Question

Find an equation of the tangent plane to the given surface at the specified point.

z = 4(x − 1)² + 4(y + 3)² + 1, (2, −1, 21)

Answer 1

To find the equation of the tangent plane to the given surface at the specified point, we need to use partial derivatives and substitute the values of x, y, and z at the specified point into the equation of the tangent plane.

Step-by-step explanation:

To find the equation of the tangent plane to the given surface at the specified point, we need to use partial derivatives. First, we find the partial derivative with respect to x, which is 8(x - 1). Next, we find the partial derivative with respect to y, which is 8(y + 3).

Finally, we substitute the values of x, y, and z at the specified point (2, -1, 21) into the equation of the tangent plane: z - z1 = (∂z/∂x)*(x - x1) + (∂z/∂y)*(y - y1), where x1, y1, and z1 are the coordinates of the specified point. This will give us the equation of the tangent plane.