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math_172_(fall_2008)_(schumaker)_second_exam_(spring07)_(key)

Math_172_(fall_2008)_(schumaker)_second_exam_(spring07)_(key)

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Unformatted text preview: Math 172: Second Semester Calculus Spring Semester, 2007 Second Exam Name: Section: ID Number: ' Instructions: Work must be shown to receive credit. No calculators. Question 1. (10 points) Let C be the curve (shown below in bold) defined by y = In x, 13x33 Write down, but do not evaluate, an integral that is equal to the ‘arclength of C. Question 2. (10 points) A spring has a natural length of 0.5 meters. It takes 5 Newtons of force to hold the spring at 1.0 meters. How much work (in Joules) is done to stretch the spring from 1.0 meters to 1.5 meters? 05 m 1.0m [WmwillC"tM!"'..'~.m'f“‘.l'['"m ........ U ”all; ”it!M.i'.’.l‘.‘.‘.?...l.'.lrl11‘.‘.l’._‘.'l'.'r.,','.4 natural length 5 Newtons chchch ((lllllllllllI3IlllllllEllillllllllllllll!lllllllllill) oooooooooooooooooo /——\ ,6) . . Frhd we 5PM mm“ X : +d+al lerg‘lh -— WQ+IJVQl [evil-“I‘M FM the. wmk w=lbk"“" Question 3. (15 points) Solve the differential equation condition y(1) = —5. Sop a; Yd“. V0 V mu“ dy dx {y2x41 subject to the initial - Y" = ‘5'! X: +C U36 int-rid! [Wdrhm +0 {)Od C ‘ ”L = —’ +6 : Solve for y '1 Z ~_l ‘5— y 6. x Question 4. (15 points) 1 Use the integral test to show that the series 2 "2,3 diverges. n=l You must demonstrate the steps of the test in detail; it is insufficient to write “this series diverges since p = 2/3 < 1.” r ,L - Ld 3(a): 3:43 5/"):141/3 Vb; X11? £460 49 y : [W 3X 3 / £460 _._ m. mgr—I) {-443 — 00 Question 5. (10 points) Consider the series 23(001)" . n=0 (A) List the first four terms in the sequence of partial sums of this series; it is perfectly valid to use decimal notation. f, = E 52 ; 3 14 3/0,0/)= 3,03 :M (B) Determine the value of the series, or show that it'diverges. (A formula along with a shortjustification will suffice.) F/m‘v’ 7’evm = CL = 3 0.0/ It [Smhnovr (”Q7120 :- V 9m /Y/</, 740‘ geomﬂk MW (WW/7“” 7% a 3 30 ~ g. a.- I /’r 07? _ ?? Question 6. (15 points) Determine whether the following series are convergent or divergent. Justify your answers. , ﬂare/n) ___ _/ (a) Ecoszm) [97’ ﬂ” = '2’;— / K" 2" Question 7. (15 points) Find the radius of convergence and interval of convergence of the following series (736)" n 00 2 n=1 Be sure to specify whether the series converges at each endpoint. ﬂ/UT ﬂ/f//CGA£€ Question 8. (10 points) Find power series representations for the following functions and determine their radius of convergence. 1 5—x (a) f (x) = War ﬂ/J/g ”Ca/0Q (b) g<x>=1n(1—2;—) ...
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