14.5 - Parameterized Surfaces. We covered this in class and there were a few homework
problems related to this, but the final will not include surface integrals over parameterized
surfaces.
Practice Problems
1. For the curve r(x, y, z) =< 2t, 1−t, t2 >, find parametric equations for the tangent line
to the curve at the time t = 3. Then find the equation of the plane that is perpendicular
to the curve at t = 3.
2. Draw cross sections of the surface x2 + y 2 /4 − z 2 /9 = 1 corresponding to z = 0 and
y = 1. Use these to graph the surface and classify it.
3. Prove that the limit of
x2 −y 2
x2 +y 2
does not exist at the origin.
4. What is the fastest direction of increase of the function f (x, y) = x2 y 3 −4y at the point
(2, −1)? What is the rate of increase from (2, −1) in the direction of 2i + 5j?
5. Find an equation for the tangent plane to the surface x2 /4 + y 2 + z 2 /9 = 3 at the point
(−2, 1, −3)
6. Find and classify the local and global extremes of 1+2xy−x2 −y 2 on the disk x2 +y 2 ≤ 1
7. Set up and evaluate a double integral to find the volume under the surface z = 2x + y
and above the triangle in the xy-plane with vertices (1, 2), (3, 0), (3, 6)
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8. Find the volume of the solid that lies under the paraboloid z = x2 + y 2 , above the
xy-plane, and inside the cylinder x2 + y 2 = 2x.
9. Evaluate the triple integral,
Z
0
3
Z
0
1
√
Z
1−z 2
zey dxdzdy
0
10. Evaluate
Z Z Z
(x2 + y 2 )dV
E
Where E is the portion of the solid cone x2 + y 2 − z 2 = 0 between the planes z = 0
and z = 2.
11. Plot the vector field F(x, y) = yi + (1/x)j by plotting the 25 vectors corresponding to
the integer coordinates inside the square [−2, 2] × [−2, 2]
12. Find div F and curl F for the vector field F(x, y) = < ex sin y, ex cos y, z >
R
13. Evaluate C xy 3 ds, where C is described parametrically by x = 4 cos t, y = 4 sin t, z =
3t.
√
14. Calculate the work done by the force field F(x, y) = x2 y 3 i − y xj in moving a particle
along the path r(t) = t2 i − t3 j, 0 ≤ t ≤ 1
15. Prove that the vector field F(x, y) = (2x cos y − y cos x)i + (−x2 sin
R y − sin x)j is
conservative. Then find its potential function and use it to calculate C F · dr, where
C is the line segment from (0, 0) to (π, 2π).
16. Prove that the vector field F(x, y) = (6x + 5y)i
R + (5x + 4y)j is conservative. Then find
its potential function and use it to calculate C F · dr, where C is the curve that traces
the ellipse x2 + y 2 /4 = 1, starting and ending at (1, 0) and going counterclockwise.
R
17. Use Green’s Theorem to calculate C ey dx + 2xey dy, where C is the square with sides
x = 0, x = 1, y = 0, y = 1 traced counterclockwise.
H
H
18. For the force field F = xi + yj, calculate C F · nds and C F · Tds, where C is the unit
circle. What do these calculations represent physically?
19. For the part of the surface z = x2 +y 2 which lies inside the cylinder x2 +y 2 = 4, find the
surface area. Then find the mass of this surface if it has density function δ(x, y, z) = z
20. Find the flux of the force field F(x, y, z) = xi − zj + yk across the surface S which is
the part of the sphere of radius 2 centered at the origin which lies in the first octant.
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21. Calculate the flux of the vector field F(x, y, z) = xyi + yzj + zxk across the surface of
the cylinder x2 +y 2 ≤ 1, 0 ≤ z ≤ 1 (including the top and bottom) using the divergence
theorem.
RR
22. Use Stokes’ Theorem to evaluate
curlFdS, where f = xyzi + xyj + x2 yzk and S
S
is the top and four sides (but not the bottom) of the cube with vertices (±1, ±1, ±1),
oriented outward.
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