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Math 251 Review 1. Find the equations of the line which passes through the point (−2, 3, −1) and is orthogonal to the plane 2x + y − 3z = 5. 2. Determine the equation of the plane which contains the point (1, −1, 2) and the line x = −1 + 2t, y = 1 + t, z = 1 + 3t. 3. Suppose that z = f (x, y), where x = 5u+2v , y = 3u+v , and that f has continuous second partial derivatives. ∂z (a) Find in terms of u, v , and the partial derivatives of f . ∂u (b) Find ∂ 2z in terms of u, v , and the partial derivatives of f . ∂u∂v 4. Find the equation of the plane tangent to the surface xy 2 z 3 = 2 at the point (2, 1, 1). 5. Use dierentials to approximate the number √ 0.99 e0.02 . 6. Determine the maximum and minimum values of f (x, y) = x2 + 2y 2 − x on the disk x2 + y 2 ≤ 1, giving the points where they occur. 2 2 7. Find the mass of the lamina that occupies the region inside the circle p x + y = 6y and outside the circle x2 + y 2 = 9. The density at (x, y) is ρ(x, y) = x2 + y 2 . 8. Set up (but don't evaluate) an iterated integral, in the order dzdydx, for ˝ E z dV where E is the region in space bounded below by z = x2 + y 2 and above by the plane 2x + 4y − z = −4. 9. Suppose that E is the region in space bounded below by the cone ˝ z = x2 + y 2 and above by the sphere x2 + y 2 + z 2 = 4. Write (but don't evaluate) x2 dV in p (a) spherical coordinates. (b) cylindrical coordinates. E 10. Evaluate the line integral and ending at (2, 2, 3). ´ y ds, where C is the line segment starting at (1, 1, 0) C 11. Find the work done by the force F(x, y) = x sin y i + y j on a particle along the curve y = x2 from (−2, 4) to (−1, 1). 12. Given F(x, y) = (1 + xy)exy i + x2 exy j (a) Show that F is conservative. (b) Find a function f such that ∇f = F. (c) Evaluate 0 ≤ t ≤ π/2. ´ C F · dr if C is the arc of the curve given by r(t) = cos t i + 2 sin t j, 13. Evaluate ˛ ex (1 − cos y) dx − ex (1 − sin y) dy C where C is the boundary of the domain D = { (x, y) | 0 ≤ x ≤ π, 0 ≤ y ≤ sin x }. 14. Write a parameterization for the part of the cylinder x2 + z 2 = 4 which lies between the planes y = −1 and x+2y+z = 8. Be sure to specify the parameter domain. 15. Compute the integral h u, v, u2 i for u2 + v 2 ≤ 4. ˜ S x dS , where S is the surface parametrized by r = 16. Find the ux of the vector eld F = x i + y j + z k across the surface z 2 = x2 + y 2 , 0 ≤ z ≤ 2. ´ 17. Evaluate C F·dr if F = h 2yz, −xy, x2 i and C is the intersection of z = 4−x2 −y 2 and z = x2 + y 2 . 18. Find the ux of F = h x, y, z i the across the surface of the region bounded by x2 + z 2 = 1, y = 0 and x + y = 3.