2311_MG1_AlgTrig_Review_solutions_F11

2311_MG1_AlgTrig_Review_solutions_F11 - MACZ311 MGl...

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Unformatted text preview: MACZ311 MGl Algebra/Trig Review Name: Sotu’l“ DV‘5 Fall 2011 Section: (circleyours) MWF 9:00 MWF 10:00 . Directions: Use all available resources, including the math lab, Tcague’s office hours, and fellow students. To justify answers and earn partial credit, show work in the spaces provided on this document. MCI is due at the start of class on Tuesday 8/30/11, and the ‘flate watch” starts at the start-of-class deadline. Papers submitted less than 24 hours late are penalized one letter grade (10%), and papers are not acceptable more than 24 hours late. E. An open box with square base x meters on each side must have volume 40 1213 (cubic meters). a) Find a function formula for S(x) , the single-variable function that expresses surface as a function of base dimension x. a”; Voiume V = X‘X'ln “Ni surface Ate“ = lPx‘L‘ Jr x'x 7.. qo=xzk (Azl-t-K- Effie-X h= L‘°/><‘ «(.0 x: Answer: S(x)= T + b) Suppose material for the sides of this box costs $3 per square meter, and material for the bottom costs $5 per square meter. Find a function formula‘for C (x) , the single-variable function that expresses total cost of materials as a function of be e dimension x. f L if To'l‘o.‘ Mo'l'eria‘ Cast" a: ‘3 3° Ctr-ea :5 5.4595 + ($. arcane :2 am use 5 “a: 3-1§Q+ 50‘?" Answer: C(x)= x + X 2. Recall that the standard equation fora circle of radius r centered at (hit) is given by 7_ L A 2 2 2 . l , 1+6... 0) == '3 (lfif’l) +(y—k) =7‘ . Full Cure e- (X‘O') xi + Y1. = C] a) Write the equation for the full circle centered at the origin with radius 3. Y1- : Cf..)('z‘ 7— '2.— Answer: X. +2 zcl / Y": Lt oi”X2' b) Let y = s(x) = “09 —— x2 . Solve the eguation in part a} for y to Show that the graph of s is the non—positive semi—circle centered at the origin with radius 3. Sketch s and state its domain and or ice mamifi‘fev- ‘ : Domain: [’3 3] of {xi ’3éxé’ag “.4 H. Rangeimw l ‘3‘; 3 5 a} ‘ 'i ‘ a m AgntS(CLtO.lce+0 LL59. ' (1+; Mtge“ or seiuhu‘ilder no‘l‘od'io‘fl 'unkhr oil no i 0) Complete the formula for the piecewise—defined function shown. The left and right pieces are linear and the middle piece is the semi—circle of radius 2 centered at the origin. ‘L. .__ 7— ' 3. Consider r(x) = xfi 3 . a) Rewrite r in radical form. Answer: r x. = ---=-'" z 2. b) Each of the non—zero x-values below produces a rational number output. Complete the table without using technology, expressing function values in exact form. Sketch the graph. Ijo_te; It will be virtually impossible to make a hand-drawn sketch 1‘0 scale, so don’t try to do this... just sketch the graph without establishing fixed scales on either axis. 6) State the domain and range of 1'. domain: ix! x ¢ 0‘ I range: l, S i 9 > o} _ Q—oo) 0) U (oft-0°) Col-too) a) b) Conclusion: lim f = lim f ( x): C) d) x~iniarmfh onur‘ wiun Yto __, 3x+t x 0:3X'l’l x=~ Consider f = 3X x 3x+1. $00 = + for xii). X. Simplify the formula for f by carrying out the. division. Answer: f z 3 “i” As x tends toward +co and —oo , the function values tend toward a specific numerical value, called a “limit”. Determine this limit by using technology to examine numerical tables and the graph of f . The result should give information about an asymptote in the graph. 3 , so the graph has l1°YlEOniEal asymptote {'1 3 . ( horizontal or vertical ‘2 ) .T*>+D'J x-—)—DD ( equation ) As x tends toward 0 from the left (negative) side of 0, the function values again get infinitely large. numerically. Determine whether the function values tend toward +00 or woo . AS x tends toward 0 from the right (positive) side of 0, the function values get infinitely large numerically. Determine whether the function values tend toward +00 or —oe. 00 Answer: iim f (x) 2 "1'50 x—> 0+ Answer: lim f = .14) 0" asymptote x: O . ( equation ) Conclusion: The graph has VQY‘Jfl all (horizontal or vertical ? ) Sketch the graph of f . If the graph is asy ptotic to a line other than a coordinate axis, sketch this asymptote using a dashed line, and 1a e he asymptote with its linear equation. Also label the x—intercept. .L 5 f(x+h)—f(x)- _ 5. Definition: For function f , the difference quotient at input x is given by I 2 a) Let f = 3 m x2 . Setup the difference quotient and simplify it completely. _Fgr fig; ‘Rmc’i‘im 'F) £(Mk) "‘ PM = inf-fix \r\ b) Let f = . Set up the difference quotient at x = 2 and Show that it simplifies to — 2: h Forfl‘gmcifion 'F anci x27.) 3 8 4(Z+\\\”f(23 __ 2+1“ 1 k 7 in I 6. Without using technology, fifid the exact value of each expression or write “undefined”. Express - answers for pans 10—11) as exact radian measures. ‘ ,- . - ‘ Ver‘l‘lt‘ao J: “fidefifled ( as: m‘fi‘o‘l‘e} “ I . - " . i. 37: -___I See Un‘c‘l c‘rrcle cos[?]— - Z ‘ ‘ 1) 31116;}— or an)?“ C) .tan[27r]: 5}“ 2%) = :2 .— 3 -- j) 36%”): uhAegv‘eA. (Ver'l‘lta-D ) fl 2 a) sin[2§£]: _- I h), tan[ l’drde. oil 2% —3_ cos (1%) — i 7 “awhjte a = 3 mm “72. * 1-; or, a] q; . :— V‘ 1) cos'1 —1 = 2"— \o mu 211- -- .ml— e) ese[2—fl)= *; =3 "L": 2 .3 56’ cos 2“. -3 , 6142"?!) 5/2. [3' 3 . _ and 3315 'm [out] fl eot Ezr— =' (05673) I {9; m) tan—3(1): E laemuse +&“%- 3 land 3 '— I ' 11. TI" .-' ' '1:- _ E 15 \v‘ E- /2 , 2_ 2. u 243: t 2 g) COS(7r)= ....I . a I 11) sec 1[__~3 )2 Eil- lnzcause. secigs -— TE ' ' ‘ . I h .‘ 11— See (ml-l- cw: le or’ jraeln (meg is m [OJ1T1’av‘d‘t'é ,7: 7. a) Sketch at least two periods of the ' b) Sketch the function y = 23in‘E x. Label the funCtion y = 008616) ' Label enough “Ck ' endpoints with exact x~ and y-coordinates, marks“ eaCh ads to indicate their _ and label enough tick marks on each axis to r Scales" . _ “211- 111” indicate their scales.- (,0; (Ex) has Period -— *. Here) 3— _ 8. Solve each equation, first over [0,271) , and then over (—00,+00) . Express answers in exact radian- measure form. When solving over [0, 27:?) you will, of course, get a finite number of solutions. The solutions over (—0o,+00) include all solutions over [0,27r) and all angles coterrninal with these, so ‘F n *7 this will be an infinite set. / “5e dqulale- omjle Ede a) Zoom—1:0 ' b) sin(2x):cosx I ZCOSX :l 25““x cosx :: (05" 055K : é , Pass-ride "In QI, 0N 25inxcosX-COSX 4": g x = If 5.71- omcl all (ole'fminal M‘le C°5K (zsw‘yfl i)” [5 o 3 , S , ' ' ,0 25'“X” . _ cosx '— _ $— __ y _ I $11: ‘hxi‘: 7.- fl- X‘= > .. 2- 7—- X ’- 6 ) 5: Answers: .‘TT . 11- 11— 5". 3-h- Over [0,27r): /3 - ’ 3 Over [0,275): "a; J 3.— , 7: ’ E If 5"” L“ 5.1: :1: Over (—co,+o:>): + in“) T + awn-i Over (—00,+00): to + 2h“) (a + 2W") 2. + NF h anq tutejer‘ ‘ I h any .mtejer . 9. Consider f (x) : xefx. a) Simplify the function formula to express f without the negative exponent, and state the domain. ‘ ' ‘ K Answer: f(x)= .25;- domain: (~00J-Ho) he cause 6 35 0 list”. a“ x ' e . b) As it tends toward +00 , the function y m e" grows much faster than y = x . Consequently, the function values for f tend toward 0 asxtends toward +00. That is, liin f (x) = O .‘Cfi‘Hfl 10. Consider f (x) mi. 1 v a) Statethedomariri Answer: W U (“+09 . b) As x tends toward +00 , the function y = x grows much faster than y 2 In x . Consequently, the function values for f tend toward i (D as x tends toward +00. That is, lim f = 'l' 90 xa+0o I c) The function f'(x):lnx”1 (In that satisfy f’(x) m 0. Show the work. is the derivative function for f . Find the exact values of x, if any, (\“KY— ' [\nK 7" i j: Answer: x= I e eltfix _____ I X =6 ...
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This note was uploaded on 12/12/2011 for the course MAC 2311 taught by Professor Teague during the Spring '11 term at Santa Fe College.

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2311_MG1_AlgTrig_Review_solutions_F11 - MACZ311 MGl...

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