How do you find the equations for the tangent plane to the surface #g(x,y)=x^2-y^2# through (5,4,9)?

Answer 1

An equation of the tangent plane of the graph of #z=g(x,y)=x^2-y^2# at the point #(x,y,z)=(5,4,9)# is #10x-8y-z=9#.

Write the equation #z=g(x,y)=x^2-y^2# in the equivalent form #G(x,y,z)=x^2-y^2-z=0#.
The gradient vector field of #G# is #nabla G(x,y,z)=(2x,-2y,-1)#. At the given point, this is #nabla G(5,4,9)=(10,-8,-1)#.

Since the gradient vector is perpendicular (normal) to the graph at this point, its components can be used as coefficients for the variables in the equation of the tangent plane (the reason for this is based on the fact that two nonzero vectors are perpendicular if and only if their dot product is zero).

The equation can be written #10(x-5)-8(y-4)-(z-9)=0#. This is equivalent to #10x-8y-z=9#.
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Answer 2

To find the equation for the tangent plane to the surface g(x,y) = x^2 - y^2 through the point (5,4,9), we can use the following steps:

  1. Calculate the partial derivatives of g(x,y) with respect to x and y.
  2. Evaluate the partial derivatives at the given point (5,4) to find their values.
  3. Use the values of the partial derivatives and the coordinates of the point to construct the equation of the tangent plane.

Let's go through these steps:

  1. The partial derivative of g(x,y) with respect to x (denoted as ∂g/∂x) is 2x, and the partial derivative with respect to y (denoted as ∂g/∂y) is -2y.

  2. Evaluate the partial derivatives at the point (5,4): ∂g/∂x = 2(5) = 10 ∂g/∂y = -2(4) = -8

  3. Using the values obtained in step 2, we can construct the equation of the tangent plane. The equation of a plane can be written as: Ax + By + Cz + D = 0

    Since the point (5,4,9) lies on the tangent plane, we can substitute its coordinates into the equation: 10(5) - 8(4) + C(9) + D = 0

    Simplifying the equation gives: 50 - 32 + 9C + D = 0

    Rearranging the terms, we obtain the equation of the tangent plane: 9C + D = -18

    Therefore, the equation for the tangent plane to the surface g(x,y) = x^2 - y^2 through the point (5,4,9) is 9C + D = -18.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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