{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 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helv etica" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 259 "" 0 "" {TEXT -1 25 "Level Curves/Contour Maps" }} {PARA 4 "" 0 "" {TEXT -1 20 "Calculus III Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 21 " \+ Required Information" }}{PARA 0 "" 0 "" {TEXT -1 69 " In this project \+ we learn about the contour maps of various surfaces." }}{PARA 0 "" 0 " " {TEXT -1 108 " The cross-section of the surface z = f(x, y) with \+ a plane parallel to xy-plane, z = k, a constant, is a " }}{PARA 0 "" 0 "" {TEXT -1 64 "plane curve. The curve obtained for one value of k \+ is called a " }{TEXT 259 11 "level curve" }{TEXT -1 26 ". Every poin t on a given" }}{PARA 0 "" 0 "" {TEXT -1 78 "level curve is at the sam e height/depth on the surface from the xy-plane. A " }{TEXT 260 11 " contour map" }{TEXT -1 13 " (or contour " }}{PARA 0 "" 0 "" {TEXT -1 85 "diagram) consists of several level curves, f(x, y) = k, projected on the xy-plane. " }}{PARA 0 "" 0 "" {TEXT -1 107 " As k incre ases the height of the cross-section of z = f(x, y) increases; thus th e contour map gives " }}{PARA 0 "" 0 "" {TEXT -1 107 "an idea of the n ature of the surface as the height or depth of the surface increases. \+ Therefore it is easy " }}{PARA 0 "" 0 "" {TEXT -1 47 "to visualize a s urface using its contour map. " }}{PARA 0 "" 0 "" {TEXT -1 94 " If the surface is a plane parallel to the xy-plane, the contour map is \+ the whole plane. " }}{PARA 0 "" 0 "" {TEXT -1 106 " If the surface is a plane perpendicular to the xy-plane, the contour map is one lin e where the plane " }}{PARA 0 "" 0 "" {TEXT -1 64 "(if necessary by ex tending its graph) intersects the xy-plane. " }}{PARA 0 "" 0 "" {TEXT -1 113 " If the surface is a plane inclined to the xy-plane, the cross-sections are lines for different values of k. " }}{PARA 0 " " 0 "" {TEXT -1 53 "In fact, the contour map is a set of parallel line s. " }}{PARA 0 "" 0 "" {TEXT -1 110 " If the surface is not a plane, the cross-sections are curves for different values of k. Hence the c ontour " }}{PARA 0 "" 0 "" {TEXT -1 26 "map is a set of curves. " }} {PARA 0 "" 0 "" {TEXT -1 113 " If k increases in equal increments, a nd the level curves are equidistant parallel lines then the surface is a " }}{PARA 0 "" 0 "" {TEXT -1 114 "plane; if the level curves are f ar apart then the surfaces rises or falls slowly; if the level curves \+ are dense, " }}{PARA 0 "" 0 "" {TEXT -1 41 "then the surface rises or \+ falls faster. " }}{PARA 0 "" 0 "" {TEXT -1 98 " The contour maps are \+ used by geologists, meteorologists, and surveyors. The mountains and o ceans " }}{PARA 0 "" 0 "" {TEXT -1 35 "are mapped using the contour ma ps. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 261 15 " Solved Examp le" }}{PARA 0 "" 0 "" {TEXT -1 84 " For each of the following equation s: 1. Draw the surface. 2. Describe the contour" }}{PARA 0 "" 0 "" {TEXT -1 101 "map algebraically. 3. Draw the contour map. 4. Descri be in words the level curves and how they are" }}{PARA 0 "" 0 "" {TEXT -1 8 "placed. " }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ (a) f(x, y) = 4" }}{PARA 0 "" 0 "" {TEXT -1 49 " \+ (b) f(x, y) = x + y" }}{PARA 0 "" 0 "" {TEXT -1 44 " (c) f(x, y) = " }{XPPEDIT 18 0 "x^2- y^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 45 " (d) f(x, y) = " }{XPPEDIT 18 0 " e^xy;" "6#)%\"eG%#xyG" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 65 " (e) f(x, y) = (x + y)/(x - y) \+ " }}{PARA 0 "" 0 "" {TEXT -1 7322 " (f) f (x, y) = y - cos(x) \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 " (g) f(x, y) = " }{XPPEDIT 18 0 "exp (1/(x^2+y^2));" "6#-%$expG6#*&\"\"\"F',&*$%\"xG\"\"#F'*$%\"yGF+F'!\"\" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 262 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n " }}}{PARA 0 "" 0 "" {TEXT -1 103 "Plotting z = f(x, y) can be done b y plot3d command, giving ranges for x and y. Note how we can title " }}{PARA 0 "" 0 "" {TEXT -1 99 "the diagram. Also when the surface is d rawn you can change the axes and drawing using the menu for " }}{PARA 0 "" 0 "" {TEXT -1 103 "the plot. You can grab the surface at any poi nt by the mouse and rotate the graph for a better view. 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The curves are rectangular hyperbolas for k > 1. " }}{PARA 0 "" 0 " " {TEXT -1 105 "For k = 1, the curve is xy = 0, i.e. the x and y axis. For 0 < k < 1, the curve is a hyperbola in the " }}{PARA 0 "" 0 " " {TEXT -1 30 "second and fourth quadrants. " }}{PARA 0 "" 0 "" {TEXT -1 104 " When x, y are both positive or both negative, the cur ves are rectangular hyperbolas in the first and " }}{PARA 0 "" 0 "" {TEXT -1 94 "third quadrants. When x and y have different signs, the curves are in the second and fourth " }}{PARA 0 "" 0 "" {TEXT -1 101 "quadrant. For this surface, the height, z, keeps increasing when bo th x, y have the same signs and " }}{PARA 0 "" 0 "" {TEXT -1 95 "their magnitude keeps increasing; the level curves are closer means the su rface rises rapidly." }}{PARA 0 "" 0 "" {TEXT -1 99 " The curves are far apart in the second and fourth quadrants. In other words, when x and y have " }}{PARA 0 "" 0 "" {TEXT -1 42 "different signs, the surf ace rises slowly." }}{PARA 0 "" 0 "" {TEXT -1 100 " For illustration, the cross-section f(x, y) = .8 and the given surface is shown in the \+ following, " }}{PARA 0 "" 0 "" {TEXT -1 57 "which can be rotated and i nspected from different angles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 " plot3d(\{exp(x*y), .8\}, x \+ = -1..1, y = -1..1, title = `cross-section of z = exp(x*y), z = .8`); " }}{PARA 13 "" 1 "" {GLPLOT3D 294 181 181 {PLOTDATA 3 "6&-%%GRIDG6%;$ !\"\"\"\"!$\"\"\"F)F&X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":4005BF0A8 B145769400401ECD67F7E3B400268660C4227364000EF9DB467DCF73FFF29EB2B7A2C0 A3FFCAC0A1B39052C3FFA61298E1E069A3FF84535933E500F3FF6546DB1BA2D123FF48 B5E3C3E81853FF2E6DA2D20C08B3FF163F5802956993FEFFFFFFFFFFFFF3FED7100FBF 669BC3FEB1660D7A223B03FE8EBEF9EAC820B3FE6EDD3122F2EA53FE5187FCE5CF8833 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" }}{PARA 0 "" 0 "" {TEXT -1 83 " There a re sudden changes as x and y approach the line y = x, i.e. the ratio , - " }{XPPEDIT 18 0 "(1-k)/(1+k);" "6#*&,&\"\"\"F%%\"kG!\"\"F%,&F%F%F &F%F'" }{TEXT -1 14 " , approaches " }}{PARA 0 "" 0 "" {TEXT -1 36 " 1 as k increases indefinitely. " }}{PARA 0 "" 0 "" {TEXT -1 102 " \+ The surface is cut up along plane y = x. The surface is the xy-plane \+ when y = -x away from (0, 0). " }}{PARA 0 "" 0 "" {TEXT -1 72 "Clearer contour map can be seen by graphing the family of curves y = - " } {XPPEDIT 18 0 "(1-k)/(1+k);" "6#*&,&\"\"\"F%%\"kG!\"\"F%,&F%F%F&F%F'" }{TEXT -1 18 " x for different " }}{PARA 0 "" 0 "" {TEXT -1 97 "value s of k. In the following the level curves for a few values of k betwe en -1 and 2 have been " }}{PARA 0 "" 0 "" {TEXT -1 85 "drawn. We hav e changed the value of the constant in order to accommodate fractions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s := \{seq(y + (1 - m/4)/(1 + m/4)*x, m = -3..6)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG<,%\"yG,&F&\"\"\"*&#F(\"\"&F(% \"xGF(!\"\",&F&F(*&#F(\"\"$F(F,F(F(,&F&F(*&#F+F1F(F,F(F(,&F&F(*&F1F(F, F(F(,&F&F(*&#F(\"\"(F(F,F(F(,&F&F(*&F:F(F,F(F(,&F&F(*&#F1F+F(F,F(F(,&F &F(F,F(,&F&F(*&#F(\"\"*F(F,F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "s := \{y, y-1/5*x, y+1/3*x, y+5/3*x, y+3*x, y+1/7*x, y+7*x, y+ 3/5*x, y+x, y-1/9*x\};" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "implicitplot(\{y, y-1/5*x, y+1/3*x , y+5/3*x, y+3*x, y+1/7*x, y+7*x, y+3/5*x, y+x, y-1/9*x\}, x=-5..5,y=- 5..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 306 220 220 {PLOTDATA 2 "6--%'CUR VESG6U7$7$$!\"&\"\"!$F*F*7$$!3q+++++++[!#%F/F+F57$F97$$!3 7**************RF/F+7$7$$!3Q*************z$F/F+F=7$7$$!3#)************ *z$F/F+7$$!3k*************f$F/F+7$7$$!3Y*************R$F/F+FH7$FL7$$!3 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" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}{PARA 0 "" 0 "" {TEXT -1 106 "The level curves on this contour map come closer together as we a pproach the origin (0, 0) on the plane. " }}{PARA 0 "" 0 "" {TEXT -1 107 "As k increases, 1 /ln(k) decreases and the radius of the circle d ecreases. This shows that the height of " }}{PARA 0 "" 0 "" {TEXT -1 78 "the surface rises rapidly as we approach the origin (0, 0, 0) f rom any side. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{PARA 0 "" 0 "" {TEXT -1 79 "_________________________________________________________ ______________________" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 257 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 8 " Problem" }}{PARA 0 "" 0 "" {TEXT -1 83 "For each of the following equations: 1. Draw the surface. 2. Des cribe the contour" }}{PARA 0 "" 0 "" {TEXT -1 101 "map algebraically. \+ 3. Draw the contour map. 4. Describe in words the level curves and \+ how they are" }}{PARA 0 "" 0 "" {TEXT -1 8 "placed. " }}{PARA 0 "" 0 " " {TEXT -1 54 " (a) f(x, y) = xy \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 " (b) f(x, y) = " }{XPPEDIT 18 0 "x^2+y^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 30 " (c) f(x, y) = 3x - 3y" }}{PARA 0 "" 0 "" {TEXT -1 25 " (d) f(x, y) = - " }{XPPEDIT 18 0 "x^2- y^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"" }{TEXT -1 5 " + 1" }} {PARA 0 "" 0 "" {TEXT -1 27 " (e) f(x, y) = y - " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 26 " (f) f(x, y) = cos" }{XPPEDIT 18 0 "sqrt(x^2+y^2);" "6# -%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*" }{TEXT -1 8 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "__________________________________ _____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 177 "MSIP Grant #P120A80089-98: \"Three Urban Calculus Reform Programs: Ad opting the Best,\" 1998-2001; MSEIP Grant #P120AA010031: \"Four Colle ges: Calculus + Enhancements\", 2001-2004 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "3 18 0" 41 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }