MATH 210 Study Guide - Final Guide: Tangent Space, Cross Product, Unit Vector
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Problem 1 solution: let f (x, y, z) = 4x2 y2 + 3z2. Find the equation of the plane tangent to the level surface. F (x, y, z) = 7 at the point (1, 3, 2). Solution: we use the following formula for the equation for the tangent plane: Fx(a, b, c)(x a) + fy(a, b, c)(y b) + fz(a, b, c)(z c) = 0 because the equation for the surface is given in implicit form. The partial derivatives of f (x, y, z) = 4x2 y2 + 3z2 are: Fx = 8x, fy = 2y, fz = 6z. Evaluating these derivatives at (4, 2, 0) we get: Problem 2 solution: let f (x, y, z) = x2 xz + xyz. (a) find the rate of change of f at the point (1, 1, 1) in the direction of the unit vector. Ying around, with position function p (t) = ht, t2, t3i, carrying a thermometer in his pocket.