( A) ( B) ( C) ( D) ( E)

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1 d Smsr Fial Exam Worksh x 5x.( NC)If f ( ) d + 7, h 4 f ( ) d is 9x + x 5 6 ( B) ( C) 0 7 ( E) divrg +. (NC) Th ifii sris ak has h parial sum S ( ) for. k Wha is h sum of h sris a? ( B) 0 ( C) ( E) divrgs k k (NC). Which of h followig sris ar divrg? 5 + 6! I. II. III ( B) ( C) ( E) I oly II oly III oly II ad III No l + 4 (NC). Which of h followig b provs h sris covrg by limi compariso s? 6! l l No!!! ( B) ( C) + ( E)

2 5. (NC) Erik is doig a mah problm, bu h go h problm wrog. Hlp him fid ou a which sp h FIRST mad a misak. Problm: Drmi whhr covrgs or divrgs. Sp : L a ad b Sp : b covrgs by Gomric Sris Ts a Sp : lim lim b Sp 4: lim Sp 5: Sic h limi is ifii, h s is icoclusiv. ( A ) Sp ( B ) Sp ( C ) Sp D Sp 4 E Sp 5 6. (NC) Which of h followig sris covrgs? 7. (NC) Which of h followig sris divrg? I. II. 0 ( B) ( C) ( E) 0 I. II. III. cos A I oly B II oly C I & III D II & III E I, II, III cos + III. ( ) 0 4 I oly II oly I ad III oly I ad II oly I, II, ad III

3 f ( x) x l ( x + ) I. f ( x) 00 ( x + ) 00 ( x + ) II. f ( x) 8 NC. For which of h followig dos lim 0? III. f ( x) x+ x+ l ( x + ) 9.(NC) Which of h followig sris covrg? cos si I. II.. π III + I ( B) II ( C) III II ad III ( E) I, II, ad III 0. Which of h followig ss prov ha h sris divrgs? I.Nh rm s II.Basic Compariso Ts wih III.Limi Compariso Ts wih I III I ad III oly D I, II,ad III E No ( A ) oly ( B) oly ( C) ( + )( 5 + ) 5 4 ( + )( + 4 ) 5 b. Wh usig h Limi Compariso Ts for, is a raioal fucio wih a coffici of ha provs is covrgc. Mawhil, b igr xpo ha hlps prov h covrgc for b c b Wha is h valu of? + ( B) ( C) ( E) is a raioal fucio wih a wih h Basic Compariso Ts. 5

4 ( NC). Which of h followig sris covrgs absoluly? ( ) l 8! I. ( ) II. ( ) III. ( ) (.) + ( B) ( C) ( E) I oly II oly I ad III I ad II II ad III ( NC). Which of h followig sris covrgs codiioally? l I. ( ) II. ( ) III. ( + ) l ( + ) ( B) ( C) ( E) I oly II oly I ad III I ad II II ad III + 4. ( NC ) Wha ar all valus of x for which h sris ( x + ) covrgs 5 ( A ) (- 7, ) ( B) (- 7, ] ( C ) ( D ) [- 7, ) ( E ) [ 7, ] 5. (NC) Ar hs sris absoluly covrg (AC), codiioally covrg (CC), ad/or divrg (D)? 4 a ( ) b ( ) + 5 ( B) ( C) ( E) + Boh CC Boh AC a -CC, b -AC a a - AC, b - D - CC, b -D

5 6. (NC) Ar hs sris absoluly covrg (AC), codiioally covrg (CC), ad/or divrg (D)? cos a Boh AC ( ) b ( ) 4 ( + ) ( ) B a - AC, b D C a - CC, b - AC a - CC, b - D E a -AC, b -CC ( ) ( x ) abou x? 7(NC). Which of h followig is h radius of covrgc of h Taylor sris ( B) ( C) ( E) 8(NC). Which of h followig is f () if f is giv by h Taylor sris + ( ) ( )? x ( B) ( C) 4 ( E) 4 9. (NC) A fucio f has Maclauri sris giv 4 4 ( ) x ( + ) x x x by ! 5! 7!! ( x) Which of h followig is a xprssio for f? A x cos x+ x x 6 B x C xsi x x D x+ x E x si x x 0. (NC) Wha is h coffici of x i h Taylor sris for x abou x 0? ( B) ( C) ( E) x

6 4 NC Wha is h coffici of x i h Taylor sris for cos x abou x 0? 0 ( B) ( C) ( E) 4 5. NC L P x 5x 8x x + 5x + 7 x b h fifh-dgr Taylor polyomial for h fucio f abou x 0. Wha is h valu of ( 4 f ) ( 0? ) A 0 B 5 C 45 D 0 E 0 (NC) A objc ravls across a coordia pla wih is posiio a im giv by h paramric fucios x cos ad y si o h irval 0 < < π. O wha irval is h fucio cocav up? A 0, B vr C 0, D, E π, π [ π] [ π] [ π π] d y 4 ( NC). If x + ad y, wha is i rms of? ( B) ( C) ( E) ( 9) ( 9) 5

7 5 (NC). Wha is h slop of h li ag o h polar curv r si θ π a h poi whr θ? A C D E ( B) 6 (NC) Th posiio of a paricl movig i h xy-pla is giv by h 5 4 paramric quaios y ad x For which im is h acclraio of h paricl 0,0?, 0,, ad ( B) 0 oly ( C), 0, ad oly 0 ad oly ( E) oly 7. NC L S b h rgio i h firs quadra ha is boudd by h π polar curvs r θ ad θ c, whr c is a cosa, 0 < c <, as show i h figur abov. Wha is h ara of S i rms of c? 4c c c c ( B) ( C) c ( E) y f ( x) 8 NC. L b h soluio o h diffrial quaio dy x y wih iiial codiio f ( 0). Wha is h approximaio of f obaid by usig Eulr's mhod ( ) wih wo sps of qual lgh, sarig a x 0? ( B) ( C) ( E).5 5 8

8 9. (NC) Th ra of chag of h volum, V, of juic i a caro wih rspc o im,, is dircly proporioal o hr-fourhs h squar roo of h volum of juic i h caro. Which of h followig is a diffrial quaio ha dcribs his rlaioship? dv dv 4 dv dv V() V ( B) ( C) V ( E) k V 4 d 4 d V d 4 d 0 (NC).Th umbr of bacria B i a pod a im x is icrasig accordig o a logisic diffrial quaio. Which of h followig could b h diffrial quaio? 0.05x 0.05x 7000 x B ( C) 0.05B (D) 0.05(7000 B) (E) 0.05 B(7000 B). (NC) Show abov is a slop fild for which of h followig diffrial quaios? A y B y C y D y E y

9 . Which of h followig sris is absoluly covrg? I oly ( B ) II oly ( C) ( ) ( + ) + +! - I. II. III. + 5 l III oly D I ad II oly E No of h abov. Which of h followig sris is absoluly covrg? I oly ( B ) II oly ( C) 4 III oly D I ad II oly E No of h abov π π si + cos + I. - II. III. (-) ( + 4 ) ( + )