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MATH 2401, PRACTICE FINAL 1. Reparametrize the curve r(t) = (3 sin t)i + (4t)j + (3 cos t)k with respect to arc length measured from the point where t = 0 in the direction of increasing t. Find the curvature at the point (0, 4π, −3). 2. Let f (x, y, x) = sin(xz 2 ) + cos(xy). (a) Find ∇f (0, −1, 2). (b) Find the directional derivative of f at (0, −1, 2) in the direction of the vector (2i − 3j + 6k)/7. (c) Find the equation of the plane tangent to the level surface sin(xz 2 ) + cos(xy) = 1 at the point (0, −1, 2). 3. Test the function 2 4 f (x, y) = + + xy x y for relative maxima and minima. 4. Show, using Lagrange multipliers, that if x, y, z ≥ 0 and x+y+z =c 3 then the maximum value of 1 (xyz) 3 is also c. This constitutes a proof of the well known inequality 1 x+y+z ≥ (xyz) 3 . 3 5. Evaluate: (a) Z 0 1 Z 1 y p 1 + x2 dxdy (b) Z Z Z x2 dxdydz, V where V is the solid that lies above the triangle with vertices (0, 0, 0), (0, 1, 0), (1, 1, 0), and below the surface z = x + y 2 . 6. Use the change of variables x = 31 (u + v), y = 31 (v − 2u) to evaluate Z Z Ω (3x + 4y) dxdy, where Ω is the region bounded by the lines y = x, y = x − 2, y = −2x, y = 3 − 2x. 7. Express Z Z Z y 2 dxdydz V as an iterated integral in spherical coordinates if V = {(x, y, z) : x ≤ 0, y ≤ 0, z ≤ 0, x2 + y 2 + z 2 ≤ 1}. 8. Evaluate the line integral Z yz dy + x dx + y dz, C where C is the line segment from (0, 1, 0) to (1, 2, 2). 9. Use Green’s Theorem to evaluate the integral Z F(r) · dr, C where F = (3x2 ey − xy 3 + 1)i + (x3 ey + sin y)j, and C is the counterclockwise oriented boundary of the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 1. 10. Let f (x, y, z) = xy 3 z 2 + sin(xy), u(x, y) = (ex + y)i + (sin y + x)j, and v(x, y, x) = yzi + (y 2 + xz)j + xyk. Compute div(∇f ) and ∇ × v. Is u conservative? If yes, find a function g such that u = ∇g. 11. Find the total surface area of the region bounded by the surfaces x2 +y 2 = 1, x2 +y 2 = z − 7, and z = 0. 2 12. Calculate the flux of the vector field v(x, y, x) = yez i + y 2 j + exy k out of the solid bounded by the cylinder x2 + y 2 = 9 and the planes z = 0 and z = y − 3.