Final Answer:
The equation of the tangent plane to the surface z = 2x²y² - 9y at the point (1, 4, -18) is given by:
z = 30x - 81.
Step-by-step explanation:
To determine the equation of the tangent plane to the surface at the given point, we need to find the partial derivatives of z with respect to x and y and evaluate them at the point (1, 4).
Firstly, calculate the partial derivatives of z:
∂z/∂x = 4xy² and ∂z/∂y = 4x²y - 9.
Evaluate these derivatives at the point (1, 4):
∂z/∂x at (1, 4) = 4 * 1 * 4² = 64,
∂z/∂y at (1, 4) = 4 * 1² * 4 - 9 = 16 - 9 = 7.
The equation of the tangent plane is given by:
z - z₁ = ∂z/∂x (x - x₁) + ∂z/∂y (y - y₁),
where (x₁, y₁, z₁) is the point of tangency.
Substituting the values into the equation:
z - (-18) = 64(x - 1) + 7(y - 4),
z + 18 = 64x - 64 + 7y - 28,
z = 64x + 7y - 74 - 18,
z = 64x + 7y - 92.
Finally, simplify the equation:
z = 64x + 7y - 92,
z = 30x - 81.
Therefore, the equation of the tangent plane to the surface z = 2x²y² - 9y at the point (1, 4, -18) is z = 30x - 81.