{"id":1043,"date":"2025-10-15T13:37:03","date_gmt":"2025-10-15T13:37:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1043"},"modified":"2025-10-23T13:18:26","modified_gmt":"2025-10-23T13:18:26","slug":"functions-of-several-variables-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/functions-of-several-variables-learn-it-2\/","title":{"raw":"Functions of Several Variables: Learn It 2","rendered":"Functions of Several Variables: Learn It 2"},"content":{"raw":"

Graphing Functions of Two Variables<\/h2>\r\n

When you graph a function [latex]z=f(x,y)[\/latex], you're working with two independent variables ([latex]x[\/latex] and [latex]y[\/latex]) and one dependent variable ([latex]z[\/latex]). For single-variable functions [latex]y=f(x)[\/latex], we use the Cartesian plane where every point has an ordered pair [latex](x,\\ y)[\/latex] associated with it. With two-variable functions, each ordered pair [latex](x,\\ y)[\/latex] in the domain maps to a real number [latex]z[\/latex], creating ordered triples [latex](x,\\ y,\\ z)[\/latex].<\/p>\r\n

The graph of a function [latex]z=f(x,y)[\/latex] is called a surface<\/strong>.<\/p>\r\n

To visualize this, imagine the [latex]xy[\/latex]-coordinate system lying flat. Every point in the domain of [latex]f[\/latex] has a unique [latex]z[\/latex]-value associated with it. If [latex]z[\/latex] is positive, the graphed point sits above the [latex]xy[\/latex]-plane. If [latex]z[\/latex] is negative, it sits below. The collection of all these graphed points forms a two-dimensional surface in three-dimensional space.<\/p>\r\n\r\n

Create a graph of each of the following functions:\r\n
    \r\n \t
  1. [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex]<\/li>\r\n \t
  2. [latex]f\\,(x,\\ y)=x^{2}+y^{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1267793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1267793933114\"]\r\n
      \r\n \t
    1. In the previous example, we determined that the domain of [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex] is [latex]\\{(x,\\ y)\\in\\mathbb{R}^{2}\\,|\\,x^{2}+y^{2}\\,\\leq\\,9\\}[\/latex] and the range is [latex]{z\\in\\mathbb{R}^{2}\\,|\\,0\\,\\leq\\,z\\,\\leq\\,3}[\/latex].\r\n\r\nWhen [latex]x^{2}+y^{2}=9[\/latex] we have [latex]g\\,(x,\\ y)=0[\/latex]. Therefore, any point on the circle of radius [latex]3[\/latex] centered at the origin in the [latex]xy[\/latex]-plane maps to [latex]z=0[\/latex] in [latex]\\mathbb{R}^3[\/latex].\r\n\r\nIf [latex]x^{2}+y^{2}=8[\/latex], then [latex]g\\,(x,\\ y)=1[\/latex], so any point on the circle of radius [latex]2\\sqrt{2}[\/latex] centered at the origin in the [latex]xy[\/latex]-plane maps to [latex]z=1[\/latex] in [latex]\\mathbb{R}^{3}[\/latex].\r\n\r\nAs [latex]x^{2}+y^{2}[\/latex] gets closer to zero, the value of [latex]z[\/latex] approaches [latex]3[\/latex]. When [latex]x^{2}+y^{2}=0[\/latex], then [latex]g\\,(x,\\ y)=3[\/latex]. This is the origin in the [latex]xy[\/latex]-plane.\r\n\r\nIf [latex]x^{2}+y^{2}[\/latex] is equal to any other value between [latex]0[\/latex] and [latex]9[\/latex], then [latex]g\\,(x,\\ y)[\/latex] equals some other constant between [latex]0[\/latex] and [latex]3[\/latex]. The surface described by this function is a hemisphere centered at the origin with radius\u00a0[latex]3[\/latex] as shown in the following graph.\r\n
      \r\n\r\n[caption id=\"attachment_949\" align=\"aligncenter\" width=\"597\"]\"A Figure 4. Graph of the hemisphere represented by the given function of two variables.<\/span>[\/caption]\r\n\r\n<\/div><\/li>\r\n \t
    2. This function also contains the expression [latex]x^{2}+y^{2}[\/latex]. Setting this expression equal to various values starting at zero, we obtain circles of increasing radius.\r\n\r\nThe minimum value of [latex]f\\,(x,\\ y)=x^{2}+y^{2}[\/latex] is zero (attained when [latex]x=y=0[\/latex]). When [latex]x=0[\/latex], the function becomes [latex]z=y^{2}[\/latex], and when [latex]y=0[\/latex], then the function becomes [latex]z=x^{2}[\/latex].\r\n\r\nThese are cross-sections of the graph, and are parabolas. Recall that the name of the graph of [latex]f\\,(x,\\ y)=x^{2}+y^{2}[\/latex] is a paraboloid<\/em>. The graph of [latex]f[\/latex] appears in the following graph.\r\n
      \r\n\r\n[caption id=\"attachment_951\" align=\"aligncenter\" width=\"585\"]\"A Figure 5.\u00a0A paraboloid is the graph of the given function of two variables.[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      To understand the shape of a surface, examine what happens when you hold one variable constant. For instance, setting [latex]x=0[\/latex] or [latex]y=0[\/latex] gives you cross-sections that are often familiar curves like parabolas or circles.\r\n\r\n<\/section>
      A profit function for a hardware manufacturer is given by\r\n

      [latex]f\\,(x,\\ y)=16-(x-3)^{2}-(y-2)^{2}[\/latex],<\/p>\r\nwhere [latex]x[\/latex] is the number of nuts sold per month (measured in thousands) and [latex]y[\/latex] represents the number of bolts sold per month (measured in thousands). Profit is measured in thousands of dollars. Sketch a graph of this function.\r\n\r\n[reveal-answer q=\"fs-id1167794933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794933114\"]\r\nThis function is a polynomial function in two variables. The domain of [latex]f[\/latex] consists of [latex](x,\\ y)[\/latex] coordinate pairs that yield a nonnegative profit:\r\n

      [latex]16-(x-3)^{2}-(y-2)^{2}\\,\\geq\\,0[\/latex]<\/div>\r\n
      [latex](x-3)^{2}-(y-2)^{2}\\,\\leq\\,16.[\/latex]<\/div>\r\n

      This is a disk of radius\u00a0[latex]4[\/latex] centered at [latex](3,\\ 2)[\/latex]. A further restriction is that both [latex]x[\/latex] and [latex]y[\/latex] must be nonnegative. When [latex]x=3[\/latex] and [latex]y=2[\/latex], [latex]f\\,(x,\\ y)=16[\/latex]. Note that it is possible for either value to be a noninteger; for example, it is possible to sell\u00a0[latex]2.5[\/latex] thousand nut in a month. The domain, therefore, contains thousands of points, so we can consider all points within the disk. For any [latex]z<16[\/latex], we can solve the equation [latex]f\\,(x,\\ y)=z[\/latex]:<\/p>\r\n\r\n

      [latex]16-(x-3)^{2}-(y-2)^{2}=z[\/latex]<\/div>\r\n
      [latex](x-3)^{2}-(y-2)^{2}=16-z[\/latex]<\/div>\r\n

      Since [latex]z<16[\/latex], we know that [latex]16-z>0[\/latex], so the previous equation describes a circle with radius [latex]\\sqrt{16-z}[\/latex] centered at the point [latex](3,\\ 2)[\/latex]. Therefore, the range of [latex]f\\,(x,\\ y)[\/latex] is [latex]\\{z\\in\\mathbb{R}\\,|z\\,\\leq\\,16\\}[\/latex]. The graph of [latex]f\\,(x,\\ y)[\/latex] is also a paraboloid, and this paraboloid points downward as shown.<\/p>\r\n\r\n\r\n[caption id=\"attachment_953\" align=\"aligncenter\" width=\"496\"]\"A Figure 6.\u00a0The graph of the given function of two variables is also a paraboloid.[\/caption]\r\n

      [\/hidden-answer]<\/p>\r\n\r\n<\/section>","rendered":"

      Graphing Functions of Two Variables<\/h2>\n

      When you graph a function [latex]z=f(x,y)[\/latex], you’re working with two independent variables ([latex]x[\/latex] and [latex]y[\/latex]) and one dependent variable ([latex]z[\/latex]). For single-variable functions [latex]y=f(x)[\/latex], we use the Cartesian plane where every point has an ordered pair [latex](x,\\ y)[\/latex] associated with it. With two-variable functions, each ordered pair [latex](x,\\ y)[\/latex] in the domain maps to a real number [latex]z[\/latex], creating ordered triples [latex](x,\\ y,\\ z)[\/latex].<\/p>\n

      The graph of a function [latex]z=f(x,y)[\/latex] is called a surface<\/strong>.<\/p>\n

      To visualize this, imagine the [latex]xy[\/latex]-coordinate system lying flat. Every point in the domain of [latex]f[\/latex] has a unique [latex]z[\/latex]-value associated with it. If [latex]z[\/latex] is positive, the graphed point sits above the [latex]xy[\/latex]-plane. If [latex]z[\/latex] is negative, it sits below. The collection of all these graphed points forms a two-dimensional surface in three-dimensional space.<\/p>\n

      Create a graph of each of the following functions:<\/p>\n
        \n
      1. [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex]<\/li>\n
      2. [latex]f\\,(x,\\ y)=x^{2}+y^{2}[\/latex]<\/li>\n<\/ol>\n