{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Title " 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 258 "" 0 "" {TEXT -1 25 "Lesson 27: Polar Graphs\n" }}}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 4 "" 0 "" {TEXT -1 23 "A. Cardioid s & Limacons" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________ __________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We're going to lo ok at a variety of cardioids, which are graph of the form" }}{PARA 0 " " 0 "" {TEXT -1 30 " y = a +- " }{XPPEDIT 18 0 "b* sin(theta)" "6#*&%\"bG\"\"\"-%$sinG6#%&thetaGF%" }{TEXT -1 11 " or y= a +- " }{XPPEDIT 18 0 "b* cos(theta)" "6#*&%\"bG\"\"\"-%$cosG6#%&theta GF%" }}{PARA 0 "" 0 "" {TEXT -1 97 "and see how the relationship among the components effects the graph.\n\n COMPARING a AND b" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "In parti cular, there are three cases : |a| = |b|. |a| > |b|, and |a| < |b|. Ea ch of these cases creates a distinctive version of the limacon." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "When |a| \+ = |b|, the graph passes through the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "This shape is known a cardioid , or heart shaped curve. Note the reference circles of radius 1 and 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "polarplot( \{1,2, 1+sin(theta)\}, theta = 0..2*Pi, scaling = con strained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "When |a| = |b|, the graph maintains some distance between it a nd the origin, resulting in a rounder, puffier plot." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "polarplot( \{1,3,5, 3+2*sin(theta)\},theta = 0..2*Pi, scaling = constrained);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "When |a | < |b|, the graph not only passes through the origin, but also part o f it folds inside itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "polarplot(\{2,3,8, 3+5*sin(theta)\} ,theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "To see all of these varieties i n one glance, execute the next block of commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "This shape is known a ca rdioid, or heart shaped curve. Note the reference circles of radius 1 \+ and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "display( polarplot( 8 + 8*cos(theta) , theta = 0..2* Pi, scaling = constrained, color = green, thickness = 3), polarplot(\{ 8 + a*cos(theta) $ a = 9..15\}, theta = 0..2*Pi, color = blue), polarp lot(\{ 8 + a*cos(theta) $ a = 1..7\}, theta = 0..2*Pi, color = red)); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 23 "CH OICE OF TRIG FUNCTION" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "There are four variations iin the format : sine, cosi ne, -sine, and -cosine. How does the choice of one of these effect the graph? Lets take a look at all four at once!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Can you decide which gra ph belongs to which? Think about what values of theta make the sine an d cosine maxima!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 124 "polarplot(\{ 8 + 7*sin(theta), 8 + 7*cos(thet a), 8 - 7*sin(theta), 8 -7*cos(theta)\}, theta = 0..2*Pi, scaling = co nstrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "polarplot( 1 0 + sin(2*Pi*theta), theta = 0..20*Pi, color = coral, \n scaling = con strained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________ ____________________________________________________" }}{PARA 4 "" 0 " " {TEXT -1 18 "B. The Rose Garden" }}{PARA 0 "" 0 "" {TEXT -1 83 "____ ______________________________________________________________________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "We're going to look polar functio ns of the form f = a sin(n ) and r = a cos(n ) which are sometimes cal led multi-petaled roses." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 26 "EVEN AND ODD NUMBER PETALS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The first distinction to \+ be made is between when n is an even or odd number." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "When n is an odd number , the resulting rose has exactly n petals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( \{9, 9*sin (5* theta)\}, theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "However, when n is e ven, the rose has 2n petals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( \{5, 5*sin(6*thet a)\} , theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Try creating some other \+ roses on your own with different numbers of petals to verify that the \+ even/odd relationship holds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "What about a single-petaled rose?" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Do you recognize t he inner shaped of the \"single petaled rose\"?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "polarplot( \+ \{9, 9*sin(theta)\}, theta = 0..2*Pi, scaling = constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 15 "SINE AN D COSINE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Although sin(x) and cos(x) will create an n-petaled roses inscrib ed in the unit circle, what is the difference between them?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The graph with the sine appears tangent to the positive x axis, while the cosine ver sion has a petal centered at the positive x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "polarplot( \+ \{sin(3*theta), cos(3*theta)\}, theta = 0..2*Pi, scaling = constrained );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "He re is an illustration of the same idea with even more petals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "polarplot(\{sin(6*theta),cos(6*theta)\}, theta = 0..2*Pi, scaling \+ = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "AMPLITUDE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "In the formula above, how does the number a, which is the amplitude in effect the graph? Here we let a =1,2,3...,12 and see how the resulting graphs look" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 113 "Each different color is a different grap h. You can see that they are inscribed in circles of radius 1,2,3,..., 12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "polarplot( \{a*cos(6*theta) $ a = 1..12\}, theta = 0. .2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 19 "C . Valentine Curves" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________________ ________________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Valentine curves - there is really no such nam e but it seemed reasonable when you take a hybrid of rings, hearts(car dioids), and flowers(roses)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "polarplot( 4 + cos(6*theta) \+ , theta = 0..2*Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 71 "polarplot( 4 + 3*sin(7*theta), theta = 0..2*Pi, sca ling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "This one wraps in on itself" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "polarplot( 3 + 7*s in(3*theta), theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Here are whole famil ies of similar curves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "polarplot( \{ 6 + a*cos(6*theta) $ a = 1. .11\}, theta = 0..2*Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "polarplot( \{12 + a*sin(7*theta) $ a = 1..12\} , theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________________ ______________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 42 "D. Familiar Shapes Disguised In Polar Form" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________________________________ _______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "There ar e many familiar shapes such as lines, circles, parabolas, and ellipses which can be expressed in polar form." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 166 "In polar coordinates, the simplest f unction for r is r = constant, which makes a circle centered at the or igin. Lets look at the graphs of r = 1, r = 2, ... , r = 20." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "This draws conc entric circles of radius 1,2,...,20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "polarplot( \{k $ k = 1..20 \}, theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We can also draw circles \+ not centered at the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "polarplot( cos(theta), theta = 0..2 *Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "polarplot( cos(theta - Pi/4), theta = 0..2*Pi, scaling = constra ined);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "...and ellipses and parabolas...." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( 1/(8 - 7*cos( theta)), theta = 0..2*Pi, scaling = constrained);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "polar plot( 1/(1 - cos(theta)), theta = 0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( 1/(3 + 2*sin(theta)), theta = 0..2*Pi, \+ scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "...even horizontal and vertical lines" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "polar plot( 2*csc(theta), theta = -2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "polarplot(2*sec(theta), theta = -2*Pi..2*Pi);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 19 "E . Spiraling Graphs" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________________ ________________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "A basic spiral is of the form r = theta." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "polarplot(theta,theta = 0..4*Pi, scaling = constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "polarplot(theta, theta = 0.. 40*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 89 "Again, a larger range of values for theta gives more chance for the graph to wrap around." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "Even more interesting gr aphs can be created using the product of theta and a trigonometric fun ction. As theta increases there is some sort of spiraling effect." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "polarplot( theta*sin(theta), theta = 0..3*Pi, scaling = constraine d);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "polarplot( theta*sin (theta), theta = 0..100*Pi, scaling = constrained);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "As we increase the rang e of values for theta, we get even more of the same." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Here is another variati on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "polarplot( 2*cos(theta) + sqrt( abs( 4*cos(theta)^2 \+ -3)), theta = 0..2*Pi, scaling = constrained, numpoints = 1000);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________ ____________________________________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 29 "F. How To Build A Better Rose" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "The so-called 'roses' above, really bo re more of a resemblance to daisies. Here is something that looks a li ttle more rose-like." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "polarplot( theta + 2*sin(2*Pi*theta), the ta = 0..12*Pi,color = red, thickness = 2 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Here are some other beautiful b otanicals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 101 "polarplot( theta + 3*sin(4*theta) - 5*cos(4*theta) , theta = 0..12*Pi,color = violet, thickness = 2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "polarplot( theta + 2*sin(2*Pi*theta) + 4 *cos(2*Pi*theta), theta = 0..12*Pi,color = green, thickness = 2 , nump oints = 1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "polarplo t( 2*cos(theta) + sqrt( abs( 4*cos(theta)^2 -3)), theta = 0..2*Pi,scal ing = constrained, numpoints = 1000 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "polarplot( cos(.95*theta), theta = 0..40*Pi,scaling = constrained, color = brown);" }}}}{MARK "0 0 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }