How can I find the equation of the plane tangent to the surface defined by the equation 7xy + yz + 4xz – 48 = 0 at the point (2, 2, 2)?

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To find the equation of the tangent plane to the surface determined by the equation:

7xy + yz + 4xz - 48 = 0

at the specified point (2, 2, 2), we can follow these steps:

  1. Determine the function: First, rewrite the equation of the surface in terms of a function:
    2. F(x, y, z) = 7xy + yz + 4xz - 48
  2. Calculate the gradient: We seek the gradient of the function F, denoted as ∇F. The gradient is a vector of the first partial derivatives of F with respect to x, y, and z:
    3.   ∇F(x, y, z) = 
  < ∂F/∂x, ∂F/∂y, ∂F/∂z >
  3. Compute the partial derivatives:
    • For ∂F/∂x:
      • ∂F/∂x = 7y + 4z
    • For ∂F/∂y:
      • ∂F/∂y = 7x + z
    • For ∂F/∂z:
      • ∂F/∂z = y + 4x
  4. Evaluate the gradient at the point (2, 2, 2):
    5.   ∇F(2, 2, 2) = < 7(2) + 4(2), 7(2) + 2, 2 + 4(2) > = < 14 + 8, 14 + 2, 2 + 8 > = < 22, 16, 10 >
  5. Write the equation of the tangent plane: The general form of the equation for the tangent plane at point (x0, y0, z0) is given by:
    6. ∇F(x0, y0, z0) • < x - x0, y - y0, z - z0 > = 0

    Substituting in the values:

    22(x - 2) + 16(y - 2) + 10(z - 2) = 0
  6. Expand and simplify the equation:
    7. 22x - 44 + 16y - 32 + 10z - 20 = 0

    Combine like terms:

    22x + 16y + 10z - 96 = 0
  7. Final equation of the tangent plane: Thus, the equation of the tangent plane at the point (2, 2, 2) is:
    8. 22x + 16y + 10z = 96

This equation describes the plane tangent to the surface at the specified point (2, 2, 2).

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