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# Algebra And Trig II MA 154

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This 0 page Class Notes was uploaded by Fiona Sipes on Sunday November 1, 2015. The Class Notes belongs to MA 154 at Indiana University Purdue University - Fort Wayne taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/233539/ma-154-indiana-university-purdue-university-fort-wayne in Mathematics (M) at Indiana University Purdue University - Fort Wayne.

## Reviews for Algebra And Trig II

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Date Created: 11/01/15

Review for the MA 154 Final Fall 2008 Date 1 Write each ofthe complex numbers 2122 and z in the form 2 re rcos 6isin6 Write 24 and z in rectangular form 2 x yi Exact please Plot and label each number on the complex plane a 21 4 39 C Z 3i d 24 32quotquot 3cos71r i3sin l e 25 e zm xEcoszTMri 2sin37 225 751 Perform the complex number arithmetic and writeyour answer in rectangular formx yi a 24 2i 43 7i b 203 in e lt1 i c i l d 2275n2l Room Eliminate the parameter and write a formula which only involves x and y Give the domain and range 2 a x2costy73cost3 b x4costy4cos t xcos2t Eliminate the parameter and write a formula which only involves x an y axt71ye bxe 5 ye os Hint Consider a double angle idmtity Write implicit formulas for each ofthe conics graphed below Give the center focal points exact any run r parametric formulas as well For part d report the equations of the asymptotes a C g ive 6 Sketch a graph of each ofthe conics Report the vertices or vertex and focal points and plot them For parts a d give the center For parts e and I give the equation ofthe directrix 2 2 2 2 ay1 b amp1 4 16 25 9 2 2 2 2 cyx1 dampamp1 25 100 1 4 e yz8x f x2riy1 7 A parabola has its vertex at 0 0 and its focus at 0 3 Give its equation 8 The ellipse shown has focal points at 1 0 and 9 0 Give its equation 9 For Question 9 a day is a 24 hour period beginning at midnight Two species of plant A and B propagate by dispersing their seeds in the wind during the height of summer On a typical day in the height of summer the wind speed wt measured in miles per hour thours a er midnight is given by the formulawt 76 cos 15 a Species A favors propagation in high winds andwill only release its seeds if the wind speed is no less than twenty miles per hour In the figure the graph ofwt is given and the time interval over which species Awill release its seeds is also indicated Complete the figure by calculating the values of the two endpoints of the interval 0 it b The seeds of species B are destroyed in high winds and hence species B has a seed release m chanism which only releases seeds ifthe wind speed is no greater than ten miles per hour For how many hours on a typical day in high summer will species B be releasing seeds 10 Find a possible formula for each a b Additional Problems from Text Section 66 17 19 23 Section 67 37 43 51 Ch 6 Tools 1 3 23 Chapter 6 Review 3 15 4347 odd 61 Section 71 5 11 17 39 Section 72 3 5 1741 45 Section 76 114 1725 Chapter 7 Review 3 5 7 9 Section 101 5 12 Section 102 3 13 14 22 Section 103 12 15 Section 104 1 25 Chapter 10 Review 5 1129 35 Section 111 725 odd 3033 Section 112 3 13 19 23 35 41 Section 113 115 odd Section 114 19 odd 19 Chapter 11 Review 1 3 5 7 9 10 Section 121 111 odd 28 Section 122 113a15 17 Section 123 1 3 17a Section 124 1 3 11a Section 125 123 odd 25 26 27 30 31 Chapter 12 Review 5 9 1719 28 a b C d b 0 Fr 25 e m 56 142 71 139 Answers to Review 21 4 4cos7r i4sin7r 48 z 5 7515450 cos7T i 50sin7T 39Oe 7m4 a 2 23 313cos13s1n 38Wr Z4 387m 3cos77r i3sin77r 371 30 73 242i737i84i73i55i 2i37i6i7212 6i 72716i 226i 22zm lt4gtul 1 1 392 7 fl gem2 210equotlt5 2gt10 210equotlt25 gt 210cos7257r isin7257r 21071 71024 To find 1 i14 write 1 iin the form re g The angle 0 714 and distance r 12 so 1 1 Egan4 Then ex7r4l4 JE14 ex7r414 21214e177r2 27ex77r27 274 7128 y 7x 3 Since x 2cos I y 73 cosI 3 solve for cos I soy 73cosI373 339 3 Therefore cos I g Using a grapher we can see that when I 0 the curve starts at the endpoint 2 0 travels on a line to the endpoint 72 6 and repeats back Note that the domain is restricted to 72 S x S 2 since 72 S 2cos I S 2 and the range is restricted to 0 S y S 6 since 0 S 73cosI 3 S 6 393 i i 4 y 4x 7 42 Solve for cos I to get cos I x7 4 and substitute therefore y 4cos I2 4x7 42 which is a parabola Using a grapher we can see that when I 0 the curve starts at the endpoint 5 4 travels on the parabola to the endpoint 3 4 and repeats back The domain is 73 S x S 5 and the range is 0 S y S 4 x yZ 71 We must write cos 2I in terms of cos I which can be done if we use the double angle identity cos 2I 2cos2 I71 Solving y 2cosI for cos I we have cosI and substituting gives x 2Z fl Simplifying gives x 2y4 2 fl 471 The domainis fl SxSl andtherange is 72SyS2 4 Eliminate the parameter and Write a formula Which only involves x and y aye b xJ or yx2xgt0 since xe 5 yer and ce 5yIS 57 0 m c m m since tx1andye e J 2 1 center 0 0 Vertices r 8 0 The focal points are c unis from the center Where C2 64 716 48 so 6 m z 693 So the focal poinw are ixE 0 or r 693 0 Parametric x 8cos ty 4sin t x7 32 y 42 9 center 3 74 There are no Vertices or focal points this is a circle Parametric x 3cos t 3y 3sin 7 4 is 1 center 73 6 Vertices 73 12 and 73 0 We have C2 36 7 9 27 so 6 m z 520 and the focal poinw are on the major axis at 73 6ix or approximately at 73 080 and 73 1120 Parametric x 3cos 7 3y 6sin 6 7 44 1 center 74 4 Vertices 74 1 and 47 The focal poinw are 6 units from the center Where C2 9 4 13 so 6 J z 361 So the focal poinw are 74 445 or 74 039 or 74 761 Asymptotes y7 4 ix4 ory15x10y715x7 2 f 64 b center 1 4 Vertices 6 74 74 74 focal points 73 74 and 5 74 We have c 42579 4 so at 1i474 center 0 0 Vertices 0 4 0 4 focal points 0r center 0 0 asymptotes y ilinxory ix Vertices 0 5 0 75 focal points mfgE or 0 1118 Va center 72 5 asymptotes y i2 Vertices 71 5 74 5 focal points 7 2i 5 5 or 424 5 024 5 p o e The vertex rs atthe ohgh Smceyxs ofsecond degree we expeet the parabolato have a hohzohtal axrs of symmetry The equatroh y 4pxhas afocusp uhrts from the vertex The graph of y xr hasp 2 so the focus rs at 2 0 The dArecmx rs the vertical hhe x 2 whreh rs 2 from the vertex f The vertex rs at 0 71 Smcexxs ofsecond degree e expeet the parabolato have avemcal axrs of symmetry 2 Here 433 so p i srrhrlar to y x m T eus 15 p r uhrts from the vertex on the WW vemcal axrs or at 0 7146 or 0 9 The duecmx 15 p 46 uhrts from the vertex so 39D 5 rt has the equanony K 771516 7 The focus 0 3 rs oh the axrs of symmetry vertrea1 so the parabola has the equatroh x2 4py wrthp so 2 12y 5 8 eehter 5 0 RUN5 s4 72 Smce 2 RUN2 7021515 52 7RISE2 2 e can solve for the RISE 4 52 7021515 RISE2 52 74 25716 9 Therefore RISE 3 7 2 2 x 5 y 1 25 9 9 a t 9 763 tl 14 237 Solve graphreany 19 526 hours Fmdwhere wt10 Thrs rs tA 2 237ahd t2 21 7635012 72 o 19 52s 10 a y4srhTx b y225m27x szizw fur the MA 154 Final MondayDecember152003100 e 3 00 p m 1 Flnd polar coordlnates ofthe polnts wlth me followlng Carteslan coordlnates a 5 0 b 5 5 a 353 b EM4 4 a Ellmlnate the parameter and wme an expllclt formulafor the curve Keen yee oszgl Flndthe endpolnts ofthe curve 5 Wme parameme equaaons forthe quarter ofan elllpse centered at 0 0 starting at 0 o and endlng at 730 5 Wme implicit formulas for eael of me eomes graphed below 73 17 339 U 7 Sketch a graph of each of the conics Report the vertices 2 2 2 2 aLL1 h 4 l6 9 25 2 2 2 2 CLL1 damp l 25 100 9 4 8 For Question 8 a day is a 24 hour period beginning at midni ht di g Two species of plant A and B propagate by spersing their seeds in the Wind during the height of summer On a typical day in the height of summer the Wind speed wt measured in miles per hour t hours a er is given by the formula wt 76cos 15 a 5 pts Species A favors propagation in high Win will o y release its seeds ifthe Wind speed is no less than twenty miles per hour In the gure the gmph of wt is given and the time interval over Which species A will release its seeds is also indicated Complete the gure by calculating the values of the tWo endpoints of the interval 0 fl f2 b 5 pts The seeds of species B are destroyed inhigh Winds and hence species B has a seed release mechanism Which only releases seeds if the Wind speed is no greater than ten miles per hour For hoW many hours on a typical day in high summer Will species B be releasing seeds 9 Find a possible formula for each a b a x3cos Paramemc equation Cartesian equation x2 y2 25 Answers to Remew b xJEosltirgtlt Egtlt7 y3sm y 25n E 5costy55mt 4 a y b 1 1 and 4 g aremanypossxbleanswers For example x3costy65mt ggzgn Anotherpossxbleanswensx 3cos2y smltz 031g x1 y LX811 Um w af r il bymlc 71 u ucnm LUJH b tamer H h H me H 4LN H CHKECK H H A h 3 2 4 z 3 a cum mm a mum m 5 5y L Hltck H 5 V75 mum LUV 703 u H 27 NM aplxlcuHH n Tim o houh hm hm m m n h Additional Problems from Text answers in back Section 62 17 21 Section 63 13 15 23 Section 64 15 Section 66 1 13 odd Section 67 5 7 17 23 31 Chapter 6 Review 29 39 41 47 Section 71 3 7 Section 72 23 27 Section 73 9 25 27 Section 74 15 Section 76 1 3 5 Chapter 7 Review 9 11 Section 82 11 53 Chapter 8 Review 1 5 7 43 49 51 Section 101 5 11 Section 102 1 7 11 13 17 Section 103 1 Section 104 3 Chapter 10 Review 11 and ShOW that the angle between any pair of vectors is 60 Section 111 1 7 10 Section 112 17 odd 11 Section 113 5 11 25 Section 114 1a 2a 3a 9111315 21 23 Section 115 121 odd Chapter 11 Review 13 5 15 17 21 23 25

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