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M137.F07.TT2.sols

M137.F07.TT2.sols - Faculty of Mathematics University of...

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Unformatted text preview: Faculty of Mathematics University of Waterloo Math 137 Term Test 2 - Fall Term 2007 Time: 7:00 — 9:00 pm. so S... LAT\C)K$ Date: November 19: 2007. AIDS: ‘PINK TIE’ CALCULATORS ONLY Family Name: __._____—___ Initials: l.D, Number: ____ Signature: Check the box next to your section: 001 B: J. Marshman (11:30 amt) 002 H. Lee (12:30 pm.) 003 M. Leifer (1:30 pm.) 004 S. Speziale (2:30 pm.) 005 P. \Vood (1:30 pm.) 006 C, Subich ( 9:30 am.) 007 I: Oancea (10:30 am.) 008 M. Best (10:30 am.) 009 D. Yang (2:30 pm.) 010 P. R011 (12:30 pm.) 011 S: Purine (9:30 am) 012 D. Park (8:30 am.) 013 I, Oancea (8:30 am.) IHUHHHD nnnnnn Your answers must be stated in a clear and logical form in order to receive full marks. Reference any theorems or rules used by their name, or the appropriate abbreviation. Useful abbreviations for this test include LSR (limit sum rule], LPR (limit product rule), LQR (limit quotient rule), LCR (limit composite rule for basic functions). the corresponding continuity rules (CSR, CPR, CQR, CCR), and derivative rules (DSR, DPR, DQR, DCR), as well as the Intermediate Value Theorem (IVT), and the Mean Value Theorem (MVT). Note: 1. Complete the information section above‘ indicating your instructor‘s name by a. checkinark in the appro- priate box. 2: Place your initials and ID No. at the top right corner of each page, 3. Tear off the blank last page of the test to use for rough work. You may use the reverse of any page if you need extra space. NIATH 137 - FALL 2007 TERM TEST #2 PAGE 2 [4] 1, a)Find:—1 if y=152ez+7r2 a:—]n(sina:+2). a: g : 2x e'y— xze"‘+- “'1 ' —- -—‘——--‘ C95“ DPR, DSR ,DCR [3] b) For m) = e"°”+(rc+1)2/:“ find mm. SL700 = erS‘X C-SUV‘VY3 + 33.. ("x“)‘i/s DSR I £033: eo. 0+ ECOH)” = E [4] c) Find the equation of the tangent line to the curve em = m — y at x— — 0 ampmmotgg’l =v e”9(aa%xfg‘)..1fig wirxmocz (>0> 9°:Oo—L5 =9 8" flow (—‘C‘ - ___,_?.—1Lfi=xy 8’ =1> 8r :1 H' l "J 2- [4] d) If f(. )— — {II for z > 0. )find the constant a such that f’(a ) = 2f(a.). l ngfwd=ochoc =b @‘g(fl“€m9¢+x-% =9nz+l Wm 2£Ca>=a> 29Cq3aema+l c» om“: The graph of 3(t) at left gives a position of a bug cum ling along a shaigh1 blanch. On the g1\ en axes skttch Hu velocity 'UU) = s'i( ) of the} 3ng and explain how xon know: (i) where the bug‘s velocity is greatest: (M 1::1‘ , wewv. Tlu. A1091. is most Posflio-e (ii) where the bug is stopped: ‘- 30-: t... ’: t5: t3 ) (ADM e‘(+)=0=1r(+) t (iii) that the bug does not end up (at t = t4) i. 4-; 1' *4 where it started (at t = 0), SliflCJz. sch-+311; 5(0) :0. MATH 137 ~ FALL 2007 TERM TEST #2 PAGE 3 [10] 2. a) Evaluate each limit. Justify your method by stating what rules/Theorems are used (e.g., limit quotient rule LQR, l‘HépitaJ’s Rule L’HR, etc.) (1)3929?” = Mm 1 °° ’ 3 ’X-am 824! (:5) HR ‘ (L ____, QLVW Lt)“ S O x4900 22’ .. . 5mm O (n) 1122) 3:2 + 255 (a 3 MR). t‘m £05“ 96m 1923:. J" _, 'x-uo 2rx+2 of» “”0 IX (9“?) V 2 (iii) lim(cosar)1/I2 OQE} f5 :1: (005 K) 'K =¥ Q’V‘H :7. 1.. Ch (COS’X) :—~0 X1, .' _ Xn‘m. e“ _, Jim BM (5'05") <2> x—a 0 L1 “ «a0 X’— 0 , . ‘ it“ M) 9m , ram 0 ’7 '4 -— C) 2. X I W60 29¢ O ‘ Lia: him -sz’x ,1 .. L . . mm = 94‘," 69MB” 9945-36” ‘ 8.. V1, 7960 2 2 “*0 “”0 ”t a L. i log CE. «3% 9.1 [3] b) Use the Squeeze Theorem to determine 11—11% |rc| sinz(;), qua“ ~16 surf-:23 5i gm aflxqéo) we Raw—o. o e nM‘Céflél gnaflvwo Smuhbl‘ao > hub whplLua 05:17.1 mule)?» 5- 1"“ ‘gcn 394.0 ‘ Si€u%\x\=‘0 MA: Moe-o a TM Sigwm W M lei/Wm do mum [5] c) (i) Use the definition of the derivative f’((]). plus L’HR, to show that f’((]) exists for the . 2 2 function HI.) = { g ln(x ) ::3 . gm = w M , as“, when r. um. Mae) 7H0 ‘ rib—o " x—ao 1—20 "177(-~ B. LIHR (D) 2: M J—-—.2 ' L6 \ 8 rx—yo Xv. “AL. =m~21 30' rxflv (ii) Using the result of (i), and other theorems as needed, Show that f is continuous on R. Since. SIG->3 Mist‘s , g. Wt ku— afaai «so (had ‘t‘i'uz. ,L 2 | Wm WM» ci‘s.) Fm X=#0) ”£00: 0‘. Qfl’i-C'X) 1A Wu» PM»? A» C POS'cT' ax. basic 8mm, #le-DCA dis. (56115“) [cud CADRfi-CCR. MATH 137 A FALL 2007 TERM TEST #2 PAGE 4 3. Consider the function f(a3) = arctan :1: + 1:3 + 1. [2] a) Using a suitable theorem, show that there is a c in [—1, 1] such that f (c) = 0. ¥(—|) :2 MLWC-l) +C-\)3+\ c. ”(Tr/4. <0 %L\\ : MCI—0M0) 4. ‘3+l _-: 2+ “/4 >0 {Mal-1‘0 80': some [M- ‘Eflfltl bra. IVT [2] b) Does the error bound for the Method of Bisection guarantee that 7 steps of this method will give the value of c in a) accurate to two decimal places (i.e., with error less than 0.01)? Explain your answer. 2mm. bemol: E 5 5-6. “$17“ an 55-95 ’SlEnTLlwfl bull-la " 2" mammal) [61,5]. OMB-3: “lgll, :r. 2 _ f l: V1. ?‘=> Ekg ;=j:_fi7 o,o\r.‘-l5_—o 3° “0» BOWUJOUAGL lacou-n To dc 8 9.5.9:. To gaff Engaoi [3] c) Use the linear approximation LOCI) to find an estimate of f(0i1). 08°60: £(ol+ §I(o)('x‘ol flame—ha +3,“ =9 $103) = l xfi 3 Sm'nu. Ho) = achmnO *- 0+ \=\ s Mk” LOCI): l l‘ DC [1] (‘1) Explain why f has an inverse f '1 on IR. - I 7. Slhu §C¥l= mil-3" >0 MIR) 9.}, ”ohms cumin mule m) ‘1 $0.0 cun‘UV'LU-{MQ- [3] e) Show that the tangent line to y = J1"1 [$10) at :r = 1 is parallel to the tangent line to y = f(:r) at :c = 0 SLBU. ggCol=l> W4 pop/Cl (0,15 cm 3=§C0 (mpmobb l’o (LOX can 5: g—‘(KL F Flam ‘ l: 4—“ l, r m ‘" — \ m , Ho £4,” 1 NVDR ( 10 (i) 3'70)». Swim. %t(o)=\, So 0.9.00 QAYI) = l, MA Math. 1100 W mu. \PMW' IVIATH 137 ~ FALL 2007 TERM TEST #2 PAGE 5 [2] [5] [2] [3] -r 'T 4, For the function f(1) = Itanm on (—1— 1—) : ‘2' 2 a) Find lim+f(:r) and 1in_ fix). z—.— 1—»— ? 2 QAKa—Ji‘k I Tam'X-w‘ob 1 So QQ3+3CV¢MX—u+.a§ Xé- ‘ 2. (Smut-140) 0J9 %-§‘I-f -, Tan/xx do iuim 2 '—’+ , SO 'xafl'”«t(—MK ___' +‘w 7. b) Find f"(r). and then show that f"(x) = 2 sec2 $0 + mtan at). / $00 = famrx + a: Aeczx (I {C70= acax i- ucqw + x.2M¢x.m.co<Ta—m1< : amqecCl—i-x'i—wnx7 c) Sketch y = z and y = tan 1 on the left axes below, and y = sec :5 on the right. ‘5 d) Use your graphs in c) to determine the intervals on which f’(a:) > 0 and on which f’(a") < 0: 7r 7r and to Show that f"(:1:) _>_ 2 for all x E (—3, 3). 1' Sum Ydmx <c> amok «mix <o gnu-géx 40 ) §6<§<o. H H >0 U “ >0 u O<3L4 {[72 \ §ICX) >0. Smut 0(me >0 an 13% Lute/«pain, amdis 00306-0) m w 3—”00: 2M0°x€l+xtfim90 >2“.sz >, 2 v e) Use the results of a.)~d) to sketch the graph of y = f ((13), indicating any local extremes, points of inflection, or asymptotes. ‘5 MATH 137 ~ FALL 200'? TERM TEST #2 PAGE 6 5. Label each statement as TRUE or FALSE in the blank provided (Use the space between questions to justify your answer Guesses will not be graded] [2] a) FD—F‘T The function f(rc) : (a: — 1)”3 has a vertical tangent line at (r = 1‘ F T- 5 S‘mo ( _ Va . 'T . Fm. 3 'x l) ,meu W‘fi _ ‘ fgimfi r' ”$100.4: +0: aox-l+ i K “COM. X l _. Wu“ \ T 0er [OOH—no awn—7| Shmacwp 09—“ [.‘C’Ck/Q g?! [2] b) If f” exists and y : f(a:) is concave up on R, then so is y = elm. O-t‘ ’K a s M‘l“ ‘ ()0 [K‘- W' [8: 8‘9 =5 3’=e&(*3' I1 £6) 1 2 (X H ‘3: e «(5433) + e§)§{)— [2] c) L The derivative f’(0) does not exist if f(m) = mlx] 5,70) = 'QLIWI» Ilil " O X-DO ,x «D ’Xao 1’- O [3] d) T The function ftp) = arcsinr + \/l‘2 — 1 is not differentiable for any real 3?. gm 1 AHA + ”24:.“ W 5N xii ~ L/V‘J W $0 “A; [Dunn LA Mifim 84“, MM (gov no‘l' Man-1.9L 34m. _.[<x<l lxl>l WM’Z. [3] e) F If f and g are both defined on the intervals [—1,0) and [0t 1]. and ling) [(1) = 0 ""°“ 1133“” "9“") 2 0‘ mm gen Cc: um to. agent . ‘Xao _— _L -'-> + 00 an cc o is . tau} K? —'I . <. ;[ M mesa gm 7 9‘) mm m swede) 3 2m mi. ’X—9 o . )4" 0 9(3 :: [LAM J... F ‘1' Mac x3 DNE [3] f) 3;“ If f’(J:) = I": a constant, for all x 2 0, then the Mean Value Theorem predicts F07. F that f is a linear function on a: 2 0, 002* 9h in. W pas-Jinnah.- MM mvr ==> ~§CV<J«§C® 3 ngcl =[4— , meg.) ___.____.————o sac-.0 [(10)— 3403 = lax. ‘ Le. gm): 6m, +— 5(0),}57 ““3 79m; went} .5 ...
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