MATH-137-1059-Test3_exam

MATH-137-1059-Test3_exam - Time 7:00 8:30 p.m Family Name...

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Unformatted text preview: Time: 7:00 - 8:30 p.m. Family Name: ID. Number: !__ Signature: Faculty of Mathematics University of Waterloo Math 137" Term Test 3 - Fall Term 2005 Date: November 21, 2005. AIDS: ‘PINK TIE’ CALCULATORS QNLY Initials: - " Check the box next to your section: DUDDDDDDUDDED Note: 1. Complete the information section above, indicating your instructor’s name by a checkmark in the appro- priate box. 2. Place your initials and Id N0. at the top right corner of each page. 3. Tear off the blank last page of the receive full marks. name, or the appropriate acronym. Section 01 Section 02 Section 03 Section 04 Section 05 Section 06 Section 07 Section 08 Section 10 Section 11 Section 12 Section 13 Section 14 test to use for rough work. C. B. Chua 11:30 am. D. McKinnon 12:30 p.m. A. Nayak 1:30 p.m. X. Liu 2:30 p.m. C. Small 1:30 p.m. - R. Khandekar 9:30 an: B. Marshman 10:34.}- 3.1... J. Verstraete 11:30 .14.-. ' D. Wolczuk 1:30 p.m'. C. Struthers 10:30 D. Wolczuk 8:30 9.9.1; P. Balka 8:30 am. _ A. Chau 2:30 p.m. ' Your answers must be stated in 'a clear and logical form in carer ‘30 | Reference any theorems Cl‘fi‘hlfis risen" t; v ‘ b/‘TV‘ V?! I} MATH 137 TERM TEST #3 PAGE 2 u,x..“m ulm .7: .L “Ag. [14] 1. a) For f(x) = 11:6”2) +ln(cosa:), find f’(0). ‘ 1‘ l I / {v .—-|—-- u “3i‘fl7gi/ (95% al 0‘ 0‘. I ‘ ’1} 4(0\~2¢ +Ot—1(°l+-—‘-—*'--Sv"lW) 3) (03¢ MOM/ b) Find the equation of the tangent line at the point (0, —1) to the curve (ilefined implicitly by my+y3=arctanz—1. 1E. M03940 - 7 f 1 ‘ Kim/0. «(0“ 3 l. .1“ t u 1 a) ’ * [email protected] v w 7 @ m : 5— <-«\~ (a) v?) 4 y(_ ‘ 1. A . U‘WVHW 3H?) : r Yw’x "' ‘. 1 1' k 0“) ((+7c)(~3+37) . z ¢ ‘‘‘‘‘‘‘‘‘‘‘‘‘ “it” \' ( . MW W {WW y audit! («W-«5 .244» {0.4) W t (4) + r(q—o) / ’2 RMYMI‘” “M W 44'“ 7:37.4/ CUJVL c) Evaluate each limit. Justify your method. _ , tan$+m2—a: (1) 321—13) sin2cc I 3/ $12M 5‘“ HM“ l/(L‘un) 3 tale 1 Simq (est-y . l '1» ’ sit: “"‘fk'wr‘ mun) last-x - 1 5:12.95 7— _ :2 i 'l '/ (11 lim 1+;1;)2x by l l ’7 1:“ 9‘) ‘5 “a” g) ( l ‘l’ [((filvimm‘c Jig/2H1: iii-'04»? ’Xpfi‘ , . . ‘ Lt“ flkCU—Ey 32%;;qu ‘5 a It)!" ‘ vi; r?’l(!+sl‘7k' ‘ LC Lip“ ?(.1) 1 41A s'fmekkaafilwlmk-fl 5’, :7 "‘ “"1 ’ ‘ l ' (um) “ ’ 5% ovw‘ Moi (with ‘ : "' K t l 1 “m d) If h(a:) = arcsina: + a: - 1, for what values of as does h’ exist? 3 “(1') :ng+ %(2‘-l\-;(Y.¥) [13(1) 1% (or) V 4X “(41) w?“ 0155* wl/w Ix >1 ml «11(4‘.» Vv'r- \ ‘v - I MATH 137 TERM TEST #3 ‘ mm 'M; a I . PAGE 3 [9] 2. Consider the function f (:13) = §z3 + x — 1. Han is so“ ' t c NW“ WW” a) Using a suitable theorem, show that f (c) = 0 for some 6 in {0, 1}. I “I hnpww cram; Hex . 3 I fi‘iX' . l = 43(5) 4 (0) ’l 2"! I (37 figs. ~$nfl€rfiuufik V‘JW» qngm' In t \ ' - .‘ kinky“ acres? .3 3 v = i L.ch “mi. NM s (04! ‘ l ‘t (0 b’ '3“) + U) l 3 {mi}, ant that “1 (2me is Ngeh‘ve 3&3“ cw. and gun‘bl‘ QnK part?“ t; “v 0W] t w Itan «m (L \0 (1' «‘ COM“ 0 q, I 3 s was terms all blazers: J- ,‘5 (k m hr b) Use the method of bisection to estimate 0 with an error of at "most 6.25. I pk “$1041. «:7 3 W 5 ~ u .C’ ( 03 '> " , “Pal/j“th .. i . M (l r 3 . ‘ i F ‘ 9) quL~ 6H 1 4/“ ‘3’“ (’9m M "0“ /‘ ‘ a,” 2 1 . . 2 h ‘ K ‘ L % L ,. flit: ". C 5 wt ‘ “ j A P u q ' /» a; ( a) = 1’4 V/,/ 5”? fiw‘ty wilful/t1 C ~ Try ‘ l 9L : 1 44 z a // L / l c) Find the linear (tangent line) approximation £366 for ‘($) so = 0 and hence estimate f(0-11)- . 1““) = (VGA) 1:" + t i/ we) =i \/ ' film»: Me) i» I‘We-“l . <5 10(1) :00 + (Mae {0(1) 2 ’l’il L/ (0(016 ‘z 9-“ "l fricon z e 0-8? M d) Explain how you know that f has an inverse f ‘1 on R. Using the Inverse Function Derivative Theorem (INVDR) or any other method, find the slope of the tangent line to y = f "1(15) at 3:“1- wkoddo 1640 mm? ‘3 [us tux lave/y, M “Q lilac-«7p :5“): (aermdlmfiac/Hw We} Y“ Magi)”; SdHCNS «m 1hr:th at. mt. u :5 x 503cm, slaw wt»; SUE-L“! “"1 M“5(¢cl“(l q”?! 6““: it; CW“ “‘3’1 Q“ :‘L W! l r i ' I w . a , l % f (“0: ‘ 3L l ) ~“ 3‘ ("W M > 249‘“) ‘ é/Ml‘ _ I “(la slo'zt a? M" 2 ARM)”; (m. h 74”“) .\ cc," «:4 (S L G") W» / J L UN a 1 SH; MATH 137 TERM TEST #3 u ‘ PAGE 4 Wlnuw—a * [10] 3. Consider the function f = xx/2 — 932. a) State the domain D of f. ngx 6 IR] —Es 7r 2.5% b) Find all critical numbers (points) of f in D. _ “7‘” 7‘ WV”. L £((1{):U\( lvxt)i+ Ix («Xe—Y'z) . l, . f ,p L. r m%7 \. (z: x . gm r- W,.., m” . I”?! r { K__._n_\_,_ .__. ~ \ l} x a K F i ' 5 Q (can, ( ‘ i I 1 ‘ rh—u l l «l. ’k 3 l i) g' (wed (Matt (1) Find the intervals on which the graph of y = f(;r) is cencave' up and concave down. 2 2 "' l A “la [ NOTE: f”(a:) = ‘9‘ (m) I 1_____._?‘ (1‘ 3) <2 — mam ' l «g. 5, (it —3 x '5. (i? . f \ O 1 M (it"s) ”' :‘310 ero'¥:tJ—3’ ( V .N Commit? ' i, is ‘ £9 :5 r. i ' ‘ U \ A q ("'4‘ ’ uni-fin (mum. at Law“ at inflict-fan. (ti-W Wk) e) Use the results of a)—d) to sketch the graph of y f: f on the given axes, indicating any 5% local extremes. f) Describe the shape of the resulting figure if you add to your sketch the graph of = —f Write the equation of the complete curve in the form 3,2 : h(x) for suitable h. it} (“(4 gqu If 7' : -£(¥) was 14:10.1 6‘0 '9ch 11lva 3%! Silk/‘1' Gm infinity fish toqu result"? inbuilt a 1 . MW '4 7 226-?" W1 ; «1 (If?)1'/ ' , r , ’ 94M ‘9’ 71354 MATH 137 TERM TEST #3 N V _ 7 PAGE 5 [7] 4. You wish to design a garden in the shape of a circular sector having radius a: and central angle 0 (in radius), as shown. The plantings you have in mind require an area of A = 64 m2. Space restrictions and design considerations require the radius a: to be no more than.10 m and no less than 2m. a) Show that, given A = 64m2, the total perimeter of the garden is P=2$+1—:§m,where2gasglo. { a % P I Z t I 4 ,4” l .1 9 2 2- 128’ 12mm P";N‘£r: ‘ '14: ' i m! ‘33:) 4 9: - , 8/ 2(H4— T'— A W ‘11?" r r° . i i“ r '2. 1-9 . r . : «Xe. w J ’kl: / do 0 P as 9 w my: 2!. “L; W / \\ NOTE: For the segment shown, the area is 52720 and the arc length is H9. b) Find the minimum length of fencing required to go around the perimeter of the garden. Justify your method. \ 1; .A': ix 9 4’ 6%:3’219 uX:?(19 ills” /" z ' 1. [X “J, ...
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This note was uploaded on 12/02/2011 for the course MATH 137 taught by Professor Speziale during the Fall '08 term at Waterloo.

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MATH-137-1059-Test3_exam - Time 7:00 8:30 p.m Family Name...

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