{"id":588,"date":"2025-10-09T15:23:27","date_gmt":"2025-10-09T15:23:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=588"},"modified":"2025-10-24T13:56:53","modified_gmt":"2025-10-24T13:56:53","slug":"functions-of-several-variables-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/functions-of-several-variables-fresh-take\/","title":{"raw":"Functions of Several Variables: Fresh Take","rendered":"Functions of Several Variables: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize a function of two variables and identify its domain and range<\/li>\r\n \t<li>Sketch a graph of a function of two variables<\/li>\r\n \t<li>Analyze functions by looking at their level curves and traces<\/li>\r\n \t<li>Identify functions of three or more variables and understand their level surfaces<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Functions of Two Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A function of two variables takes an ordered pair [latex](x, y)[\/latex] as input and produces a single output [latex]z[\/latex]. It's written as [latex]z = f(x, y)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Instead of mapping one number to another like [latex]y = f(x)[\/latex], you map pairs of numbers to a single output. Each point [latex](x, y)[\/latex] in the domain produces exactly one value [latex]z[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Domain and Range<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Domain<\/strong>: The set of all ordered pairs [latex](x, y)[\/latex] for which the function is defined<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Range<\/strong>: The set of all possible output values [latex]z[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding the Domain<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Look for the same restrictions as single-variable functions:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Square roots need non-negative expressions: [latex]\\sqrt{9 - x^2 - y^2}[\/latex] requires [latex]9 - x^2 - y^2 \\geq 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Denominators can't be zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithms need positive arguments<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The difference? These restrictions now define regions in the [latex]xy[\/latex]-plane, not intervals on a line.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding the Range<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Pick a value [latex]z[\/latex] and ask: can I find [latex](x, y)[\/latex] that makes [latex]f(x, y) = z[\/latex]? The range includes all [latex]z[\/latex] values where the answer is yes.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Identify restrictions on [latex]x[\/latex] and [latex]y[\/latex] to find the domain. Test boundary and interior points to determine the range. Sketch the domain region in the [latex]xy[\/latex]-plane when helpful.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain and range of the function [latex]f\\,(x,\\ y)=\\sqrt{36-9x^{2}-9y^{2}}[\/latex].[reveal-answer q=\"fs-id1167793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793933114\"]\r\nThe domain is the shaded circle defined by the inequality [latex]9x^{2}+9y^{2}\\,\\leq\\,36[\/latex], which\u00a0has a circle of radius [latex]2[\/latex] as its boundary. The range is [latex][0,\\ 6][\/latex].\r\n\r\n[caption id=\"attachment_991\" align=\"alignnone\" width=\"342\"]<img class=\"size-full wp-image-991\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/28231310\/4-1-tryitans1.jpeg\" alt=\"A circle of radius two with center at the origin. The equation x2 + y2 \u2264 4 is given.\" width=\"342\" height=\"347\" \/> Figure 3.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186147&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=UaWAuFLTy98&amp;video_target=tpm-plugin-8c5jetnr-UaWAuFLTy98\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.1_transcript.html\">transcript for \u201cCP 4.1\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Graphing Functions of Two Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">The graph of [latex]z = f(x, y)[\/latex] is a surface in three-dimensional space. Each point [latex](x, y)[\/latex] in the domain gets a height [latex]z[\/latex], creating ordered triples [latex](x, y, z)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Picture the [latex]xy[\/latex]-plane lying flat. For each point [latex](x, y)[\/latex] in the domain, the function assigns a [latex]z[\/latex]-value. Positive [latex]z[\/latex] means the point sits above the plane. Negative [latex]z[\/latex] means it sits below. All these points together form a surface.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Common Surfaces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Hemisphere<\/strong>: [latex]g(x, y) = \\sqrt{9 - x^2 - y^2}[\/latex]<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex]x^2 + y^2 \\leq 9[\/latex] (disk of radius 3)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex]0 \\leq z \\leq 3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shape: Upper half of a sphere<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Paraboloid<\/strong>: [latex]f(x, y) = x^2 + y^2[\/latex]<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All of [latex]\\mathbb{R}^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex]z \\geq 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shape: Bowl opening upward<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Understanding the Shape<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Look at cross-sections by holding one variable constant:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Set [latex]x = 0[\/latex]: You get [latex]z = f(0, y)[\/latex], often a familiar curve<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set [latex]y = 0[\/latex]: You get [latex]z = f(x, 0)[\/latex], another familiar curve<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set [latex]z = c[\/latex]: You get level curves showing where the surface has height [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Identify the domain. Find key cross-sections. Determine what happens at boundaries. Recognize familiar surface types like spheres, paraboloids, or planes.<\/p>\r\n\r\n<\/div>\r\n<h2>Level Curves<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A level curve shows all points where a function [latex]f(x, y)[\/latex] equals a constant value [latex]c[\/latex]. It's like a contour line on a topographical map\u2014points on the same curve have the same \"height.\"<\/p>\r\n<p class=\"whitespace-normal break-words\">For a function [latex]z = f(x, y)[\/latex] and a value [latex]c[\/latex], the level curve is the set of points satisfying [latex]f(x, y) = c[\/latex]. Each different value of [latex]c[\/latex] gives a different curve in the [latex]xy[\/latex]-plane.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Example: Hemisphere<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]g(x, y) = \\sqrt{9 - x^2 - y^2}[\/latex], set [latex]g(x, y) = 2[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Solve: [latex]\\sqrt{9 - x^2 - y^2} = 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Square: [latex]9 - x^2 - y^2 = 4[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify: [latex]x^2 + y^2 = 5[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">This is a circle of radius [latex]\\sqrt{5}[\/latex]. Different [latex]c[\/latex] values give concentric circles.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Reading Level Curves<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Curves close together<\/strong>: Function changes rapidly (steep terrain)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Curves far apart<\/strong>: Function changes slowly (gentle slope)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Contour Maps<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A <strong>contour map<\/strong> shows multiple level curves together. It gives a two-dimensional view of a three-dimensional surface.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Vertical Traces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">While level curves live in the [latex]xy[\/latex]-plane, <strong>vertical traces<\/strong> show slices through the surface:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Set [latex]x = a[\/latex]: Get a curve in the [latex]yz[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set [latex]y = b[\/latex]: Get a curve in the [latex]xz[\/latex]-plane<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Vertical traces help visualize the surface's shape from different angles.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Choose a value [latex]c[\/latex]. Set [latex]f(x, y) = c[\/latex]. Solve for the relationship between [latex]x[\/latex] and [latex]y[\/latex]. Repeat for several [latex]c[\/latex] values to build a contour map.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">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].[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<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[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<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<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\" data-mce-fragment=\"1\"><\/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><\/section>\r\n<h2>Functions of More Than Two Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Functions can have three or more independent variables. A function [latex]f(x, y, z)[\/latex] takes a point in three-dimensional space and maps it to a single output value.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding the Domain<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Use the same restrictions as before:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Square roots need nonnegative radicands<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Denominators can't be zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithms need positive arguments<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Example<\/strong>: [latex]f(x, y, z) = \\frac{3x - 4y + 2z}{\\sqrt{9 - x^2 - y^2 - z^2}}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The denominator requires [latex]9 - x^2 - y^2 - z^2 &gt; 0[\/latex], or [latex]x^2 + y^2 + z^2 &lt; 9[\/latex]. This is a ball of radius 3 (without the surface).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Level Surfaces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For functions of three variables, level curves become <strong>level surfaces<\/strong>. A level surface is the set of all points where [latex]f(x, y, z) = c[\/latex] for some constant [latex]c[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Pattern<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">As you add variables, level sets increase in dimension:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Two variables \u2192 level curves (1D)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three variables \u2192 level surfaces (2D)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Four variables \u2192 level hypersurfaces (3D)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain of the function [latex]h\\,(x,\\ y,\\ t)=(3t-6)\\,\\sqrt{y-4x^{2}+4}[\/latex].[reveal-answer q=\"fs-id2168793953124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2168793953124\"]\r\n<p style=\"text-align: center;\">[latex]\\text{domain}\\,(h)=\\{(x,\\ y,\\ t)\\in\\mathbb{R}^{3}|y\\,\\geq\\,4x^{2}-4\\}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the equation of the level surface of the function\r\n<p style=\"text-align: center;\">[latex]g\\,(x,\\ y,\\ t)=x^{2}+y^{2}+z^{2}-2x+4y-6z[\/latex]<\/p>\r\ncorresponding to [latex]c=2[\/latex], and describe the surface, if possible.\r\n\r\n[reveal-answer q=\"fs-id2168793959124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2168793959124\"]\r\n<p style=\"text-align: center;\">[latex](x-1)^{2}+(y+2)^{2}+(z-3)^{2}=16[\/latex] describes\u00a0a sphere of radius [latex]4[\/latex]\u00a0centered at the point [latex](1,-2, 3)[\/latex].<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186149&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=OoAPC9MiAhw&amp;video_target=tpm-plugin-yx2gcocw-OoAPC9MiAhw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.5_transcript.html\">transcript for \u201cCP 4.5\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize a function of two variables and identify its domain and range<\/li>\n<li>Sketch a graph of a function of two variables<\/li>\n<li>Analyze functions by looking at their level curves and traces<\/li>\n<li>Identify functions of three or more variables and understand their level surfaces<\/li>\n<\/ul>\n<\/section>\n<h2>Functions of Two Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A function of two variables takes an ordered pair [latex](x, y)[\/latex] as input and produces a single output [latex]z[\/latex]. It&#8217;s written as [latex]z = f(x, y)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Instead of mapping one number to another like [latex]y = f(x)[\/latex], you map pairs of numbers to a single output. Each point [latex](x, y)[\/latex] in the domain produces exactly one value [latex]z[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Domain and Range<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Domain<\/strong>: The set of all ordered pairs [latex](x, y)[\/latex] for which the function is defined<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Range<\/strong>: The set of all possible output values [latex]z[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Finding the Domain<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Look for the same restrictions as single-variable functions:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Square roots need non-negative expressions: [latex]\\sqrt{9 - x^2 - y^2}[\/latex] requires [latex]9 - x^2 - y^2 \\geq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Denominators can&#8217;t be zero<\/li>\n<li class=\"whitespace-normal break-words\">Logarithms need positive arguments<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The difference? These restrictions now define regions in the [latex]xy[\/latex]-plane, not intervals on a line.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Finding the Range<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Pick a value [latex]z[\/latex] and ask: can I find [latex](x, y)[\/latex] that makes [latex]f(x, y) = z[\/latex]? The range includes all [latex]z[\/latex] values where the answer is yes.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Identify restrictions on [latex]x[\/latex] and [latex]y[\/latex] to find the domain. Test boundary and interior points to determine the range. Sketch the domain region in the [latex]xy[\/latex]-plane when helpful.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain and range of the function [latex]f\\,(x,\\ y)=\\sqrt{36-9x^{2}-9y^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793933114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793933114\" class=\"hidden-answer\" style=\"display: none\">\nThe domain is the shaded circle defined by the inequality [latex]9x^{2}+9y^{2}\\,\\leq\\,36[\/latex], which\u00a0has a circle of radius [latex]2[\/latex] as its boundary. The range is [latex][0,\\ 6][\/latex].<\/p>\n<figure id=\"attachment_991\" aria-describedby=\"caption-attachment-991\" style=\"width: 342px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-991\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/28231310\/4-1-tryitans1.jpeg\" alt=\"A circle of radius two with center at the origin. The equation x2 + y2 \u2264 4 is given.\" width=\"342\" height=\"347\" \/><figcaption id=\"caption-attachment-991\" class=\"wp-caption-text\">Figure 3.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186147&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=UaWAuFLTy98&amp;video_target=tpm-plugin-8c5jetnr-UaWAuFLTy98\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/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\/CP4.1_transcript.html\">transcript for \u201cCP 4.1\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Graphing Functions of Two Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The graph of [latex]z = f(x, y)[\/latex] is a surface in three-dimensional space. Each point [latex](x, y)[\/latex] in the domain gets a height [latex]z[\/latex], creating ordered triples [latex](x, y, z)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Picture the [latex]xy[\/latex]-plane lying flat. For each point [latex](x, y)[\/latex] in the domain, the function assigns a [latex]z[\/latex]-value. Positive [latex]z[\/latex] means the point sits above the plane. Negative [latex]z[\/latex] means it sits below. All these points together form a surface.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Common Surfaces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Hemisphere<\/strong>: [latex]g(x, y) = \\sqrt{9 - x^2 - y^2}[\/latex]<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Domain: [latex]x^2 + y^2 \\leq 9[\/latex] (disk of radius 3)<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex]0 \\leq z \\leq 3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shape: Upper half of a sphere<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Paraboloid<\/strong>: [latex]f(x, y) = x^2 + y^2[\/latex]<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Domain: All of [latex]\\mathbb{R}^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex]z \\geq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shape: Bowl opening upward<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Understanding the Shape<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Look at cross-sections by holding one variable constant:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Set [latex]x = 0[\/latex]: You get [latex]z = f(0, y)[\/latex], often a familiar curve<\/li>\n<li class=\"whitespace-normal break-words\">Set [latex]y = 0[\/latex]: You get [latex]z = f(x, 0)[\/latex], another familiar curve<\/li>\n<li class=\"whitespace-normal break-words\">Set [latex]z = c[\/latex]: You get level curves showing where the surface has height [latex]c[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Identify the domain. Find key cross-sections. Determine what happens at boundaries. Recognize familiar surface types like spheres, paraboloids, or planes.<\/p>\n<\/div>\n<h2>Level Curves<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A level curve shows all points where a function [latex]f(x, y)[\/latex] equals a constant value [latex]c[\/latex]. It&#8217;s like a contour line on a topographical map\u2014points on the same curve have the same &#8220;height.&#8221;<\/p>\n<p class=\"whitespace-normal break-words\">For a function [latex]z = f(x, y)[\/latex] and a value [latex]c[\/latex], the level curve is the set of points satisfying [latex]f(x, y) = c[\/latex]. Each different value of [latex]c[\/latex] gives a different curve in the [latex]xy[\/latex]-plane.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Example: Hemisphere<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For [latex]g(x, y) = \\sqrt{9 - x^2 - y^2}[\/latex], set [latex]g(x, y) = 2[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Solve: [latex]\\sqrt{9 - x^2 - y^2} = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Square: [latex]9 - x^2 - y^2 = 4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]x^2 + y^2 = 5[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">This is a circle of radius [latex]\\sqrt{5}[\/latex]. Different [latex]c[\/latex] values give concentric circles.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Reading Level Curves<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Curves close together<\/strong>: Function changes rapidly (steep terrain)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Curves far apart<\/strong>: Function changes slowly (gentle slope)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Contour Maps<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A <strong>contour map<\/strong> shows multiple level curves together. It gives a two-dimensional view of a three-dimensional surface.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Vertical Traces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">While level curves live in the [latex]xy[\/latex]-plane, <strong>vertical traces<\/strong> show slices through the surface:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Set [latex]x = a[\/latex]: Get a curve in the [latex]yz[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">Set [latex]y = b[\/latex]: Get a curve in the [latex]xz[\/latex]-plane<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Vertical traces help visualize the surface&#8217;s shape from different angles.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Choose a value [latex]c[\/latex]. Set [latex]f(x, y) = c[\/latex]. Solve for the relationship between [latex]x[\/latex] and [latex]y[\/latex]. Repeat for several [latex]c[\/latex] values to build a contour map.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id2167793933114\">Show Solution<\/button><\/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<figure id=\"attachment_992\" aria-describedby=\"caption-attachment-992\" style=\"width: 492px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption id=\"caption-attachment-992\" class=\"wp-caption-text\">Figure 5.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id2168793933124\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/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\" data-mce-fragment=\"1\"><\/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<\/section>\n<h2>Functions of More Than Two Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Functions can have three or more independent variables. A function [latex]f(x, y, z)[\/latex] takes a point in three-dimensional space and maps it to a single output value.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Finding the Domain<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Use the same restrictions as before:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Square roots need nonnegative radicands<\/li>\n<li class=\"whitespace-normal break-words\">Denominators can&#8217;t be zero<\/li>\n<li class=\"whitespace-normal break-words\">Logarithms need positive arguments<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Example<\/strong>: [latex]f(x, y, z) = \\frac{3x - 4y + 2z}{\\sqrt{9 - x^2 - y^2 - z^2}}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The denominator requires [latex]9 - x^2 - y^2 - z^2 > 0[\/latex], or [latex]x^2 + y^2 + z^2 < 9[\/latex]. This is a ball of radius 3 (without the surface).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Level Surfaces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For functions of three variables, level curves become <strong>level surfaces<\/strong>. A level surface is the set of all points where [latex]f(x, y, z) = c[\/latex] for some constant [latex]c[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Pattern<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">As you add variables, level sets increase in dimension:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Two variables \u2192 level curves (1D)<\/li>\n<li class=\"whitespace-normal break-words\">Three variables \u2192 level surfaces (2D)<\/li>\n<li class=\"whitespace-normal break-words\">Four variables \u2192 level hypersurfaces (3D)<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain of the function [latex]h\\,(x,\\ y,\\ t)=(3t-6)\\,\\sqrt{y-4x^{2}+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id2168793953124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id2168793953124\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\text{domain}\\,(h)=\\{(x,\\ y,\\ t)\\in\\mathbb{R}^{3}|y\\,\\geq\\,4x^{2}-4\\}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the level surface of the function<\/p>\n<p style=\"text-align: center;\">[latex]g\\,(x,\\ y,\\ t)=x^{2}+y^{2}+z^{2}-2x+4y-6z[\/latex]<\/p>\n<p>corresponding to [latex]c=2[\/latex], and describe the surface, if possible.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id2168793959124\">Show Solution<\/button><\/p>\n<div id=\"qfs-id2168793959124\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex](x-1)^{2}+(y+2)^{2}+(z-3)^{2}=16[\/latex] describes\u00a0a sphere of radius [latex]4[\/latex]\u00a0centered at the point [latex](1,-2, 3)[\/latex].<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186149&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=OoAPC9MiAhw&amp;video_target=tpm-plugin-yx2gcocw-OoAPC9MiAhw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/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\/CP4.5_transcript.html\">transcript for \u201cCP 4.5\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-588","chapter","type-chapter","status-publish","hentry"],"part":426,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/588","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/588\/revisions"}],"predecessor-version":[{"id":1758,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/588\/revisions\/1758"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/588\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=588"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=588"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=588"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=588"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}