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152RevMidI-1 - MATH 152 FALL 2009 CALCULUS II FOR THE...

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Unformatted text preview: MATH 152, FALL 2009 CALCULUS II FOR THE MATHEMATICAL AND PHYSICAL SCIENCES REVIEW QUESTIONS FOR MIDTERM I Please read the following instructions carefully BEFORE you start. 0 Do NOT turn over this page until you are told to do so. In the meantime fill in your name and section number below. 0 NO textbooks, notes or other study materials are permitted in the exam room. NO calcu~ lators or other electronic items may be used. Cell phones must be TURNED OFF and not simply silenced. 0 You have 80 MINUTES to complete this exam. 0 Write neatly and in the spaces provided. Additional scrap paper will be provided upon request. Rough work on scrap paper will NOT be graded but will be collected at the end of the exam. 0 Finally, please read the questions CAREFULLY and follow the hints and instructions. 1...“... W V Question Points available Points scored T pmbalob most. malisfi‘cl ‘ Question 1. (A) Verify that tan2 6 2 sec2 6 — 1. (B) Compute the indefinite integral / tan3 9 secs 9 d9. (A3 we: 5W . C038 30 SficZQ ~ M029 -: “L.“ H $028 : (”$016 C0819 (0519 COSZE : c0326 , r2 5 ’Z - (DEB 9an 3m (9m 8-; ”9 jam saga OW :— chcZQ «flfime 31139 (16 m3 CA3 I j (um wow Sew WKCG 0M: Sfcgtzmfictfi : jML‘*MZ &U /\ 5:: : ,, ~ g} + C state How WSW/diam S 3 dearb ‘, dcmur firgd’ . ( ; J; 8&36 ”L 8<1C3<9 + C (W at? Question 2. (A) Sketch the solid S obtained by rotating the region underneath the graph of f (x) =2 x3 + 2x2 over [0, 1] about the vertical line x = —1. (B) Compute the volume of S. Draw any disks or shells that you may be using, and clearly mark their radius and height as appropriate. [dock your axes “ $ Label W With/J and ” wish-t o? m 3M! (87 Using M13: mime? sum; r=>c+1 ,mgwr 0t sheik: bzxfile Mm volume V: Z’F f,(’>i+\)(%‘+2fi\dx o ‘3 2“ j xiiéxfl’bd’ an O s 3‘? : 2T: {2L “LE-xi +31% 5 L1 5 rao : l- § 1‘ M (S 1“ ‘1 iii 1'- i935 b‘doggi’ajkflmpi‘fi 30 QNWW W‘- Question 3. Compute the definite integral 2 / x3ex dx. 0 aw S‘Zc‘rfim Whoo‘ ikagrOdW-OYI (03 Park): b (56% .Qgian' ‘2 \quzlg J? l(7L): 37L?“ 7) 1L jg M Q &% 9l(%):€.% a (7t) :61 : Xsflx 17:0 “‘ 3 £1730,“ 01% “13:11 H1) =2“ 3‘(M:e% gm): ex = #6“; v 3(x2e1lio ~2jfnewx) 1 Wex~$xleqL5l7:o 4— 6 flack Hwkx New“ 3((wkex 9(“Mf6 U : 6L%€1~3xlex>lu:a + 6 (Kelli; m flax 0.x) am can 0% can/Se ~€UCLW€ emh'e; and «510 M Com pedm‘im m ”WA ’. 1:; 6%()c3—37L1+6K~Q>12 {—1 TL :0 : e1 (X—tzuzréJS—Hwép) = 262%; Question 4. Calculate the work required (in Joules) to build a pyramid with square base of side length 100m and height 50m, assuming that the construction material has density 2000 kg/m3. [You may take the gravitational constant g to equal 10kg m/sz. Show your work, including how you build up your integral] Cmfidf 301W Wrok pyramid oi mags 3C; Uowm is \/;=(25(313)2nA9; = ”(S—OMO'LAB; mass LS ML = 13th = LU) (Sb-3171133: i . V - ~ Hm + boom \s Wfi Mug-h; om hi 5051’me K 0U WA "'1 L016} WW ail as r q. 2000 vIO- [SO-301 3i Ab; "tom ooovk b WM‘ f (A); : 30,000 '2: (SO/«532 gCAgi i=\ L" MW Question 5. Evaluate the indefinite integral Thymmwmfismgfihwwu 1 NOW, Used? “'Smgigns‘. (Masha/bid be cm goth/ac 3W /' 01¢ each W conmohhé‘dfrpvtmam dx x2+9' X r ”étané? d n23 8639 d8 ’2 Win JranzGJrI = See 9 Question 6. Evaluate the integral / 5x2—8x——1 md. x3—3x2—l-x—3 x WW obaominodrbv‘c flab/thaw“) =(7LZHM'24'3) ‘n‘rfigmfim b3 WAC/Hand PM: W~_AX+E JrC '.... “ (MHOFM) KZH ” 24/3 Tm $19“ 11> moms "imp/M: \ cmd Sth M be used 11> 0%on 61th 1, QWH3MM‘QJr Cg 12+|:/ @1319, .QBWCCGJ hOJrQ/J‘ W (7L7+l3(vty3§ awkwmvfi fir "s‘} f Wm M M5v2H—sztoc => (.1 Sifimj 7L~O ~l: ~3P3+2 z) B:\ gg-fing 7L=l 2 5"g’\ == (A+\)(~2}+2.2 :7 4+1: ”2A+Z (arbi’n/ZL%) & 90mm» x C 003a mg m z) A z 3 Um'bgefir Qflw'mfiwM ' ‘ SO jSXZJB/xw‘ CW: 3%” dxfluj 9’ OW, Question 7. Recall that the trapezoidal formula 1 TN 2 EAx[f(a) +2f(a+Ax) +2f(a+2Ax) + - - - +2f(a+ (N—1)Ax) +f(b)] approximates the integral f: f (x) dx to within an error of at most K2(b — Ll)3 12N2 ' (A) Write down (but do NOT evaluate) an expression for T4 in the case of the integral 4 2 (1:0 lb: LN Mcq I 2/ Sate—x /5 dx. 30 AK: l 0 v J" 4 , ,‘K , = z [0+205e, 4 +2.loe%+ +«2 xs’e 4% 206 6‘6"} (B) In the diagram below, clearly mark the approximating area and all the information necessary to apply the trapezoidal rule. (C) Given the additional information in the diagram below, how large do you have to take N to obtain an approximation TN to I that is correct to within 10—6? rcad ha gumbo“ meta“ and mam 8w: (Jove (FoHaN M ms’mwix’ms K2:— mM WWW ,m K1: max Wet Wipe "val 7C9 [MIG] K6 CON) , ((04% c7 HIS'SflG ()4 4N7- 3) N’W (3 don'ta) midi/d X’QO ...
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