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Math 215 \Vinter 2012
Exam 2
Name: Kevv^ ___^ _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 mmImum, you will be expelled from and fail this exam if you cheat. Further disciplinary measures are possible. So don't do it.
C I.· .pOin.. t.. s ~:J
L~l iJ
! Problem 21 --- 16
L. i Problem 31· 16.
Problem 12
Problem 5 25
Problem 6 15
--
Problem 1 (4 x ;1 16 points): Multiple choice. No partial credit.
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---~~~~~~~~~ """--~~~~~~~~ ,"~~~~ ~-~~~~~~~~ - .......... "", -- .......... "", (^) ""~~~~~ ~~~~~~~~~ **---........ "", - ................... "" ""'---**^ --~~~~~~ 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?
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) C1?fF(x,yt) c) G(x,y) d) None of them perform any work.
Problem 2 (8 +;1 + 4=16 points): Consider the 2-dimensional force field F y^3 i + (2e 2y^ + 3xy2)j.
i) Is F conservative? vVhy or \vhy not? If it is, find a potential function f(x, y) whose gradient is F.
F ~ <. ~ 11) =) Po=- $~ t ; Q)(~ 5~ ~ =)Ji"'J, c",ru.""A...]
+x::- 'fJ~ =) +'" X '(/ ~ c. ~)
~ Cz [x} +r:: 2/"1 + 5)( ~ ~ -=) + '" e Z{j -I- (^) )(~
'
---- - J 2, If{xl~) ::::^ e^ ~^ )(
3
--J
~ 2 2, '2 ~ Ckcb: t^ '"^ ~^ ~ t ~ e r 3x'O
ii) Find the work done by the force field F in moving an object from P(O, 1) to Q(1,2) via the path y x 2 + 1.
FToLI i^ F,^ d;^ ~^ +(1^1 2 )^ --f(~^ I) 13( L
::: e
f- Iin 2) - flo t !)
{'(~t) - 1(rI 2/ ==
+ a
- -e - 0
Problem 3 (16 points): Consider the utility function
U(x, y) = X^2 / 3 yl/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
maximuCUlvaJue of ~? ('f;)~ 22;/ I-/iu.-< 'P("/~) ~ /2)( + 15'g
OV\ s 1v-&-JVI r 0 (rg 2./:
/ .l- ( l:.)~ {[-e) 1> 9 £1- =: «0 It) ';:> V (^) U := t..... 3 >< / .I V / 0 /
/2)(1- It; 'ff ::- '
23 t{;) ~ ~ !2A i (-~)
I ;; IsA / A,,-----
) _ .-L 1}4: r - -L (-K J'~ IJ 2 ~t.\e~J :!~ (^) - is X' - 4-) ~
let; 6 'x
f Ie (x) ~^ ~^ fix^ ~^ It; ::::::=-) ~^ :K^ Cj^ ~)
(2 x + (<) ( if ) ~ 2? u 5r... bf. ,'v-. j." 0-Y/..· .. ./
/By ~ 2eD
2'2.0 _
:= -~ (i>
~ [If/((;) -=:fj)(T}? (^) ~
[ilt;Xf
u~)~ 01 U; (^) Ief- (^) t'{ fJij
Y
~
iv) Write (but do not evaluate) an iterated integral in Cartesian coordinates which gives the mass of the
lamina. I ~t I I d
- ~x !r lIVt -= V )(?..(- t I:;> f 0 It)
v) Write (but clo !lot evaluate) an iterated integral in polar coordinates which gives the mass of the lamina.
'if Sih £
Vh
lL 50 d~ d{j
vi) Choose one of t.he above integrals to evaluate.
Sa~ f'ke Jv d.e := f~ !)J^ 'h^ f}^ JI:J
v
~ C 05 e J
~
"2- (^) - _(O-I) ~J]J
0==
01
=-(i]i')f ~ 1f{(O)'1'!>-(ltr'5} rCft~
q;f ef e 57) {- iJ:J =
