How can I find the equation of the tangent plane to the surface defined by z = 3y² – 2x² + x + 2 at the point (1, 1, 3)?

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To find the equation of the tangent plane to the given surface at the specified point, we can follow these steps:

  • 1. Identify the surface function: The surface is given by the equation z = 3y² - 2x² + x + 2.
  • 2. Compute the partial derivatives: We need the partial derivatives of the function with respect to x and y.
  • Find ∂z/∂x:

    • ∂z/∂x = -4x + 1

  • Find ∂z/∂y:

    • ∂z/∂y = 6y

Evaluate the partial derivatives at the point (1, 1):

  • ∂z/∂x at (1, 1): ∂z/∂x = -4(1) + 1 = -3
  • ∂z/∂y at (1, 1): ∂z/∂y = 6(1) = 6

3. Write the equation of the tangent plane: The equation of the tangent plane at a point (x₀, y₀, z₀) can be expressed as:

z = z₀ + ∂z/∂x(x₀, y₀)(x - x₀) + ∂z/∂y(x₀, y₀)(y - y₀)

Substituting the known values:

z = 3 + (-3)(x - 1) + 6(y - 1)

4. Simplifying the expression:

z = 3 - 3x + 3 + 6y - 6

Thus, we can combine like terms:

z = -3x + 6y + 0

5. Final equation of the tangent plane:

The equation of the tangent plane at the point (1, 1, 3) is:

z = -3x + 6y

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