final.soln

final.soln - 2 1 A CALCULUS Monica Vazirani Nev-1g; 2007 ....

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 1 A CALCULUS Monica Vazirani Nev-1g; 2007 . Du \O K, Midtermfix: w ILL Name: lu l (\O ID: Section: 1. (10 points) Calculate the following derivatives, using any method you choose (power rule, product rule, quotient rule, chain rule, logarithmic differentiation, memorization, etc). Even though it is multiple choice, you MUST show your work for full credit. (If it is a memorized formula, state so.) Circle the letter of the correct answer and / or write it on the line __ provided. a. % sin(cos :r) (A) wig; (B)El— sinzrcos(cos:c) (C) —tanx (D) sinarcos(cos:r) cos(cos:c) (F) sec2 :r (G) cos2 a: — sinax be LL: Coax , \\:‘~;l/)u Q l /" CE}? ‘ (Li! Cl“ :— <.‘_o3LL'('-‘*‘$m%) ‘2 (O‘k<COS x\('**$"‘>9 (5X 7.. « Sa‘nx (‘usKwSKX b. dim- sin(Cos_1x) C (A)a: (B) sin( -1) (go); -I (D) cos(Cos'1:r) (E) \/—1—x2 g; x - > D“ : Charo/é; \i :4; 1-)?“ ' ,__,.__ \I‘x" Name: ID: Section: 2 2. (35 points) Determine whether the following limits exist, and if so, evaluate them: Even though it is multiple choice, you MUST show your work for full credit. CWOQ )WQ \Q\\C/ .3»: WNW \\ on Aug \MQ. a. limm_,0+ (Pita ‘1 (A) DNE (B) ln3 (C) 0 (D) 1 (E) 00 @fin‘l gm QMVMX \ Nka Ll ; Q: 3”" i“! v y m on. Kim fl Q m: e LIMA ‘ V/~ , (MAO 4 l, owl; we it q Al- l m S x Q \ \[ ¢ l 34x j‘wkl \VM /_..——~—~ ’- 1:; f (’1, A/ILI (03 K O -\ X40k ‘K an \ W + : e' “(JMKK (C136 x m 4 WW» "“ \ 2m —— (Aw (B)DNE (we ((mo) (moo \S ufilmocw 0‘33” VT? so Aha \lm\\ \3 1’ 1 is W‘IW - O 9“ wwvm (—7“an ~ 0 \X SQCX \3 C\~S\C\Ac\\ CVYKAV‘V-CNS ("1 xio SO “‘r \W‘ w (03243— 3;: , :1: N Name: ID: Section: 3 d. limm_,0+a:ln(:c) xv (A) ~§ (B) DNE (c) —e (D)1 (E) —oo (F) o) ‘1 ~ QR, \ X l/y‘ \M Mm _ ‘7 a 5—5;» - L — x 3("0; \‘qflfiox X ),.—\Q X X fig? ~35: e. limzqs A {2%) (B) 31% (c) «@742 (m (E) 0 (F) oo (G)1 \JIJX ‘ mw( L\€A HAW/n0 C‘kLfiK-“U‘Q \“I\"‘\"“ ‘3 (b L‘A‘fioy‘ V v I I N / 4.; .L l ( L O - , A; ; M 1+ x 9 1t ( S) ,4 SL & (D f. limzfio—m—fi—Tfm __Q , {I (A) (8)1 (D) 0 (E) 3 (F) 2V3 (G) DNE \. M (ML, \ 1 \-\M ><('§x2.+\2x+§ ' a (3) H) mm N2 s we fiwg * .- \W GEM * “3 {ME = x; ' Xfifl‘o Wflxa \L ‘ \';~ (9 L' Woo \«(JJ Name: ID: g. limzn_w§§% D ., (A) 6 (13) —oo (c) —% (9) (E) —6 (F) 1 \ Li “145‘ ll/)\‘ 4; -0 *9 a I M ” “M” ' an o” 5 3. (10 points) Consider the function What value of a (if any) makes f a continuous function? D 5x1n(ac4) a ifx7é0 ifx=0. (A) 5 (B) In 4 (C) 1 D_)’0/' (E) not possible \>\r\,\ SX /\v‘xq &\"\ \(z-s 0 X «3) L) B D (“K "\ Xfl‘” x j C) “ '\\\\C.x: L ’ IQ 7"” 0 X740“ A , q R x 30 \ ’fl D K /k "\ ” C, x w- o A k v\ k g ‘ C K I 0 Section: Name: ID: Section: 5 Il‘ CT \m'; me “Hi L03 lov‘b'fia. WYAQ Mord Art: 3%. V '3‘)“ I “3) ‘ 4. (25 points) Consider the function 9(33) 2 41'3 z'z—Z a. On your own, find where g is positive and negative. (It will help you sketch the graph later. I. a 5 _. ) (km) 1 “if p F flfi—rwro-l ~ + (x—(?:)(_K+\2-\ “(z 0 (2. b. Find all asymptotes. If there is MORE than one asymptote, circle (and/or list) ALL 0 rrect answers. A C- Y; @91th asymPtOte at 93 = fl (B) horizontal asymptote at y = 4 ( C: I ertical asymptote at x = —\/§ (D) horizontal asymptote at y = —4 (E) oblique asymptote at y = 8:1: @ii‘soblique asymptote at y = 430 (G) oblique asymptote at y = —a: + 8 no asymptotes \)’\r\ ‘71 '31:) \\W\ 2 -» QC) \l’V\ *on‘) ' .30 ll‘“ 1 h ("Q xfa+<i x») \r V» r ‘1 x—«x —» a" n d {‘7 Q)‘ “"TQ \J\Pl\~‘”“ \1 Q1” *1( r38 “K 4 v ’\\‘ x- ’ \3 K \h\'\4\ C r (3:: ll N\ 00‘] i." " "Jr-J y iflxlbq—VQ- Ckl\c V,\CJ like ( 'chv;¥lq:L Xvi. / (P 3 5‘ 33y « ('5')? ‘ 5 - a .5“ 9‘“ -\ (“1‘1 is Ck; (ADVCIKH Lk RN“ ohm : k \ALQ x___m__m_ww__, ,. :1 [ix + 8: 3O (7/X i 3 X1.” )A’Liéi Alix //KK XL”; XE). X "3% CA ‘ (NJ “vi, CLEAN dim-ll c. Find all intervals Where g(m) is increasing. If there is MORE than one interval, circle ' ALL correct answers. E F H m (o. «5) (B) (0, «6) (c) (—«6, «6) (D) (—oo, 45) QED/ (moo) F) —oo, 46) (G) Nico) (H) (—oo, #6) <1) (46, fl) 1 .3 A (X’L/Qgtflg Z 93le 1 fixfl 2? ("L _ g (r * (XL; 931. (x7323; (x -q 7K ‘ \ + #Jjfl“ .... .. \/ C) TL h a \ £9 ,‘L \ ‘l (0 \vxL. a Q Cgek ling J' i (1 Va. :1, 2% “X4 3, Name: ID: (1. Find all critical points and local extrema. If there is MORE than one critical point, circle ALL correct answers. d V ,N\ (A) local minimum at a: = 0 f B) local minimum at a: 2 \f6- (C) local minimum at x = —\/(_3 ' (D local maximum at x = -—\/5 (E) local maximum at a: = —\/§ Wlocal maximum at :L‘ = 0 e. Find all intervals where g(x) is concave down If there is MORE than one interval, circle Section: 6 correct answers. 1 C. (KM/(0M?) (B) (x/Zoo) (C), —oo,—\/§) (D) (45,0) , H M 6/00 > «I X 19% (58—23 W . . 1 I _ ’ I .L\ 1 ‘ ‘4_ 3‘ gym: q Hfiflflaflegg);th 1 q_(w\({xj3x)--E§L K (X2327 (X7: QB”; k Ear/IA :5 2 3) ‘37; g 2 I /’\(<ct k 2 IC’ ’xvzfl-ififlfi “A (V: r (a X+Qfiv : /é ><(2<Z'+(o) XL" 2. w V _ \ V: ( SO (x q (KL/SLY 7:. (“Wimp (5' $qu 0 VL dud n U Q ((lfixu I‘ \& f. On the graph below, sketch the function g, also drawin (with a dark dot) critical and in ection points. ' . / Name: ID: Section: 7 5. (10 points) A spherical iron ball with radius 5m is coated with a layer of ice of uniform thickness. If the ice melts at a rate of 8 m3/s at What rate is the thickness of the ice changing when it is 2m thick? E: M (A) (B) 7:; (c) a (D) 8« (CE) —& LXL:-$§ Rd Ac 4 an; iv a: “ Cw 4 fl 2‘ .r 9‘ ,F ,.V CC 7 , .K ‘r‘ A M . an 3(s+q 3.0 x LK—ui Ma: :5“ _ 5:. - (“H 6. (10 points) Find the equation of'the tangent line to the curve defined by ' a _ \5/5‘/singx g y _ 621(2x + 3)2 W 25 (A) —'2'15=—53~1) (B)y-1§15=*25($“1) (c) jig—awn D —1=ix—1 E ——=—lm—1 Fnoneoftheabove y 25 25 5 E LKUCCVVW \'\, \,\” 3 m(X_V\L\ (“it (h:.VC:: (Mg)? ‘4 C‘ l‘, . M6” 4 3mm + Q- 1x ’ Q/QaleQ 9L 3 0V 3 X 3‘ smgx “53:: GT L i \ —\ Li 4 _lg — \ A _ x ”’ 0 ~ — -"::" _ _.w_c ‘ ___Z or gsi D L 01 s 15 mg Name: ID: Section: 8 Q a lush) V g (B - q 0M”; 7. (10 points) Find the dimensions (height and radius) of the cylinder of largest volume that can be inscribed in a sphere of radius 5. / @jradius = 5—3/5; height = % (B) radius 2 ég; height = 39 (C) radius = 5435; height = % (D) radius 2 ¥; height = V: “(Lk as ,r 3 TV X c113 § (E) none of the above El“ w as: a 78: 331% c , v 1m \l V? 33“ a “b )1 5 k . 4_ No. L\ fl ,; a (As » grfi 5-7-- firwg Vi) ‘ rg/(g g/W‘S X\V\1mi $3: at (1)4 QPL Xw” A?) ' 27:13; 2 2C a ~ )W L (“Waugh 3" (QC\\\:‘)': x; 3~\—<‘ ‘ The function f = 3335+ 2m + tanx has how many local extrema (maxima/minima)? i. fl f \Jlu )’ 17' : . ‘ T (or @Bwpoints) 7) L i“ Q 5%; <2 . X. "a. Q7 [Explain briefly why] (A) one (B) two (C) three Inbne ' Mo 2 35734,; a Seczx >, (Haw) 7 C) £1“ch x CM g“ QQ LAY? \,\_5\_:> 5f v-C‘ f((\ T ii! rn‘ K; ,6?" C'QI‘ACN (Qw 0‘ C ‘7h-i V‘ dc, ) /-’\‘\N»> c Cox \f\ 4‘ V?,\ \xcL QQL \ v i~—- {Q g « $6.0: SLZX —Sx~~§ 'Mx 9“) 2 b h (L V_ , r. ‘ a my 7 L» ( - ' in. «1 K56 (“(3733: xvi—“m “36 W; \ 4 -:S r: )0 \L- R?) \‘H * “A A :Q g. b. [How many solutions (zeros) does sin2 a: —— 3x = 5 have? [Explain briefly why] ’4 Q (A) most one (B) at most two (C) at most three ‘_ _ ‘ ""QAOU\J>SNQ. « > K D i S\V\\ : Cm ml ,‘K ' fix 7 S. ‘ \ \.\ K" L‘L‘K' 41AM“; plQAm S i C Q S /"\ I Lihrfim ( K (A WA‘AFQJ‘ KMKASL U Rik.) o \ of \ l .i i ) ‘ - i ._ & L4 gun “Qt \ (A) V O <MAX (“x q) [it-l ‘l \ \ivxx é‘ )4 q i n) x i‘ i So Jung/:1 I / i a , \K 0:" \gcx p H) ‘0 0’L'L"W"‘K3 ~l ’: kazxi‘rv é " \‘2, SQSLAK (as X é— ’L I , «\\(x\i(riQ»v—if' ') (’ pl V— Name: ID: Section: 9 9. (15 points) Alice shoots a paper Clip up into the air, so its height is given by the equation 3(t) = 3% — 4t2 where t is time in seconds, and the height is given in meters. 5 a. How long doeshitgr ke the paperclip to reach its maximum height? b (A) —4 @3) (C) 8 (D) 16 (E) 32 (F) 64 x” I ’ kk 1 ‘E\ \ r: r +- - vm; s’m v» Ag:— >5 fl 3 ‘3 W E ) “wr’w ( é flax _' l mu. K [M w d 7- . x __ i 3 s \< \ gm“: \ sci) — H_L[(Y,LD b. With What speed does it leave Alice’s hand? _ (A) —4 (B) 4 (C) 8 (D) 16 ()E) 32 m); )(F) 64 \x. a, I V( o) ‘ 3m ‘ ’5‘?) '7; 1:), m): c. What is the highest it will go? F /, \ (A) —4 (B) 4 (C) 8 (D) 16 32 63;)ng SON) ? (9“ M. Name: ID: Section: 10 \ LA . ; <06” 3“ mun/$3”) ’“D S‘VK 10. (10 points) Fill in the following FIVE blansz in the following problem, which involves using the precise definition of limit. A Verify that .i/ / lim4:1:—3=:7j ,fOI‘€=1. (1) m—«r2 We are given 6 = 1. Now we begin our scratchwork to find a 6 that works. We want lf(a:) —- LI < 6, and we calculate: // |4x—3— :3 |<1 (2) <=> —1<4x—8<1 e 7<4$<9 ¢> 7/4<:c<9/4 <=> —1/4<:z:—-2< Vi . (3) Thus a good choice of 6 is \ , 6 = COA>AM¢~\ ‘ exckkhq )(V‘q h Kl/q (8 0Q "\‘00 \ Since all the arrows above are <=>, it means that whenever 0 < lfv — 9—! < 6 (5) we for certain deduce — L| < 1. ...
View Full Document

This note was uploaded on 10/07/2008 for the course MATH 21A taught by Professor Osserman during the Spring '07 term at UC Davis.

Page1 / 10

final.soln - 2 1 A CALCULUS Monica Vazirani Nev-1g; 2007 ....

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online