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prelim1_fa07_solutions

prelim1_fa07_solutions - (“x A 7{if ”a i‘ “ Your...

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Unformatted text preview: (“x ' A 7 {if ”a. i‘ .. “ . \ Your name: ”7 : KZS 1‘: Your TA’s name: Your section number and / or day and time: Math 191, Prelim 1 Thursday, Sept 27th, 2007. 7:30 ~ 9:00 PM This exam should have 8 pages, with 5 problems adding up to 100 points. The last two pages are blank and can be used as scrap paper for computations and checking answers. No calculators or books allowed - You may have one 8.5X11 formula sheet. To improve your chances of getting full credit (or maximum partial Problem 1: /20 credit) and to ease the work of the graders, please: , _, _ Problem 2: . / 20 a write clearly and legibly; Problem 3: /20 ' your answers; Problem 4: / 20 o simplify your answers as much as possible; Problem 5; /20 o explain your answers as completely as time and space allow. TOTAL: /100 Academic Integrity is expected of all students of Cornell Uni— versity at all times, whether in the presence or absence of members of the faculty. Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination. W Signature of the Student [20] 1. Evaluate the following: w/3 2,; (a)/ 3325111503 da: :6 {1% WWW ~7r/3 ? 22 ”Lg W 4%? , 2 V, E 3,” L ”31>3i g) ‘Evvzléxaicrcw (3 gmwim : ~L @wvg, ‘7) 3i? 3 9 E ,4? Z? : <3 w: - Wm , o 0 \, ”a; b 1 1 1d () 3;; 45% mafia hiwzdiu X, 3(3/ 5 r g 3 2 »—\ yak/{Mi 2% flJifgrQ I Vlfl r‘r( 2*;(33125 \M i Q 7 ”Vt; 4C 3 > ' «K d POST W {E ' __ _2 ._7 _~ (“Mg ._. a L (c)dm \/1 tdt , QL g‘éxfljq‘é‘w 4‘”; M ,2; CU (M; M 5} a ._ M 1’ ’ 1””Le3x. géAC/M’Ba M \jirhwf’ 1M3”) ; ~gwx21xmyzzl W m ”£2 (1 932 (d) If f(t) is differentiable andmsinzn: dm/O f(t)dt, f(m) “‘P 'L ‘L *2 (y gem :, c} 84635;; 2 fig) @321 .; 16%) 1x {M 3 M ”A {$45 “« WM h 1, 3" RI”) .1: giw‘x‘) :: W6 \ w} ajww 1 “Ma c; “M a\‘§ WA: 6\ 3‘3"" _ $271 sint for 0 < t W) (a) In terms of fit) and its derivative(s) find an expression for the arc length of the curve . 2. Consider a parametric function defined by 55(t) : 7"(t) cost and y(t) 20] I , | e k 1 S k 0 O 1 MM . h m we t n m e C t t m .1 S n a I O ,1 t y C S S W ,,,,, m 10 \AL) AOL! .1“ e; e m e o w n gig M a n C C .11 n p cm _ _ F ewe? M a d 1 1 Wu Hf; S S (.\ l.\ 2 fizz. e .1 .11 2 .m \./ \./ t a a.” i t. U Max :59}; .efl m m m y <_ 5/ {23/ a e 0 ,e :we , A a :0. {gay ,,. effggiaégwue / ) e, , w Figure 1: Cardioid _~4_* 3. For the region enclosed by y : 212: ~ .792 and the X~axis evaluate the following: (a) Sketch the region. ‘_ -— 5 —- [20] 4. Find the area between the curves f(.’l)) : 4332 + 43: and g(m) : 293. (a) Sketch the curves and shade the desired area. 3 n(n+1) n k2“ n(n+1)(2n+1) ikzg : <n(n+1)>2 PF H H Pr H H FM _6_ 5, A dam is built on a flood—prone river. At the bottom of the dam is an emergency release gate shaped like a parabola with a flat top (enclosed by the curves 3/ 2 4502 and y : 1). If the force on the gate exceeds llOkN, the gate will open and spill water until the force has dropped to lOOkN. The goal of this problem is to find the water level When the gate first opens and when it closes. For '11) use the following: the density of water is lOOOkg/m3 and the acceleration due to gravity is glem/s2 (a) Draw the gate and dam. Label the height of the water above the bottom of the gate. (b) Derive an expression to describe the force on a thin strip of the gate AA. AF : (Wetw‘l’JQ’ASHj B'CAchQ‘DSmfloce) ~ AA BF : 93 (““9153 Mold 332w AF: 83 (1‘ j)/.\A AA: ngbbjflxéjfelgdj (c ) Use t e answer in b to find the total force on the gate due to water pressure. \ 1 F : jolt C 0583 (k‘sfi) 334:, I 0 053333505 ‘ ijz’sl (e) Find the water level When the gate first opens. h: K NO \600 +§Z* 2 l6 \aoo (f) Find the water level when the gate closes. ...
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