PRACTICE EXAM 2
General Exam Information
Just like with the first exam, you may make a reference sheet for yourself of one side of an
8.5 x 11 paper. You may put anything you’d like on the sheet. I would recommend going
through this practice exam, and for anything you end up having to look up in the book or
your notes as you are solving the problems, that material should go on your review sheet.
This exam covers sections 12.6 through 13.9 in the book. I will list here some formulas
and portions of those sections that we did NOT cover, and that you will NOT be responsible
for. You can assume anything NOT on this list is fair to ask on the test.
- 12.7 ”Differentials” - we did not discuss this explicitly, although the concept was actually
on the previous practice exam: you can approximate a function near a point by its tangent
plane at that point.
- 12.8 ”Two or More Constraints” - we only discussed problems with 1 constraint function.
- 13.3 ”x-simple and y-simple” - I did not define these in class. I implicitly worked with
them in my examples when I described how to turn a region into limits of integration.
- 13.4 ”Probability Integral” - we did not discuss any of the probability density function
applications in class. The integral given in this section is one I showed you how to do in
class using a clever trick, but you do not need to remember how to do this for the test.
- 13.5 - You do not need to know anything from this section except the formulas and how
to apply them.
- 13.7 - ”Multiple Random Variables” - again ignore this, we did not discuss probability
density functions.
Practice Problems
1. Given that w = xey/z , x = t2 , y = 1 − t, and z = 1 + 2t, find dw/dt. Clearly indicate
the use of the chain rule.
2. Given that u = x2 + yz, x = pr cos θ, y = pr sin θ, z = p + r, find
this at p = 2, r = 3, θ = 0.
3. Find
dy
dx
du
.
dr
Then evaluate
in the equation cos(x − y) = xey using implicit differentiation.
4. Find an equation for the tangent plane to the surface x2 + y 2 − z 2 = 13 at the point
(5, 2 ,4).
5. Write the second order Taylor Polynomial for the function f (x, y) = sin(x2 + 2y) at
the point (0, 0, 0). Use it to evaluate the limit
lim(x,y)→(0,0)
1
sin(x2 + 2y)
x2 + y 2
6. Find all maxima, minima, and saddle points of f (x, y) = x4 + y 4 − 4xy + 1. Classify
these points using the Second Partials Test.
7. Use lagrange multipliers to find the points on the surface z 2 = xy + 1 that are closest
to the origin.
8. Find and classify the extreme values of f (x, y) = 2x2 + 3y 2 − 4x − 5 subject to the
constraint x2 + y 2 ≤ 16.
9. Approximate the volume under the function f (x, y) = xy and above the rectangle
R = [0, 6] × [0, 4] by splitting R into six equal squares and sampling the function at
the center of each square. Then find the volume exactly using an iterated integral.
10. Evaluate
1
Z
x2
Z
(x + 2y)dydx
0
0
11. Change the order of integration in the previous problem to dx dy, then evaluate the
new integral.
12. Sketch the region of integration implied by the limits of the integral
Z √25−y2
3
Z
f (x, y)dxdy
4
0
13. Evaluate the integral by first sketching the region of integration and then changing the
order of integration.
3
Z
Z
9
y cos(x2 )dxdy
y2
0
14. Use a change of variables to polar coordinates to find the volume of the region inside
both the cylinder x2 + y 2 = 4 and the ellipsoid 4x2 + 4y 2 + z 2 = 64
15. Find the center of mass of the triangular region with vertices (0, 0), (1, 0), (0, 2) if it
has a density function δ(x, y) = 1 + 3x + y.
16. Set up (but do not evaluate!) a double integral whose value is the surface area of the
surface z = 1 + 2x + 3y + 4y 2 over the rectangular region [1, 4] × [0, 1].
17. Evaluate the triple integral,
Z
2
Z
x
Z
x−y
2xzdzdydz
0
1
y
2
18. Sketch the region of integration implied by the limits of the integral
1
Z
Z
1−x
2−2z
Z
dydzdx
0
0
0
19. Sketch the region of integration implied by the limits of the integral
Z
1
Z
0
1
√
Z
1−y
dzdydx
x
0
Then use your sketch to rewrite the integral with the order dxdzdy.
RRR p
20. Evaluate
x2 + y 2 dV , where S is the region inside the cylinder x2 + y 2 = 16
S
and between the planes z = −5 and z = 4.
21. Find the volume of the solid that lies above the cone φ = π/3 and below the sphere
ρ = 4 cos φ. (Draw a picture first!)
22. For the transformation given by x = u2 − v 2 , y = 2uv, find the image of the uv-plane
square [0, 1] × [0, 1] in the xy-plane.
23. Use a change of variables u = x + y, v = x − y to evaluate the integral
Z Z
e(x+y)/(x−y) dA
R
Where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, −2), (0, −1).
24. Derive the polar, cylindrical, and spherical change of variable formulas using the Jacobian.
3