{"id":3909,"date":"2022-04-05T18:27:17","date_gmt":"2022-04-05T18:27:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=3909"},"modified":"2022-10-29T01:10:17","modified_gmt":"2022-10-29T01:10:17","slug":"level-curves","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/level-curves\/","title":{"raw":"Level Curves","rendered":"Level Curves"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Sketch several traces or level curves of a function of two variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIf hikers walk along rugged trails, they might use a topographical map that shows how steeply the trails change. A topographical map contains curved lines called\u00a0<em>contour lines<\/em>. Each contour line corresponds to the points on the map that have equal elevation (Figure 1). A level curve of a function of two variables [latex]f\\,(x,\\ y)[\/latex] is completely analogous to a counter line on a topographical map.\r\n\r\n[caption id=\"attachment_955\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-955\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27164628\/4-1-6.jpeg\" alt=\"This figure consists of two figures marked a and b. Figure a shows a topographic map of Devil\u2019s Tower, which has its lines very close together to indicate the very steep terrain. Figure b shows a picture of Devil\u2019s Tower, which has very steep sides.\" width=\"975\" height=\"534\" \/> Figure 1.\u00a0(a) A topographical map of Devil\u2019s Tower, Wyoming. Lines that are close together indicate very steep terrain. (b) A perspective photo of Devil\u2019s Tower shows just how steep its sides are. Notice the top of the tower has the same shape as the center of the topographical map.[\/caption]\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nGiven a function [latex]f\\,(x,\\ y)[\/latex] and a number [latex]c[\/latex] in the range of [latex]f[\/latex], a\u00a0<strong>level curve of a function of two variables<\/strong> for the value [latex]c[\/latex] is defined to be the set of points satisfying the equation [latex]f\\,(x,\\ y)=c[\/latex].\r\n\r\n<\/div>\r\nReturning to the function [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex], we can determine the level curves of this function. The range of [latex]g[\/latex] is the closed interval [latex][0,\\ 3][\/latex].\u00a0First, we choose any number in this closed interval\u2014say, [latex]c=2[\/latex].\u00a0The level curve corresponding to [latex]c=2[\/latex] is described by the equation\r\n<p style=\"text-align: center;\">[latex]\\sqrt{9-x^{2}-y^{2}}=2[\/latex].<\/p>\r\nTo simplify, square both sides of this equation:\r\n<p style=\"text-align: center;\">[latex]9-x^{2}-y^{2}=4[\/latex].<\/p>\r\nNow, multiply both sides of the equation by [latex]-1[\/latex] and add [latex]9[\/latex] to each side:\r\n<p style=\"text-align: center;\">[latex]x^{2}+y^{2}=5[\/latex].<\/p>\r\nThis equation describes a circle centered at the origin with radius [latex]\\sqrt{5}[\/latex]. Using values of [latex]c[\/latex] between [latex]0[\/latex] and [latex]3[\/latex]\u00a0yields other circles also centered at the origin. If [latex]c=3[\/latex],\u00a0then the circle has radius [latex]0[\/latex], so\u00a0it consists solely of the origin. Figure 2\u00a0is a graph of the level curves of this function corresponding to [latex]c=0,\\ 1,\\ 2[\/latex], and [latex]3[\/latex].\u00a0Note that in the previous derivation it may be possible that we introduced extra solutions by squaring both sides. This is not the case here because the range of the square root function is nonnegative.\r\n\r\n[caption id=\"attachment_956\" align=\"aligncenter\" width=\"342\"]<img class=\"size-full wp-image-956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27164820\/4-1-7.jpeg\" alt=\"Three concentric circles with center at the origin. The largest circle marked c = 0 has a radius of 3. The medium circle marked c = 1 has a radius slightly less than 3. The smallest circle marked c = 2 has a radius slightly more than 2.\" width=\"342\" height=\"347\" \/> Figure 2. Level curves of the function\u00a0[latex]\\small{g(x,y)=\\sqrt{9-x^{2}-y^{2}}}[\/latex], using [latex]\\small{c=0,1,2}[\/latex], and [latex]\\small{3}[\/latex] ([latex]\\small{c=3}[\/latex] corresponds to the origin).[\/caption]A graph of the various level curves of a function is called a <strong>contour map<\/strong>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Making a Contour Map<\/h3>\r\nGiven the function [latex]f\\,(x,\\ y)=\\sqrt{8+8x-4y-4x^{2}-y^{2}}[\/latex], find the level curve corresponding to [latex]c=0[\/latex]. Then create a contour map for this function. What are the domain and range of [latex]f[\/latex]?\r\n\r\n[reveal-answer q=\"fs-id1167793033114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793033114\"]\r\nTo find the level curve for [latex]c=0[\/latex], we set [latex]f\\,(x,\\ y)=0[\/latex] and solve. This gives\r\n<div style=\"text-align: center;\">[latex]0=\\sqrt{8+8x-4y-4x^{2}-y^{2}}[\/latex].<\/div>\r\n&nbsp;\r\n\r\nWe then square both sides and multiply both sides of the equation by [latex]-1[\/latex]:\r\n<div style=\"text-align: center;\">[latex]4x^{2}+y^{2}-8x+4y-8=0[\/latex].<\/div>\r\n&nbsp;\r\n\r\nNow, we rearrange the terms, putting the [latex]x[\/latex] terms together and the [latex]y[\/latex] terms together, and add [latex]8[\/latex] to each side:\r\n<div style=\"text-align: center;\">[latex]4x^{2}-8x+y^{2}+4y=8[\/latex].<\/div>\r\n&nbsp;\r\n\r\nNext, we group the pairs of terms containing the same variable in parentheses, and factor [latex]4[\/latex] from the first pair:\r\n<div style=\"text-align: center;\">[latex]4(x^{2}-2x)+(y^{2}+4y)=8[\/latex].<\/div>\r\n&nbsp;\r\n\r\nThen we complete the square in each pair of parentheses and add the correct value to the right-hand side:\r\n<div style=\"text-align: center;\">[latex]4(x^{2}-2x+1)+(y^{2}+4y+4)=8+4(1)+4[\/latex].<\/div>\r\n&nbsp;\r\n\r\nNext, we factor the left-hand side and simplify the right-hand side:\r\n<div style=\"text-align: center;\">[latex]4(x-1)^{2}+(y+2)^{2}=16[\/latex].<\/div>\r\n&nbsp;\r\n\r\nLast, we divide both sides by [latex]16[\/latex]:\r\n<div style=\"text-align: center;\">[latex]\\large{\\frac{(x-1)^{2}}{4}+\\frac{(y+2)^{2}}{16}=1}[\/latex].<\/div>\r\n&nbsp;\r\n\r\nThis equation describes an ellipse centered at [latex](1,\\ -2)[\/latex].\u00a0The graph of this ellipse appears in the following graph.\r\n<div>[caption id=\"attachment_957\" align=\"aligncenter\" width=\"417\"]<img class=\"size-full wp-image-957\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27165134\/4-1-8.jpeg\" alt=\"An ellipse with center (1, \u20132), major axis vertical and of length 8, and minor axis horizontal of length 4.\" width=\"417\" height=\"422\" \/> Figure 3.\u00a0Level curve of the function\u00a0[latex]\\small{f(x,y)=\\sqrt{8+8x-4y-4x^{2}-y^{2}}}[\/latex] corresponding to [latex]\\small{c=0}[\/latex][\/caption]<\/div>\r\n&nbsp;\r\n\r\nWe can repeat the same derivation for values of [latex]c[\/latex] less than [latex]4[\/latex].\u00a0Then, the equation of an ellipse centered at (1,\u22122) becomes\r\n<div style=\"text-align: center;\">[latex]\\large{\\frac{4(x-1)^{2}}{16-c^{2}}+\\frac{(y+2)^{2}}{16-c^{2}}=1}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nfor an arbitrary value of [latex]c[\/latex].\u00a0Figure 4\u00a0shows a contour map for [latex]f\\,(x,\\ y)[\/latex] using the values [latex]c=0,\\ 1,\\ 2[\/latex], and [latex]3[\/latex]. When [latex]c=4[\/latex], the level curve is the point [latex](-1,\\ 2)[\/latex].\r\n<div>[caption id=\"attachment_959\" align=\"aligncenter\" width=\"417\"]<img class=\"size-full wp-image-959\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27165511\/4-1-9.jpeg\" alt=\"An series of four concentric ellipses with center (1, \u20132). The largest one is marked c = 0 and has major axis vertical and of length 8 and minor axis horizontal of length 4. The next smallest one is marked c = 1 and is only slightly smaller. The next two are marked c = 2 and c = 3 and are increasingly smaller. Finally, there is a point marked c = 4 at the center (1, \u20132).\" width=\"417\" height=\"422\" \/> Figure 4.\u00a0Contour map for the function\u00a0[latex]f(x,y)=\\sqrt{8+8x-4y-4x^{2}-y^{2}}[\/latex], using the values [latex]c=0,1,2,3[\/latex], and [latex]4[\/latex].[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind and graph the level curve of the function [latex]g\\,(x,\\ y)=x^{2}+y^{2}-6x+2y[\/latex] corresponding to [latex]c=15[\/latex].\r\n\r\n[reveal-answer q=\"fs-id2167793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2167793933114\"]\r\nThe equation of the level curve can be written as [latex](x-3)^{2}+(y+1)^{2}=25[\/latex],\u00a0which is a circle with radius [latex]5[\/latex] centered at [latex](3,\\ -1)[\/latex].\r\n\r\n[caption id=\"attachment_992\" align=\"aligncenter\" width=\"492\"]<img class=\"size-full wp-image-992\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/28231410\/4-1-tryitans2.jpeg\" alt=\"An circle of radius 5 with center (3, \u20131).\" width=\"492\" height=\"497\" \/> Figure 5.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother useful tool for understanding the <strong><span id=\"term154\" data-type=\"term\">graph of a function of two variables<\/span><\/strong> is called a vertical trace. Level curves are always graphed in the [latex]xy[\/latex]-plane,\u00a0but as their name implies, vertical traces are graphed in the [latex]xz[\/latex]- or [latex]yz[\/latex]-planes.\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nConsider a function [latex]z=f\\,(x,\\ y)[\/latex] with domain [latex]D\\subseteq\\mathbb{R}^{2}[\/latex].\u00a0A <strong><span id=\"term155\" data-type=\"term\">vertical trace<\/span><\/strong> of the function can be either the set of points that solves the equation [latex]f\\,(a,\\ y)=z[\/latex] for a given constant [latex]x=a[\/latex] or [latex]f\\,(x,\\ b)=z[\/latex] for a given constant [latex]y=b[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Vertical Traces<\/h3>\r\nFind vertical traces for the function [latex]f\\,(x,\\ y)=\\sin{x}\\cos{y}[\/latex] corresponding to [latex]x=-\\frac{\\pi}{4}, 0[\/latex], and [latex]\\frac{\\pi}{4}[\/latex], and [latex]y=-\\frac{\\pi}{4}, 0[\/latex], and [latex]\\frac{\\pi}{4}[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1267793033114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1267793033114\"]\r\nFirst set [latex]x=-\\frac{\\pi}{4}[\/latex] in the equation [latex]z=\\sin{x}\\cos{y}[\/latex]:\r\n<div style=\"text-align: center;\">[latex]z=\\sin{\\big(-\\frac{\\pi}{4}\\big)}\\cos{y}=-\\frac{\\sqrt{2}\\cos{y}}{2}\\approx{-0.7071\\cos{y}}[\/latex].<\/div>\r\n&nbsp;\r\n\r\nThis describes a cosine graph in the plane [latex]x=-\\frac{\\pi}{4}[\/latex]. The other values of [latex]z[\/latex] appear in the following table.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 19.5499%;\">[latex]c[\/latex]<\/td>\r\n<td style=\"width: 80.4501%;\"><em><strong>Vertical Trace for<\/strong><\/em>[latex]x=c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.5499%;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80.4501%;\">[latex]z=-\\frac{\\sqrt{2}\\cos{y}}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.5499%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 80.4501%;\">[latex]z=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.5499%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80.4501%;\">[latex]z=\\frac{\\sqrt{2}\\cos{y}}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn a similar fashion, we can substitute the [latex]y[\/latex]-values in the equation [latex]f\\,(x,\\ y)[\/latex] to obtain the traces in the [latex]yz[\/latex]-plane, as listed in the following table.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 19.7264%;\">[latex]d[\/latex]<\/td>\r\n<td style=\"width: 80.2736%;\"><em><strong>Vertical Trace for<\/strong><\/em>[latex]y=d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.7264%;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80.2736%;\">[latex]z=\\frac{\\sqrt{2}\\sin{x}}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.7264%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 80.2736%;\">[latex]z=\\sin{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.7264%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80.2736%;\">[latex]z=\\frac{\\sqrt{2}\\sin{x}}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe three traces in the [latex]xz[\/latex]-plane\u00a0are cosine functions; the three traces in the [latex]yz[\/latex]-plane\u00a0are sine functions. These curves appear in the intersections of the surface with the planes [latex]x=-\\frac{\\pi}{4},\\ x=0,\\ x=\\frac{\\pi}{4}[\/latex] and [latex]y=-\\frac{\\pi}{4},\\ y=0,\\ y=\\frac{\\pi}{4}[\/latex] as shown in the following figure.\r\n\r\n[caption id=\"attachment_961\" align=\"aligncenter\" width=\"899\"]<img class=\"size-full wp-image-961\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27165937\/4-1-10.jpeg\" alt=\"This figure consists of two figures marked a and b. In figure a, a function is given in three dimensions and it is intersected by three parallel x-z planes at y = \u00b1\u03c0\/4 and 0. In figure b, a function is given in three dimensions and it is intersected by three parallel y-z planes at x = \u00b1\u03c0\/4 and 0.\" width=\"899\" height=\"424\" \/> Figure 6.\u00a0Vertical traces of the function [latex]f(x,y)[\/latex]\u00a0are cosine curves in the [latex]xz[\/latex]-planes(a), and sine curves in the\u00a0[latex]yz[\/latex]-planes (b).[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDetermine the equation of the vertical trace of the function [latex]g\\,(x,\\ y)=-x^{2}-y^{2}+2x+4y-1[\/latex] corresponding to [latex]y=3[\/latex], and describe its graph.\r\n\r\n[reveal-answer q=\"fs-id2168793933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2168793933124\"]\r\n[latex]z=3-(x-1)^{2}[\/latex]. This function\u00a0describes a parabola opening downward in the plane [latex]y=3[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186148&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Fqz94Q9khdA&amp;video_target=tpm-plugin-eossxc24-Fqz94Q9khdA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.3_transcript.html\">transcript for \u201cCP 4.3\u201d here (opens in new window).<\/a><\/center>Functions of two variables can produce some striking-looking surfaces. The following figure shows two examples.\r\n\r\n[caption id=\"attachment_963\" align=\"aligncenter\" width=\"973\"]<img class=\"size-full wp-image-963\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27170438\/4-1-11.jpeg\" alt=\"This figure consists of two figures marked a and b. In figure a, the function f(x, y) = x2 sin y is given; it has some sinusoidal properties by increases as the square along the maximums of the sine function. In figure b, the function f(x, y) = sin(ex) cos(ln y) is given in three dimensions; it decreases gently from the corner nearest (\u20132, 20) but then seems to bunch up into a series of folds that are parallel to the x and y axes.\" width=\"973\" height=\"437\" \/> Figure 7.\u00a0Examples of surfaces representing functions of two variables: (a) a combination of a power function and a sine function and (b) a combination of trigonometric, exponential, and logarithmic functions.[\/caption]","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Sketch several traces or level curves of a function of two variables.<\/li>\n<\/ul>\n<\/div>\n<p>If hikers walk along rugged trails, they might use a topographical map that shows how steeply the trails change. A topographical map contains curved lines called\u00a0<em>contour lines<\/em>. Each contour line corresponds to the points on the map that have equal elevation (Figure 1). A level curve of a function of two variables [latex]f\\,(x,\\ y)[\/latex] is completely analogous to a counter line on a topographical map.<\/p>\n<div id=\"attachment_955\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-955\" class=\"size-full wp-image-955\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27164628\/4-1-6.jpeg\" alt=\"This figure consists of two figures marked a and b. Figure a shows a topographic map of Devil\u2019s Tower, which has its lines very close together to indicate the very steep terrain. Figure b shows a picture of Devil\u2019s Tower, which has very steep sides.\" width=\"975\" height=\"534\" \/><\/p>\n<p id=\"caption-attachment-955\" class=\"wp-caption-text\">Figure 1.\u00a0(a) A topographical map of Devil\u2019s Tower, Wyoming. Lines that are close together indicate very steep terrain. (b) A perspective photo of Devil\u2019s Tower shows just how steep its sides are. Notice the top of the tower has the same shape as the center of the topographical map.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>Given a function [latex]f\\,(x,\\ y)[\/latex] and a number [latex]c[\/latex] in the range of [latex]f[\/latex], a\u00a0<strong>level curve of a function of two variables<\/strong> for the value [latex]c[\/latex] is defined to be the set of points satisfying the equation [latex]f\\,(x,\\ y)=c[\/latex].<\/p>\n<\/div>\n<p>Returning to the function [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex], we can determine the level curves of this function. The range of [latex]g[\/latex] is the closed interval [latex][0,\\ 3][\/latex].\u00a0First, we choose any number in this closed interval\u2014say, [latex]c=2[\/latex].\u00a0The level curve corresponding to [latex]c=2[\/latex] is described by the equation<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{9-x^{2}-y^{2}}=2[\/latex].<\/p>\n<p>To simplify, square both sides of this equation:<\/p>\n<p style=\"text-align: center;\">[latex]9-x^{2}-y^{2}=4[\/latex].<\/p>\n<p>Now, multiply both sides of the equation by [latex]-1[\/latex] and add [latex]9[\/latex] to each side:<\/p>\n<p style=\"text-align: center;\">[latex]x^{2}+y^{2}=5[\/latex].<\/p>\n<p>This equation describes a circle centered at the origin with radius [latex]\\sqrt{5}[\/latex]. Using values of [latex]c[\/latex] between [latex]0[\/latex] and [latex]3[\/latex]\u00a0yields other circles also centered at the origin. If [latex]c=3[\/latex],\u00a0then the circle has radius [latex]0[\/latex], so\u00a0it consists solely of the origin. Figure 2\u00a0is a graph of the level curves of this function corresponding to [latex]c=0,\\ 1,\\ 2[\/latex], and [latex]3[\/latex].\u00a0Note that in the previous derivation it may be possible that we introduced extra solutions by squaring both sides. This is not the case here because the range of the square root function is nonnegative.<\/p>\n<div id=\"attachment_956\" style=\"width: 352px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-956\" class=\"size-full wp-image-956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27164820\/4-1-7.jpeg\" alt=\"Three concentric circles with center at the origin. The largest circle marked c = 0 has a radius of 3. The medium circle marked c = 1 has a radius slightly less than 3. The smallest circle marked c = 2 has a radius slightly more than 2.\" width=\"342\" height=\"347\" \/><\/p>\n<p id=\"caption-attachment-956\" class=\"wp-caption-text\">Figure 2. Level curves of the function\u00a0[latex]\\small{g(x,y)=\\sqrt{9-x^{2}-y^{2}}}[\/latex], using [latex]\\small{c=0,1,2}[\/latex], and [latex]\\small{3}[\/latex] ([latex]\\small{c=3}[\/latex] corresponds to the origin).<\/p>\n<\/div>\n<p>A graph of the various level curves of a function is called a <strong>contour map<\/strong>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Making a Contour Map<\/h3>\n<p>Given the function [latex]f\\,(x,\\ y)=\\sqrt{8+8x-4y-4x^{2}-y^{2}}[\/latex], find the level curve corresponding to [latex]c=0[\/latex]. Then create a contour map for this function. What are the domain and range of [latex]f[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793033114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793033114\" class=\"hidden-answer\" style=\"display: none\">\nTo find the level curve for [latex]c=0[\/latex], we set [latex]f\\,(x,\\ y)=0[\/latex] and solve. This gives<\/p>\n<div style=\"text-align: center;\">[latex]0=\\sqrt{8+8x-4y-4x^{2}-y^{2}}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>We then square both sides and multiply both sides of the equation by [latex]-1[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]4x^{2}+y^{2}-8x+4y-8=0[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>Now, we rearrange the terms, putting the [latex]x[\/latex] terms together and the [latex]y[\/latex] terms together, and add [latex]8[\/latex] to each side:<\/p>\n<div style=\"text-align: center;\">[latex]4x^{2}-8x+y^{2}+4y=8[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>Next, we group the pairs of terms containing the same variable in parentheses, and factor [latex]4[\/latex] from the first pair:<\/p>\n<div style=\"text-align: center;\">[latex]4(x^{2}-2x)+(y^{2}+4y)=8[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>Then we complete the square in each pair of parentheses and add the correct value to the right-hand side:<\/p>\n<div style=\"text-align: center;\">[latex]4(x^{2}-2x+1)+(y^{2}+4y+4)=8+4(1)+4[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>Next, we factor the left-hand side and simplify the right-hand side:<\/p>\n<div style=\"text-align: center;\">[latex]4(x-1)^{2}+(y+2)^{2}=16[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>Last, we divide both sides by [latex]16[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\large{\\frac{(x-1)^{2}}{4}+\\frac{(y+2)^{2}}{16}=1}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>This equation describes an ellipse centered at [latex](1,\\ -2)[\/latex].\u00a0The graph of this ellipse appears in the following graph.<\/p>\n<div>\n<div id=\"attachment_957\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-957\" class=\"size-full wp-image-957\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27165134\/4-1-8.jpeg\" alt=\"An ellipse with center (1, \u20132), major axis vertical and of length 8, and minor axis horizontal of length 4.\" width=\"417\" height=\"422\" \/><\/p>\n<p id=\"caption-attachment-957\" class=\"wp-caption-text\">Figure 3.\u00a0Level curve of the function\u00a0[latex]\\small{f(x,y)=\\sqrt{8+8x-4y-4x^{2}-y^{2}}}[\/latex] corresponding to [latex]\\small{c=0}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>We can repeat the same derivation for values of [latex]c[\/latex] less than [latex]4[\/latex].\u00a0Then, the equation of an ellipse centered at (1,\u22122) becomes<\/p>\n<div style=\"text-align: center;\">[latex]\\large{\\frac{4(x-1)^{2}}{16-c^{2}}+\\frac{(y+2)^{2}}{16-c^{2}}=1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>for an arbitrary value of [latex]c[\/latex].\u00a0Figure 4\u00a0shows a contour map for [latex]f\\,(x,\\ y)[\/latex] using the values [latex]c=0,\\ 1,\\ 2[\/latex], and [latex]3[\/latex]. When [latex]c=4[\/latex], the level curve is the point [latex](-1,\\ 2)[\/latex].<\/p>\n<div>\n<div id=\"attachment_959\" style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-959\" class=\"size-full wp-image-959\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27165511\/4-1-9.jpeg\" alt=\"An series of four concentric ellipses with center (1, \u20132). The largest one is marked c = 0 and has major axis vertical and of length 8 and minor axis horizontal of length 4. The next smallest one is marked c = 1 and is only slightly smaller. The next two are marked c = 2 and c = 3 and are increasingly smaller. Finally, there is a point marked c = 4 at the center (1, \u20132).\" width=\"417\" height=\"422\" \/><\/p>\n<p id=\"caption-attachment-959\" class=\"wp-caption-text\">Figure 4.\u00a0Contour map for the function\u00a0[latex]f(x,y)=\\sqrt{8+8x-4y-4x^{2}-y^{2}}[\/latex], using the values [latex]c=0,1,2,3[\/latex], and [latex]4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find and graph the level curve of the function [latex]g\\,(x,\\ y)=x^{2}+y^{2}-6x+2y[\/latex] corresponding to [latex]c=15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id2167793933114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id2167793933114\" class=\"hidden-answer\" style=\"display: none\">\nThe equation of the level curve can be written as [latex](x-3)^{2}+(y+1)^{2}=25[\/latex],\u00a0which is a circle with radius [latex]5[\/latex] centered at [latex](3,\\ -1)[\/latex].<\/p>\n<div id=\"attachment_992\" style=\"width: 502px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-992\" class=\"size-full wp-image-992\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/28231410\/4-1-tryitans2.jpeg\" alt=\"An circle of radius 5 with center (3, \u20131).\" width=\"492\" height=\"497\" \/><\/p>\n<p id=\"caption-attachment-992\" class=\"wp-caption-text\">Figure 5.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Another useful tool for understanding the <strong><span id=\"term154\" data-type=\"term\">graph of a function of two variables<\/span><\/strong> is called a vertical trace. Level curves are always graphed in the [latex]xy[\/latex]-plane,\u00a0but as their name implies, vertical traces are graphed in the [latex]xz[\/latex]&#8211; or [latex]yz[\/latex]-planes.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>Consider a function [latex]z=f\\,(x,\\ y)[\/latex] with domain [latex]D\\subseteq\\mathbb{R}^{2}[\/latex].\u00a0A <strong><span id=\"term155\" data-type=\"term\">vertical trace<\/span><\/strong> of the function can be either the set of points that solves the equation [latex]f\\,(a,\\ y)=z[\/latex] for a given constant [latex]x=a[\/latex] or [latex]f\\,(x,\\ b)=z[\/latex] for a given constant [latex]y=b[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Vertical Traces<\/h3>\n<p>Find vertical traces for the function [latex]f\\,(x,\\ y)=\\sin{x}\\cos{y}[\/latex] corresponding to [latex]x=-\\frac{\\pi}{4}, 0[\/latex], and [latex]\\frac{\\pi}{4}[\/latex], and [latex]y=-\\frac{\\pi}{4}, 0[\/latex], and [latex]\\frac{\\pi}{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1267793033114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1267793033114\" class=\"hidden-answer\" style=\"display: none\">\nFirst set [latex]x=-\\frac{\\pi}{4}[\/latex] in the equation [latex]z=\\sin{x}\\cos{y}[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]z=\\sin{\\big(-\\frac{\\pi}{4}\\big)}\\cos{y}=-\\frac{\\sqrt{2}\\cos{y}}{2}\\approx{-0.7071\\cos{y}}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>This describes a cosine graph in the plane [latex]x=-\\frac{\\pi}{4}[\/latex]. The other values of [latex]z[\/latex] appear in the following table.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 19.5499%;\">[latex]c[\/latex]<\/td>\n<td style=\"width: 80.4501%;\"><em><strong>Vertical Trace for<\/strong><\/em>[latex]x=c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.5499%;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80.4501%;\">[latex]z=-\\frac{\\sqrt{2}\\cos{y}}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.5499%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 80.4501%;\">[latex]z=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.5499%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80.4501%;\">[latex]z=\\frac{\\sqrt{2}\\cos{y}}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In a similar fashion, we can substitute the [latex]y[\/latex]-values in the equation [latex]f\\,(x,\\ y)[\/latex] to obtain the traces in the [latex]yz[\/latex]-plane, as listed in the following table.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 19.7264%;\">[latex]d[\/latex]<\/td>\n<td style=\"width: 80.2736%;\"><em><strong>Vertical Trace for<\/strong><\/em>[latex]y=d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.7264%;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80.2736%;\">[latex]z=\\frac{\\sqrt{2}\\sin{x}}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.7264%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 80.2736%;\">[latex]z=\\sin{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.7264%;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80.2736%;\">[latex]z=\\frac{\\sqrt{2}\\sin{x}}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The three traces in the [latex]xz[\/latex]-plane\u00a0are cosine functions; the three traces in the [latex]yz[\/latex]-plane\u00a0are sine functions. These curves appear in the intersections of the surface with the planes [latex]x=-\\frac{\\pi}{4},\\ x=0,\\ x=\\frac{\\pi}{4}[\/latex] and [latex]y=-\\frac{\\pi}{4},\\ y=0,\\ y=\\frac{\\pi}{4}[\/latex] as shown in the following figure.<\/p>\n<div id=\"attachment_961\" style=\"width: 909px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-961\" class=\"size-full wp-image-961\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27165937\/4-1-10.jpeg\" alt=\"This figure consists of two figures marked a and b. In figure a, a function is given in three dimensions and it is intersected by three parallel x-z planes at y = \u00b1\u03c0\/4 and 0. In figure b, a function is given in three dimensions and it is intersected by three parallel y-z planes at x = \u00b1\u03c0\/4 and 0.\" width=\"899\" height=\"424\" \/><\/p>\n<p id=\"caption-attachment-961\" class=\"wp-caption-text\">Figure 6.\u00a0Vertical traces of the function [latex]f(x,y)[\/latex]\u00a0are cosine curves in the [latex]xz[\/latex]-planes(a), and sine curves in the\u00a0[latex]yz[\/latex]-planes (b).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Determine the equation of the vertical trace of the function [latex]g\\,(x,\\ y)=-x^{2}-y^{2}+2x+4y-1[\/latex] corresponding to [latex]y=3[\/latex], and describe its graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id2168793933124\">Show Solution<\/span><\/p>\n<div id=\"qfs-id2168793933124\" class=\"hidden-answer\" style=\"display: none\">\n[latex]z=3-(x-1)^{2}[\/latex]. This function\u00a0describes a parabola opening downward in the plane [latex]y=3[\/latex].\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186148&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Fqz94Q9khdA&amp;video_target=tpm-plugin-eossxc24-Fqz94Q9khdA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.3_transcript.html\">transcript for \u201cCP 4.3\u201d here (opens in new window).<\/a><\/div>\n<p>Functions of two variables can produce some striking-looking surfaces. The following figure shows two examples.<\/p>\n<div id=\"attachment_963\" style=\"width: 983px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-963\" class=\"size-full wp-image-963\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27170438\/4-1-11.jpeg\" alt=\"This figure consists of two figures marked a and b. In figure a, the function f(x, y) = x2 sin y is given; it has some sinusoidal properties by increases as the square along the maximums of the sine function. In figure b, the function f(x, y) = sin(ex) cos(ln y) is given in three dimensions; it decreases gently from the corner nearest (\u20132, 20) but then seems to bunch up into a series of folds that are parallel to the x and y axes.\" width=\"973\" height=\"437\" \/><\/p>\n<p id=\"caption-attachment-963\" class=\"wp-caption-text\">Figure 7.\u00a0Examples of surfaces representing functions of two variables: (a) a combination of a power function and a sine function and (b) a combination of trigonometric, exponential, and logarithmic functions.<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3909\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 4.3. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 4.3\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3909","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3909","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3909\/revisions"}],"predecessor-version":[{"id":5896,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3909\/revisions\/5896"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/3909\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=3909"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=3909"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=3909"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=3909"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}