Question

Find an equation of the tangent plane to the given surface at the specified point. z = 2(x − 1)^2 + 4(y + 3)^2 + 2, (3, −2, 14)

Answer 1

To find the equation of the tangent plane to the given surface at the specified point, you need to find the partial derivatives of the surface equation and evaluate them at the given point. Then, substitute the values into the equation of the tangent plane and simplify to get the final equation.

Step-by-step explanation:

To find the equation of the tangent plane to the given surface at the specified point (3, -2, 14), we need to find the partial derivatives of the surface equation and evaluate them at the given point. The equation of the tangent plane can be written in the form z = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b), where f(a, b) is the value of the surface equation at point (a, b) and fx(a, b) and fy(a, b) are the partial derivatives of the surface equation with respect to x and y, respectively.

1. Take the partial derivative of z with respect to x: fx(x, y) = 4(x - 1)
2. Take the partial derivative of z with respect to y: fy(x, y) = 8(y + 3)
3. Plug in the values of x, y, and z into the equation of the tangent plane: z = f(3, -2) + fx(3, -2)(x - 3) + fy(3, -2)(y + 2)
4. Simplify the equation to get the final form of the tangent plane.

Substituting the values, we get z = 14 + 4(3 - 1)(x - 3) + 8(-2 + 3)(y + 2). Simplifying further, the equation of the tangent plane is z = 14 + 8(x - 3) - 16(y + 2), or z = 8x - 16y - 24.