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exam1sol 17C - Math 17C(Fall 2008 Kouba Exam 1 Please PRINT...

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Unformatted text preview: Math 17C (Fall 2008) Kouba Exam 1 Please PRINT your name here : ________________________________________________________ Your Exam ID Number ____________ 1. PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO. 2. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE T 0 COPY ANSWERS FROM ANOTHER STUDENT’S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER STUDENT TAKE YOUR EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION. 3. No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM. 4. 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. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important. 5. Make sure that you have 7 pages, including the cover page. 6. You will be graded on proper use of limit, derivative, and integral notation. 7. You have until 11:50 a.m. sharp to finish the exam. 1.) (4 pts. each) Use known formulas to determine the Laplace transform for each of the following functions y : f (t) b. t : 2t+ ‘4t os2t )f() 6 e 0 34—4 _ _ __L. ———~_—— 5( E63 {“101 01%} -. fife—“chalk ((324%) = 5~2 4' (SH/fax" c.) f(t) = 58int ~— sin 5t iib'Mf-WOY’} = EstiMch—atfmsf} : 5.035 _ 514:1 2.) (5 pts. each) Find the inverse Laplace transform L‘1 {F (3)} for each of the following Laplace transforms F(s). s 1 S l a.) F(3): 32+25 + 8+25 '2. 524—5; ‘f’ $4.51; ___9, - - 1‘: at [EFCSB’ : 004.6% + e '25 (5+3 :A(S+()+g<5.;{] 3+3 3+3 _ A + _8_ Msmz 5-. 3A»A:% b) F(S):s2—s—2 = Cs~2j®+0 ' 5-9. 5+1 M591: arm-.312?) 573’ % l 2% ._.g .. _____ ‘— ____. _ 5- .9: ‘ 5-; sue; {[email protected]}~—3—e *3?— c)F(s)— 3+2 __ 54-1 _ 54—1 82+28+10 ‘ 52+as+l+fi ‘ Cs+l)l+3l_ _ (3-H 4- l ~ 5+! 4. _L, 3 fl (5+!)K4'37‘ ‘ @+I)2+39~ 3 (5102431 ~i— I -(: up; A ~ 4 - 16 j 1 txflifilo-éfiois Ml : M “(A'2‘2 LLL i—S-E/A] SA” 6' 5 o 4.) (12 pts.) Use a Laplace transform then an inverse Laplace transform to solve the following differential equation: y”— — 12732 + 1% and 31(0) 2 1, y’(0) = 2 3’49”}: IaiEf‘}+l;£Eé3> '” $1437} S‘éj 1%]L311 éf—IQF 3W} 3—%— 4. 54—2 —6 (.1 J, 3. gliy}: 55' 5‘4 + 3 4—59“ ~ 1L. 3‘ i. .J. _ 55+ +2 5“ Jr 5 + 2 5" ~52 5.) (14 pts.) Use a Laplace transform then an inverse Laplace transform to solve the following systems of differential equations. (1:; =932 , x1(0):0 XEX‘IEZ (jix'ZE ———7 L22 2—331 , “(0)21 Xixalg : ' 6(5st d} [5169}: Lo - “5ng E 13795? = 5°fixxifi (W67 saws yo) - was swear} ~ was; ~95%fi&}+ikktl @Mdifigzl—e i993: 5T?“ xlei/vdc Min ~I W- 6.) (8 pts.) The position (51:1,:c2) of a particle at time t is given parametrically by the following. Eliminate t and write the path as an equation in only $1 and x2 . Then sketch the graph of the path in the 9:1 xg-plane, indicating the direction of motion of the particle. 1131—152 x2=t6+1 for ——1<t<2 x=tfi [email protected])+/= Xf+/ W X01: ><,3+( ~fizz—l ‘- XIZI, xok: ‘1 CSTART) {:o‘ X,=Oj X0121 Bifié‘gfion t: 82: x,:<// X&=C25/ GSA/p 7.) (15 pts.) Solve the following system of differential equations with initial conditions. Write your answer in matrix (vector) form. X’:<__21 31)X and m1(0):0,x2(0)=1 i¢*¢A-i14:ro.,§ Ari [%7* a,) :o-—e (Aa3)0\»1):0 “[email protected]/ :3 j Poq/l:: 3m (:4 AIthO/éflX—‘a' H 01~ Mg}, ”—0 M X01:_é (Way—axlzxatt 4o. XZCQK:[:ibttii40¢7V“””j?“g v,: ‘1 1 3' ,V -, 5m CA'AIQX 0 WXH EVE—Li}N ['0 $211,, XI‘FXAvO 40M KOLZ+ 8.) (9 pts.) The position ($1, :02) of a particle at time t is given parametrically by {:51 = t2 + 2t x222t—t2 forOStfiQ' and its graph is given below. For t 2 1 determine a.) and plot the point ($1,302). I QI/Xfitczjl) LOP _ dxsléol: ‘R-XfJ/fwle b.) the [slope of the graph. 3 g ‘ X, I a? {- *1— 9‘ X 0 ' ' Wl Z ‘9‘ : ~ ~ 6) X I v . . c.) and sketch :direction vector; VW W [32% . A’VC 9.) (5 pts.) The point (0,0) is an equilibrium for the following system of differential equations. Determine if (0, 0) is an unstable or stable equilibrium. Then categorize (0, 0) as a saddle, sink, or source. X,:(4 —5)X MCA~AIj :o ——> 0—3 ' qu: :3’41 l :- C‘HJC‘W) w “’ Ac‘fi A34 ”6’ (”/0/ 14 W The following EXTRA CREDIT PROBLEM is worth 10 points. This problem is OP- T IONAL )Derive a formula for £{y” ’} consisting of the terms £{y}, y(0 ), y’(0), and y” (0) . (187” } GHQ/”)7? : Soiiy”}vj"€0) ; 5[saiiygv5y(02~J‘Co)]—J”Co) :. s3a<{y§- 34w) v Sy’co) ~J”<o) ...
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