Math 53 Worksheet - Partial Derivatives
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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?
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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.
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