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Math 215 Winter 2012
Exam 2
Name: Lab section:
Instructions:
- The exam consist of 6 problems for a total of 100 points. Please look through the exam booklet and make sure it has twelve pages. The next to last page is a list of formulas which may be useful. The last page is blank and is to be used as scratch paper. You may tear both of those pages apart from the rest of the exam.
- The exam duration is 90 minutes.
- No calculators are allowed.
- The first problem is multiple choice. For multiple choice problems there is no partial credit. For all other problems, show all your work to receive full credit.
- Make sure your answers are clearly marked (circled or boxed).
- Do not cheat. At a minimum, you will be expelled from and fail this exam if you cheat. Further disciplinary measures are possible. So don’t do it.
Points Possible
Problem 1 16
Problem 2 16
Problem 3 16
Problem 4 12
Problem 5 25
Problem 6 15
Total 100
Problem 1 (4 × 4 = 16 points): Multiple choice. No partial credit.
E (x,y) F (x,y) G (x,y)
Consider the 2-dimensional vector fields E(x, y), F(x, y), and G(x, y) shown above. Suppose you know that exactly two of the above vector fields are conservative.
i) Which makes the most sense?
a) E and F are conservative. b) E and G are conservative. c) F and G are conservative. d) None of the above make a bit of sense.
ii) Which vector field performs the most work in moving a particle from the origin to the point (1, 1) via a straight line? (Choose the best answer)
a) E(x,y) b) F(x,y) c) G(x,y) d) None of them perform any work.
Problem 2 (8 + 4 + 4=16 points): Consider the 2-dimensional force field F = y^3 i + (2e^2 y^ + 3xy^2 )j.
i) Is F conservative? Why or why not? If it is, find a potential function f (x, y) whose gradient is F.
ii) Find the work done by the force field F in moving an object from P (0, 1) to Q(1, 2) via the path y = x^2 + 1.
iii) Find the work done by the force field F in moving an object from Q(1, 2) to P (0, 1) via a straight line.
Problem 3 (16 points): Consider the utility function
U (x, y) = x^2 /^3 y^1 /^3 ,
which represents the quality of a bowl of ice cream which contains x grams of chocolate ice cream and y grams of vanilla. Suppose that chocolate ice cream costs 12 cents per gram, and vanilla costs 15 cents per gram, and that you have $2.20 (220 cents) to spend. Use Lagrange multipliers to determine the amount of chocolate and the amount of vanilla you should buy to have the best bowl of ice cream possible. What is the maximum value of U?
Problem 5 (3 + 4 + 4 + 4 + 4 + 5 = 25 points): Consider the lamina consists of the half of the disk
x^2 +
y −
which lies in the first quadrant. Suppose that the density of the lamina is given by
ρ(x, y) =
x^2 + y^2
i) Sketch the region in R^2 occupied by the lamina.
ii) Describe this region by some inequalities using Cartesian coordinates.
iii) Describe the same region by some inequalities using polar coordinates.
iv) Write (but do not evaluate) an iterated integral in Cartesian coordinates which gives the mass of the lamina.
v) Write (but do not evaluate) an iterated integral in polar coordinates which gives the mass of the lamina.
vi) Choose one of the above integrals to evaluate.
iii) Evaluate the integral in part ii).
You may find some of the following formulas useful (but probably not all of them)
- sin^2 (x) + cos^2 (x) = 1 and cos(2x) = cos^2 (x) − sin^2 (x)
- sin(2x) = 2 sin(x) cos(x) and sin^2 (x) =
1 − cos(2x) 2
1 + cos(2x) 2
- cos(π/3) = 1/2 and sin(π/3) =
2 /2 and sin(π/4) =
3 /2 and sin(π/6) = 1/2.
- cos(π/2) = 0 and sin(π/2) = 1.
- cos(0) = 1 and sin(0) = 0.
- (^) dxd sin(x) = cos(x).
- (^) dtd cos(t) = − sin(t).
- Polar coordinates x = r cos θ , y = r sin θ.
- Spherical coordinates
x = ρ cos θ sin φ , y = ρ sin θ sin φ , z = ρ cos φ , r = ρ sin φ.
- The Jacobian (“extra factor in the integrand”) of the transformation from Cartesian to polar coordinates: r.
- The Jacobian (“extra factor in the integrand”) of the transformation from Cartesian to spherical coor- dinates: ρ^2 sin φ.
- Center of mass of a 2-d object is (x, y), where
x =
m
D
xρ(x, y)dA , y =
m
D
yρ(x, y)dA ,
m is total mass, ρ is density.
