Question

Find the equation of the tangent plane to f (x, y) = x² − 2xy +2y² having slope 2 in the positive x direction and slope 4 in the positive y direction.

Answer 1

To find the equation of the tangent plane to f(x, y) = x² - 2xy + 2y² with given slopes, differentiate the function with respect to x and y, then substitute the point of tangency into the equation of the plane.

Step-by-step explanation:

The equation of the tangent plane to f(x, y) = x² - 2xy + 2y² with the given slopes can be found using partial derivatives. Firstly, differentiate the function with respect to x and y to obtain fx(x, y) = 2x - 2y and fy(x, y) = -2x + 4y respectively. Next, substitute the point of tangency (x₀, y₀) into the equation of the plane, which is given by z - f(x₀, y₀) = fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀). Finally, simplify the equation to find the equation of the tangent plane.