Math215Winter2012
Exam2
Name:Lab section:
Instructions:
?The exam consist of6problems for a total of100points.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 is90minutes.
?No calculators are allowed.
?The?rst 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
Problem116
Problem216
Problem316
Problem412
Problem525
Problem615
Total100
Problem1(4×4=16points):Multiple choice.No partial credit.
E(x,y)F(x,y)G(x,y)
Consider the2-dimensional vector?elds E(x,y),F(x,y),and G(x,y)shown above.Suppose you know that exactly two of the above vector?elds 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?eld 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.
iii)Which vector?eld performs the most work in moving a particle along a closed loop surrounding the origin in the counterclockwise direction?(Choose the best answer)
a)E(x,y)
b)F(x,y)
c)G(x,y)
d)None of them perform any work.
iv)Which could be the gradient of the function f(x,y)=?x2+2y2+1?
a)E(x,y)
b)F(x,y)
c)G(x,y)
d)All of the above
e)None of the above
Problem2(8+4+4=16points):Consider the2-dimensional force?eld F=y3i+(2e2y+3xy2)j.
i)Is F conservative?Why or why not?If it is,?nd a potential function f(x,y)whose gradient is F.
ii)Find the work done by the force?eld F in moving an object from P(0,1)to Q(1,2)via the path y=x2+1.
iii)Find the work done by the force?eld F in moving an object from Q(1,2)to P(0,1)via a straight line.
Problem3(16points):Consider the utility function
U(x,y)=x2/3y1/3,
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 costs12cents per gram,and vanilla costs15cents per gram,and that you have$2.20(220cents)to https://www.doczj.com/doc/239064738.html,e 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?
Problem4(3+9=12points):Consider the improper integral
1 0 1
√
x
cos(y2)
√
x
dy dx.
i)What makes this an improper integral?
ii)Compute this improper integral using the proper techniques for improper integrals.
Problem5(3+4+4+4+4+5=25points): Consider the lamina consists of the half of the disk
x2+
y?
1
2
2
≤
1
4
which lies in the?rst quadrant.Suppose that the density of the lamina is given by
ρ(x,y)=
1
x2+y2
.
i)Sketch the region in R2occupied 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.
Problem6(4+5+6=15points)
Consider the integral
1
0 √1?z2
√1?y2?z2
x3+xy2+xz2
x2+y2
dx dy dz.
i)Sketch and describe in words the region in R3over which the integration takes place. ii)Rewrite this integral using spherical coordinates.
iii)Evaluate the integral in part ii).
You may?nd some of the following formulas useful(but probably not all of them)?sin2(x)+cos2(x)=1and cos(2x)=cos2(x)?sin2(x)
?sin(2x)=2sin(x)cos(x)and sin2(x)=1?cos(2x)
2
?cos2(x)=1+cos(2x)
2
?cos(π/3)=1/2and sin(π/3)=√
3/2.
?cos(π/4)=√
2/2and sin(π/4)=
√
2/2.
?cos(π/6)=√
3/2and sin(π/6)=1/2.
?cos(π/2)=0and sin(π/2)=1.
?cos(0)=1and sin(0)=0.
?d sin(x)=cos(x).
?d
dt
cos(t)=?sin(t).
?Polar coordinates
x=r cosθ,y=r sinθ.
?Spherical coordinates
x=ρcosθsinφ,y=ρsinθsinφ,z=ρcosφ,r=ρsinφ.
?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:ρ2sinφ.
?Center of mass of a2-d object is(x,y),where
x=1
m
D
xρ(x,y)dA,y=
1
m
D
yρ(x,y)dA,
m is total mass,ρis density.
Scratch paper.