Name: _:I)_..__""""-\-'--._:D~_.,c._('_;~=---· _____ PID: ______ _

TA: ___________ Sec. No: ___ Sec. Time: __ _

Math 20C. Final Exam June 13, 2013

Tum off and put away your cell phone. No electronic devices may be used during this exam. You may use a two-sided page of notes, but no books or other assistance during this exam. Read each question carefully, and answer each question completely. Show all of your work, justify each step, and state any theorems or non-trivial results used from this class; no credit will be given for unsupported answers. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification.

I # II Points I Score I

1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10

I :E II 9o

1. (10 points) Find the value of the limit or show that it does not exist.

1 2-

- f!_,(V\.. (_ \'C?J~ '/ ( \~~) "L

1:~ y... 1::)

\~ X t.. +-"'j 1. -

(~I'(JJ--J}(O,O l \-)D

J~~ -z.. CrdSBS\V\.8 \

('--:::.>(:7

l\Lk7, ~~ --t"'e 'fo\ar '('''~ .. :\~.J). 1

)

..Q,~ !l~"L -=-D

l~ .~)->(y, vI -:/ L-tjl.-

2. (10 points)

(a) Find an equation of the plane that contains the point P = ( 1, 1, -1) and the line tangent to the curve f(t) = (2t2

, 2t- 1, t2 + 1) at t = 1.

}.1-t)= ~(0 -tt r'cJ):::: (;J, l,z-; -+..t.<t..t,'-,'-""

~-; &-? -::?-d-:::- (1,0}- 3/

~ ~-=(b) W,J'j 0 -3

(b) Find the point of intersection of the plane from part (a) and the line f(t)=(t+2 , t+1 , t+5). -z: <(~(:t\ !:1L.f\ lCtl)

I I

"511-t-e ~o ""-

~~~(~·'_)_D_,_Lt_}_, _~_ ~ ?o,\1\.+ ·

3. (10 points) Find an equation that represents the s . . the angle between the tangent plane to the ra h ~tfo(f pOI~s m th~ xy-plane where the plane x + y- 2z = 10 is Z!: g P o x, y) - 10- x - y2 + 2xy and

4"

~0 rv>l>t\ +o ~j~ec -p'vl._ : Vl,-::> .( ~~~~ • - \)

:::: <- d-~ -t-2 ~ , - 7~ ~-;f..')( ) - \"--:>

(- z )1. ~ t~) + 1- 2. ::1~ Z.;e..) ;- d

u ~ (~-"l~";t·-tt?o.r'Z,t.~z. 4 \

--

I --- '3

I -to

( 0.~\e.. ~"'" opk~ ,-z.:, \ Ci "'f) \er ~V\. '1\t)r.fV\. ol <=. )

~ \ - ---- u {~ (~,<-!~}~+\

4. (10 points) An ant walks along a hot plate, represented by the disk in the xy-plane, x2 + y2 ~ 4. The temperature of the hot plate is given by T(x, y) = 40 - x2

- y (measured in degrees Celsius). Suppose the velocity of the ant is given by

iJ( t) = ( 4t3' 2t) ' 0 ~ t ~ 1

(t measured in seconds) , and the initial position of the ant is r(O) = (0, -1).

(a) What is the temperature at the ant's location at timet= 1?

Q o.;c-\;oV\. = \' £:t ') "'- <~;•.f ( 1. ~ J.-t. = < -/:_ ~\ I t. 7_ '7 + < c_ I I c__.,_ -;_

, < L' 'c_ t J -:: "h.ol ..,... l_o ,-1 ') ::>'). ["(t) -:.. ( t..'·' ) -1.:'·- \ '?] r-[~)-:: ~\,0)

\c\,o):: Yo-\

(b) At time t = 1, what direction should the ant walk to avoid the heat as fast as possible?

\;'.1 ~..,, -t"k ,_ .. _-r s ~,;.,. 1 J. w « 1 ~t "'- OJ. ('ft.. -1 ·"' i\ - \7 T I o , • , .

-?fie,.,\ -::: ( -t.x, -lilt ,,,., :::1<(;7' I>]

(c) What will the instantaneous rate of change of the temperature be if the ant walks in the direction from part (b)?

---- - \s ~Is

5. (10 points) Find the global extreme values and their locations of f(x , y) = (x2-y2)e- x

2-y

2 on theclosedand ,boundeddomainD ~ {(x,y) 1 x2+y2::; 1}.

(Make sure you check the whole boundary, not just the critical points on the boundary).

~ ~ j I '\;,~?,~1._ ( \;..'~-"Z :Z.) D , v-r :: <...., l_ :1 ')(. - ~ "" ( ~.,_-:;j .. "! ;c-1

- 2 'j - 'Z::5 t~ -z._~"" J ...e -~

-::: < ~ '1. {_ I - .. /'-+~.,_) (/"" ~ ~?. ~ ( I t- i. --z._ 'jz.) (2 'l.. '!.?.. )

-d- j c \ -\- 'f. "f..- 'j.,) -:::: 0 • ""I"f' '/. 01~-*0 J ) =' ~7..-'6 "'1. ~-);

• ::r{l ~~C) J X(\-;('-) -;..c> -:::.') )('::.01-i:.\

'(\0 $0\ IV\,

c.JII \j c ,.; -\-. p+s [ (o1e>J1

( :t 1,o'l ) Co,~ l ) J

-~0·)'~ ( ceo+,~r'II-L \ ~(c o~t.f, -9(1.7.~} d.'

<6' { t) -:: (:. ~ Cct.l-6· .,.j- - ;) 5•'-~ ~) e( -::;. - £.t e I ~-t !!';•vri ;;:; D

::::::z) cc'Q-b 6t'A -L - 0 -:;: "'> i ;::;. 0-~ ':f) iT, ~

~) e':.c\re.-- vJ.~ a::CW a..t- [-!:. \ 1 o I , (_ 0,1:: I \ od op "L

I •

6. (10 points) Use the method of Lagrange Multipliers to find the dimensions of the rectangle with maximal area that fits inside the ellipse (~) 2 + (t) 2 = 1. (a and bare fixed constants)

r---+--~ [ y.\ '') Ar~ -::- 'ne16 \\+ • \..N )'~+~

J ~c., ... +

~-f-·1 M ·, -ze_

-::: ( ?'1 ") ( '2. >'-} =:o L\ )(.. ~

A , u."'~ ~'o~~+ +o ~ -=- (_{)-z -\- l ~ )"!.__\ ~ 0

~~~, \lA-:'>--v~ ~') <. Lt~,LI~)::; '>--- < ::._, ~ ') ~ ?-~:_7. :::- ), = ~3;"L -:;:-) l~J'-=== (~ )'-

L-Jl6-~'vL-== d. x :::: Gc.

h.~:-~"'-\- ., 'd- 'j ""' fi_ 'o

7. (10 points) Let D be the region in the plane determined by the equations x 2 + y2 s; 1, x + y 2': 1. Compute the following integral:

j j (x2 + y2) -3f2dA

D

' -{_

\

.-L-

;f~~ ~'

\'~c..oSo-\"~ c'-16 )~)

t~-- '):'

»----~::. c ~ [- ,-ll ~ /){;) )o u~,VJ

_ G -0- 1.)- (_o-l -o}

--

8. (10 points) Let W be the region in the first octant above the domain 0 ::; x ::; 1, 0 ::; y ::; x between the surfaces z = 0 and z = xy. Find the average height of the points in W.

9. (10 points) A watermelon is flung at an angle of i (measured from the x-axis) towards

a hungry humpty dumpty who is sitting on top of a wall. The wall is 10 meters away and 10J3- 1 meters tall.

(a) What initial velocity is needed so that the watermelon lands in the hands of the hungry humpty dumpty? (Ignore wind resistance).

Vo-:= ~o (~b,:S,'<?) -::= So (~I 'i '/ & : c( 0 I -ct.~')

Vc~) oz < o ,-C.. .t?.+..; + S.,l~.A. ~')-:: ( ~ ,-q~ H q: s..,) l((..r) ~t+)

~ C / ~ L Q.~ ,'l US ..L "-..... L\'1\;-\,cc\ ~·-hof\.. O~f~'V\ '\ \ t "\. ') - ~ \fC.~)d--t "' a- "'{;) - ?- "Co + ~ C){, / u J

So ~0 \0 ~ i.Lt) ~ ~-/;;, -='> -/;; -;:. ~

\O 13 -I ., ~ ( ~) ~ - ~ ~s~~ -r 10~ ='--> <:;. .. 1..~ ¥ (7_o) ~ ?) So= fqf ·do -=:-") Yo--=- \OW fqi ( -1_, ~ ")

- ~<€. ~ .z.o - '2-

(b) Suppose humpty dumpty has a great fall before the watermelon reaches him. If the watermelon continues on the same path, at what time will it reach its maximal altitude?

VJ(:t):; - a; f. "l. + S \bfq:"<&t I

--::::::) C,\': .+ . --p+ · T S

s rr v;;-~ =- o sn; ~Ct.<g