Final answer:
To find the equation of the tangent plane to a surface at a specified point, you need to find the partial derivatives of the surface with respect to x and y, evaluate these partial derivatives at the specified point, and use these values to write the equation of the tangent plane. In this case, the equation of the tangent plane to the surface z = 6(x - 1)² + 5(y + 3)² + 3 at the point (2, -2, 14) is z - 14 = 12(x - 2) + 10(y + 2).
Step-by-step explanation:
To find the equation of the tangent plane, we can start by finding the partial derivatives of the given surface with respect to x and y. Taking the partial derivative with respect to x, we get:
∂z/∂x = 12(x - 1)
Taking the partial derivative with respect to y, we get:
∂z/∂y = 10(y + 3)
Now we can evaluate these partial derivatives at the specified point (2, -2, 14):
∂z/∂x = 12(2 - 1) = 12
∂z/∂y = 10(-2 + 3) = 10
Next, we can use the values of the partial derivatives and the coordinates of the specified point to write the equation of the tangent plane:
z - z₀ = ∂z/∂x (x - x₀) + ∂z/∂y (y - y₀)
where (x₀, y₀, z₀) are the coordinates of the specified point. Plugging in the values, we get:
z - 14 = 12(x - 2) + 10(y + 2)