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# exam%204%20green - Exam 4(Spring 2007 Name ’{€€/V...

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Unformatted text preview: Exam 4 (Spring 2007) Name ’ {€€/V\ Score /100 Circle the correct answer. Each problem is worth 4 points. °° —1 " 3n 1. Determine if the series 2( 2) 2 converges absolutely, converges conditionally, or diverges. n=l n — Jam 1;». :C) so War/L416? sum; @1/ converges absolutely n LL a E 6) converges conditionally “A00 . . — J— J—- ‘ c. diverges La’r' b“’ n 2"" DI V an d. cannotbedetermined 1,“;3 50 in 541.7. DIV n—w . . . w (—1)"Sn2 .. . 2. Determme 1f the serles ZS— converges absolutely, converges condltlonally, or d1verges. ":1 n +3. g3}. ~o 50 warn-4*" series CU converges absolutely L n. n 543 " “a b. converges conditionally V‘ "’°° I , c U A c. diverges LCT.‘ lg n : 7,“? L35 54; U d. can not be determined M 5 n2 , Li ; 5 a \$36 3. Determine the convergence set for ( x all real numbers a n : Lil)“ A b. (‘1,1) 2H0“; IL, 0- (0’2) ' ’47. d. (—2,0) ‘0 4"” 4. Determine the equation of a parabola in standard position with a focus at (~2, O) . e? a. x2=8y ajrl/PX D b. 362:” F Pie? c. y2=8x y2=—8x 5. Determine the equation of an ellipse in standard position with a focus at (0,—4) and a vertex at (0,6). .2 x2 v 1.. 3:, a + —1 a 9 36 20 (a F L, a x y _ z 2 :2 g Q:® Q:b—lc __ Q c i+ﬁ=l F + 52 36 v , :2 x2 2 JOwlo d ———+l—=1 36 52 Exam 4b, page 1 of 6 2 2 . . x y __ 6. Determmethe focme—z—S—l. OZ: OLZJ/ L2 a. (i3,0) \_ x, w J‘// b.(Qi® ﬁ>\i<:\ 03,/&4g93w:14/ C, @3 (Mo) /\~9 -=\ d. (mix/E) 7. Determine the vertices (x302 +(y12)2 =1. a. (4,—2), (—2,'—2) H-132) b. (1,1), (1,—5) r, C (2,2), (—4,2) VP] t [291) d. (~1,5), (—1,—1) Show all work for full credit for the following problems. Each problem is worth 5 points unless labeled otherwise. 8. Approximate the error when only the ﬁrst 4 terms are used to evaluate i (—1) 2n . (4 points) n=1 "2+5 +k Bfror é )5 +€fmi J’5 :,A2:11 5°55 30 3 (3x)2 (3903 (3904 9. Determine the convergence set for (3x)+ 2 + 3 + 4 +.. a / Qﬂf' fl fL , (305+? .1: g ;/r 2714’ ’ My}! [l/rvv 77/ (391)" p 4,; r M n—baa ,1 __L )3X/A/ ==> 'lg—lkgxé/ —; €434.43 3(‘é 4:1): U<y=~é'£ A :én av , ,L 3'4 K ,1 Ck?’3~Z/l—f’):1 DIV 3 ’ 3 Exam 4b, page 2 of 6 cosx 10. Determine the power series representation for f (x) = through the x4 term and simplify all l—x coefficients ' .2 ’1 q ,1 L—...< 14149.; 3+y+--' 9931‘: w5¢(/flf): 0 if}; ) H75 74 ) /’ 1 z 9 '__ :/[/Jv+yz+r3+yz‘r~>7;.[/+31+>£+. >+')Z£_‘;U+,,,)+ =/+x+x1+x+x+ 3-é,2’+ «#024,; .7 3 v :v‘X‘I’L .29. 1.3.54. ﬂ' / g+£+§y+ 11. Determine the power series representation for f (x) = sinh(2x) through the x4 term and simplify all 3 5 . , CV ,- r Stnhﬁ" "L 5! 11' coefﬁcients. (4 points) 3 smMm): (4)431» @3347)” 1L.” 0 :O?3(+%3+--- sum/20:07! + \$3 “‘ 5/2 ) 12. Determine the power series representation for f (x) = (1 + x through the x4 term and simplify all °°CHETTZM\$ / i (772 + (:‘M t ( :z)x3+ my? 5/1 5 5 '2 L2. 5 5/ —’ (at ~20 5/1 ; "" '— Z/ 0? 5 56‘ i.a)(§-—3) ’5 5/2 , fig"), 3% /5 (9’1): W‘ gab/IE3 [Z ’ :2?” ' 7" ‘ 2r " 4/ W Exam 4b, page 3 of 6 13. Determine the Taylor polynomial for f (x) = In x for a =1 through the x4 term and simplify all coefﬁcients. ‘F[¥) mar) K C, ¥,f"5;7_‘z'_f_ 1L F7094 ‘P’ (’0‘ ' XL #0?” - ﬁll/LR);09Y’3; \$5 (‘0) ’9? “,9 40 (I); «(a '9, wan / w /——-— / ﬁlek/(w) * 2, 1 4/ 3 f r ~/ / “kW—k 0",) (9:104'7" Q7" [7‘ 3 7 /’3" 4 2 14. Determine the Maclaurin series for f (x) = sin x through the x4 term and simplify all coefﬁcients. Singx ; (st/WVBQ: (sin XXﬁZnK) 3! 3 3 - ,, ,. 39, , / 2(z/%+ 5 a W ) 9 2 r // y / ﬁ— L’,’ ,, r (a Q 9 ’ a; // / L .f/ an X 3 J” 3c2-5005c 15. Determine a good bound for /§_§:§f§3i/g/ééai ioEC/é/3g/#Z75—Ze;3¢ £3a9+%:09705 with c in the interval [2,9]. W Z a 5 3‘9 +54 _. /34Z/5W Lgééfeil g/jfL/iifliaﬁé f1 4% b / a Exam 4b, page 4 of 6 16. Determine the equation of the line tangent to y2 = —6x at the point (—3, 35). Solve the equation for y. 181 9': "/6; , ’3 '2 ,. r 5 L -‘ ’- 3 3; 3 Q a; X V31 V5537?) a;1¥-§.+3¢i{@_) .’ ﬂ, J5 y; V31 ’ ﬂ ~— ’ :5" ""9 9 M5 17. Name the following conics and include the orientation. (4 points) 2 2 2 2 i—Z— 1 b. x—+y—_1 12 20 12 20 18. Determine the equation of the set of points, the sum of whose distances ﬁ'om (4, 0) and (—4, O) is 12. F @;4 avg: b27L d2 (1 :1 072:: 3(0;ba*/é 2 o? b 97020 at b9, 0? 2 L» 7" ﬁ/ : / 3 (0 (£0 2 2 19. Determine the equation of the line tangent to xj+22l7=1 at the point (J8,—3). Solve the equation for y. 2233-; ﬂay—51170 «'BSJECK'E) 9 077 3 I e , (3+ "JE 34 w k 0? ' yo 9 07‘? I 3:J574’9 ’9’ 9 / Jé’g';0 0L”: J6 Exam 4b, page 5 of6 20. Determine the conic (or limiting form) and center of 9x2 + 4 y2 + 72x —16y +124 2 0. 907+ 8w/é) 69/3944»; +4) :- 729 + ?'/bw\$’- l/ M +ﬁ::/ Uﬂ/47M6géfse 4% 9 New 21. Determine the equation of the parabola with focus at (3, —7) and directrix at y =1. F8" 7495'9103 -1 OF )0’47’ Wag???) U3): '9'753’4) V (3:3) 014232; 75 10a *3) 22. Determine the equation of the hyperbola with vertices at (3, 2) and (3,—4) and a focus at (3,3). :5 4(3)°2%) “l: 4/ if: :/ b’Zz 4&6, W a 1 01¢ Zﬁee‘der Hum/W) " (1'3) 7,, J’I) :/ L: 7(Lﬁdagktgua\ *7 ? az+b21¢z 0, +5919 10737 Known Series 1 (1 - x)2 2 3 4 2 3 4 1 1—=1+x+x2+x3+x4+... =1+2x+3x2+4x3+... —x ln(1+x)=x—x—+x——x—+... ex=1+x+x—+x—+x—+... 2 3 4 2! 3! 4! . x3 x5 x7 x2 x4 x6 s1nx=x——+———+... cosx=1——+———+,,_ 3! 5! 7! 2! 4! 6! 3 5 7 2 4 6 sinhx=x+ZC——+—)—C—+—J~C—+... coshx=1+i+i+x_+m 3! 5! 7! 2! 4! 6! 3 5 7 tan'lx=x—x—+f——x—+... (l+x)p =1+ p x+ p x2+ p x3+... 3 5 7 1 2 3 Exam 4b, page 6 of 6 ...
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## This note was uploaded on 02/25/2010 for the course MATH 201 taught by Professor Kim during the Spring '07 term at SIU Carbondale.

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exam%204%20green - Exam 4(Spring 2007 Name ’{€€/V...

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