{"id":95,"date":"2022-02-22T19:28:40","date_gmt":"2022-02-23T00:28:40","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=95"},"modified":"2022-03-08T15:17:52","modified_gmt":"2022-03-08T20:17:52","slug":"unit-2-tangent-plane","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-2-tangent-plane\/","title":{"raw":"Unit 2: Tangent Plane","rendered":"Unit 2: Tangent Plane"},"content":{"raw":"

The Concept<\/h2>\r\nFor a 2D curve [latex]y=f(x)[\/latex], there is at most one tangent line<\/strong> to a point [latex](x_0, y_0)[\/latex] on the curve.\u00a0 The equation of tangent line<\/strong> to 2D curve [latex]y=f(x)[\/latex] at point [latex](x_0, y_0)[\/latex] is\r\n

[latex]y=y_0+f'(x_0)(x-x_0)[\/latex].<\/p>\r\nThe tangent plane<\/strong> in 3D is an extension of the above tangent line in 2D. For a 3D surface [latex]z=f(x,y)[\/latex], there are infinitely many tangent lines to a point [latex](x_0, y_0, z_0)[\/latex] on the surface; these tangent lines lie in the same plane and they form the tangent plane at that point.\r\n\r\nRecall that two lines determine a plane in 3D space. Thus, one usually uses two special tangent lines to determine a tangent plane and these two tangent lines are related to the partial derivatives (i.e., [latex]f_x[\/latex] and [latex]f_y[\/latex]) of the surface function [latex]z = f(x,y)[\/latex]. The equation of the tangent plane<\/strong> to surface [latex]z = f(x,y)[\/latex] at point [latex](x_0, y_0, z_0)[\/latex] is\r\n

[latex] z = z_0 + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y-y_0)[\/latex].<\/p>\r\n\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below to understand the tangent plane[footnote]Made with <\/span>GeoGebra<\/a>, licensed Creative Commons <\/span>CC BY-NC-SA 4.0.<\/a>[\/footnote]. Follow the steps below to apply changes to the plot and observe the effects:\r\n
    \r\n \t
  1. Input a 3D surface function in the function box in the plot. The function can be a single variable function or a double variable function.<\/li>\r\n \t
  2. Adjust point [latex]P[\/latex] using the sliders or by dragging the point on the graph below.<\/li>\r\n \t
  3. The tangent plane equation will be depicted on the plot.<\/li>\r\n<\/ol>\r\n[h5p id=\"8\"]\r\n\r\n
    <\/header>\r\n
    <\/header>\r\n
    \r\n
    \r\n

    Self-Checking Questions<\/h2>\r\n<\/header>\r\n
    \r\n\r\nCheck your understanding by solving the following questions[footnote]Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax, Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).[\/footnote]:\r\n
      \r\n \t
    1. Find the equation of the tangent plane to the surface defined by the function [latex]x^2+10xyz+y^2+8z^2=0,P(\u22121,\u22121,\u22121)[\/latex]<\/span><\/li>\r\n \t
    2. Find the equation of the tangent plane to the surface defined by the function [latex]h(x,y) = ln(x^2) + y^2[\/latex] at Point [latex](x_0,y_0) = (3,4)[\/latex].<\/span><\/li>\r\n<\/ol>\r\n
      [h5p id=\"9\"]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n<\/div>","rendered":"

      The Concept<\/h2>\n

      For a 2D curve [latex]y=f(x)[\/latex], there is at most one tangent line<\/strong> to a point [latex](x_0, y_0)[\/latex] on the curve.\u00a0 The equation of tangent line<\/strong> to 2D curve [latex]y=f(x)[\/latex] at point [latex](x_0, y_0)[\/latex] is<\/p>\n

      [latex]y=y_0+f'(x_0)(x-x_0)[\/latex].<\/p>\n

      The tangent plane<\/strong> in 3D is an extension of the above tangent line in 2D. For a 3D surface [latex]z=f(x,y)[\/latex], there are infinitely many tangent lines to a point [latex](x_0, y_0, z_0)[\/latex] on the surface; these tangent lines lie in the same plane and they form the tangent plane at that point.<\/p>\n

      Recall that two lines determine a plane in 3D space. Thus, one usually uses two special tangent lines to determine a tangent plane and these two tangent lines are related to the partial derivatives (i.e., [latex]f_x[\/latex] and [latex]f_y[\/latex]) of the surface function [latex]z = f(x,y)[\/latex]. The equation of the tangent plane<\/strong> to surface [latex]z = f(x,y)[\/latex] at point [latex](x_0, y_0, z_0)[\/latex] is<\/p>\n

      [latex]z = z_0 + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y-y_0)[\/latex].<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the 3D plot below to understand the tangent plane[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Input a 3D surface function in the function box in the plot. The function can be a single variable function or a double variable function.<\/li>\n
      2. Adjust point [latex]P[\/latex] using the sliders or by dragging the point on the graph below.<\/li>\n
      3. The tangent plane equation will be depicted on the plot.<\/li>\n<\/ol>\n
        \n
        <\/div>\n<\/div>\n
        <\/header>\n
        \n
        <\/header>\n
        \n
        \n
        \n

        Self-Checking Questions<\/h2>\n<\/header>\n
        \n

        Check your understanding by solving the following questions[2]<\/sup><\/a>:<\/p>\n

          \n
        1. Find the equation of the tangent plane to the surface defined by the function [latex]x^2+10xyz+y^2+8z^2=0,P(\u22121,\u22121,\u22121)[\/latex]<\/span><\/li>\n
        2. Find the equation of the tangent plane to the surface defined by the function [latex]h(x,y) = ln(x^2) + y^2[\/latex] at Point [latex](x_0,y_0) = (3,4)[\/latex].<\/span><\/li>\n<\/ol>\n
          <\/p>\n
          \n