Problem 8 - Entrance Test

Find the equation of the tangent plane to the surface z = x^2 + y^2 at the point (1, 2, 5).

A. z = 2x + 4y - 5B. z = 2x + 4yC. z = x + 2y - 5D. z = x + 2y

Correct: A

Let f(x, y) = x^2 + y^2. Then fx = 2x and fy = 2y. At (1, 2), fx(1, 2) = 2 and fy(1, 2) = 4. The equation of the tangent plane is z - f(1, 2) = fx(1, 2)(x - 1) + fy(1, 2)(y - 2) => z - 5 = 2(x - 1) + 4(y - 2) => z - 5 = 2x - 2 + 4y - 8 => z = 2x + 4y - 5.