Math 53 Worksheet - Partial Derivatives p 1. If f (x, y) = 3 x3 + y 3 , find fx (0, 0). 2. The ellipsoid 4x2 + 2y 2 + z 2 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 2). 3. Suppose you need to know an equation of the tangent plane to a surface S at the point P (2, 1, 3). You don’t have an equation for S but you know that the curves r1 (t) = h2 + 3t, 1 − t2 , 3 − 4t + t2 i r2 (u) = h1 + u2 , 2u3 − 1, 2u + 1i both lie on S. Find an equation of the tangent plane at P . 4. Are there any points on the hyperboloid x2 − y 2 − z 2 = 1 where the tangent plane is parallel to the plane z = x + y? 5. One side of a triangle is increasing at a rate of 3 cm/s and a second side is decreasing at a rate of 2 cm/s. If the area of the triangle remains constant, at what rate does the angle between sides change when the first side is 20 cm long, the second side is 30 cm, and the angle is π/6? 6. Suppose that the directional derivatives of f (x, y) are known at a given point in two nonparallel directions given by unit vectors u and v. Is it possible to find ∇f at the point? If so, how would you do it? √ 7. Show that the function f (x, y) = 3 xy is continuous and the partial derivatives fx and fy exist at the origin, but the directional derivatives in all other directions do not exist. 1
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