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Unformatted text preview: SAMPLE Math 16C Sec 2 (Malkin) Name:
Final Exam Student ID:
June 12th 2008 Signature: DO NOT TURN OVER THIS PAGE
UNTIL INSTRUCTED TO DO SO! Write your name, student ID, and signature NOW! NO NOTES, CALCULATORS, OR BOOKS ARE ALLOWED.
NO ASSISTANCE FROM CLASSMATES IS ALLOWED. Read directions to each problem carefully. Show all work for full
credit. In most cases, a correct answer with no supporting work
will NOT receive full credit. Be organized and neat, and use
notation appropriately. You will be graded on the proper use of
derivative and integral notation. Please write legibly!  Student’s Score Maximum possible Score I 1— we  Page 1 of 12 SAMPLE _________________________._.__—————————————— 1. (a) (2 points) Verify that y = 2e~t + 10 is a solution of the differential equation 1/ + y = 10.
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(b) (4 points) Use any method to ﬁnd the general solution of the differential equation
x2?! _ 2y2 = myz g, 311% :8%dm=j%1¥3\idbk (34%). U6?
l my — y = :1:(ln(x))3..\’
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on EYES SAMPLE 2. (a) (2 points) Consider the function f(a:) = «$2 + 3/2 — 9. Find the domain of f
and sketch the domain. x2+y2+z2—2m+4y—6z+10=0.
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ﬂ M—e racial,“ is 1 wai PL WIS (1,43). (c) (6 points) Consider the suface z = 9 —— x2 — y2. Determine and sketch the
crytrace, wz—trace, and the yz—trace for this surface. % Page 3 of 12 S AMPLE
3. (a) 2 points) Find all the second order partial derivatives of f (as, y) = 3x2—xy+2y3. < 41 (obj 4”: lo 41:) z \ {b aLylﬂ)’ J33 ‘2 mg) by c'l.
( 4 points) Find all of the critical points of f (x, y) = m3 — 3:1: + 23/2 —— 8y. (b)
(Hint: there are two critical points). *“l ﬁlth Bibi} I: O (’3 (blVLx’BOQ :3 3L:
' c \‘l' ' 8 :1 (l 7/) :(L .
l3 :3 ‘ 3 go, Med are ﬂ 71M initial/l fawn (ll) WA (All). (c) (4 points) The function h(x, y) = 23:23; — 21:2 — y2 + 3 has three critical points:
(0,0), (1,1), and (1,1). Classify each of the critical points of h. Page 4 of 12 SAMPLE 4. A solution containing 1 pound of salt per gallon ﬂows into tank at the rate of 5
gallons per minute and the well—stirred mixture ﬂows out of the tank at the rate of
4 gallons per minute. The tank initially holds 50 gallons of water containing no salt.
Let S be the number of pounds of salt in the tank at time t minutes. (a) (2 points) Set up a differential equation that describes the rate at which the
amount of salt S in the tank changes at time t. ~ . 5 \ ,
as its... milﬂjﬁf’fxéilﬁ. bf (in ~0Ud': 8" 501$ / 5 “M g
m (b) (6 points) Find the particular solution to the differential equation (i.e. solve the
differential equation and determine all the constants). ﬁg 5:; ‘ ELS £23.35“
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(c) (1 point) How much salt is in the tank after 30 minutes. f: (BOSSHSMQCDWSl ngtljh 5‘9). (d) (1 point) How much salt will be in the tank as 25 becomes inﬁnitely large? A} tﬁw (S490 Page 5 of 12 SAMPLE 5. (10 points) Use the method of Lagrange multipliers to minimize the function f(w,y,z) =902+312+z2 subject to the constraints m—y+2z=3and3x+y—z=0. F}; t "A 1/0 (:5 %=?>DC +3 .
9, (ﬁe) “twigj +1; :0 =3 "Lué +‘2(3M:'»;°'
(2) Lime + $330 A"? neat  L ‘1; 9)) t o :2) ac— iii?”ng ’% CO L V’A i “(Xv
:L‘B3Q .9 jute. x) 3, :Q (13 E). v l 0M0! E : gown (":95 i . ‘ Page60f 12
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0 1'2 (a) (1 point) Sketch the region R over which we are integrating. 6. Consider the following integral: Page 7 of 12 SAMPLE __’__________________.———————————— 7. (a) Determine the nth term (starting with n = 1) of each of the following sequences: i. (2 points) 4, 9, 16, 25, 36, 49, 64, '1.
 q »
a“ , (m   1 3 5 7 9 11 13
ll. pomts) "I,5,—g,'ﬁ,m,ﬁ,——504m.m n ‘an.
(An?<.'\3 T (b) Determine whether each series converges or diverges. You must justify your
answer and state which test you are using. i. (2 oints) 22:13:21. “ _ _ (
p Z Cthzinb genes We serif; Di “5% «b {famine/the mm 3:63: mm \A: “2 > ,\ . ii. (2 points) 22:0 544,72. / . .. \ x a
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\ lvws Page 8 of 12 SAMPLE 8. (5 points) Find the radius of convergence and interval of convergence of the following power series:
0° 2 Z + 5)". n=1 You do NOT need to say what happens at the end points of the interval. JUN \Qw ll’r 3M» um“ \[ML/Q‘W bUrSl). Wee
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2—va W “:1? f “40° ' (\MA)‘.
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7% 07f mire/91M Ce 7.! [—cﬁ, 06) Page 9 of 12 SAMPLE 9. (a) (5 points) Find the Maclaurin series for f = cos(:c) by using the fact that 2 22:0 éﬁgﬁatzn‘” and ad; (sin(m)) = cos(x). d (‘R 0‘ Ed“ MAM
caste) teatime :3; L émx 5 0° d V)
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—,— :12; ll“ vx—uo Cl“>\' (b) (5 points) Approximate the deﬁnite integral fol 1n(:r + 1)da: using the 3rd degree Taylor polnomial for 1n(:c + 1) centered at O. Hint: ln(x + 1) = 2:1 #x".
X? A \a \ \As‘ '17; —
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go mam ‘eijc—lgmfam )0 30“ 13 .72.,\ h” 9 5° ., Page 10 of 12 SAMPLE 10. (10 points) Complete 2 iterations of Newton’s method to approximate a zero of the
function f = x3 + 3:3 —— 1. Use :51 = 1 as an initial guess. Express your answer as a fraction. 3L: 42L: \ at» xszkuv 51(w\>33(n'3x&*;
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2: \31‘1‘Jr3 z W~\f’54’5
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jg) MM Dams art“ (xlt'r szAiWBUzQ—g Page 11 of 12 SAMPLE //// 11. Consider an 8 by 8 checkerboard. Place 1 penny on square 1; place 2 pennies on
square 2; place 4 pennies on square 3; place 8 pennies on square 4; place 16 pennies
on square 5; and etc. How many pennies are needed to cover all the squares on the checkerboard? 1% ZWVI (JWh/Mj 7mm myz)‘ MW [ﬁlm/cc! a}: .lr (73 n / K 2%” z 2 9~ . a} aim/JIM mm/ 01 “‘0 aria \y‘Ll aal amo/P;la b} ‘_ uM‘ \ \ﬁlblf)
ﬁvw/ Z A ELF/La( 26%.\. Z z l" ( “ l’L : Lwl 3:203 lulu/A 5% 1‘3“
S A _ l“ ; ‘ ,\ 6;) q; / 3:. Q; —,\ K V) m V) 36:3 hf) Q J; m V\ V\ ~k LA“ Wig: z 3L Z €“1\wte\w:_ij
i ‘/ \ S3» h ‘rmlmmcmm :D‘agsrnakaqac +O _UL+CAC_ “(Salaam 4%”) ‘j’gbfdm 1 Xlzmzbd'bc'dm, : LXL picnﬂr mammw Page 12 of 12 ...
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This note was uploaded on 06/15/2011 for the course CAL 3 taught by Professor Smith during the Spring '11 term at Arkansas State.
 Spring '11
 Smith

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