MA 174: Multivariable Calculus Final EXAM (practice) NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper.

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1 MA 174: Multivariable alculus Final EXAM (practice) NAME lass Meeting Time: NO ALULATOR, BOOK, OR PAPER ARE ALLOWED. Use the back of the test pages for scrap paper. Points awarded 1. (5 pts). (5 pts). (5 pts). (5 pts) 3. (5 pts). (5 pts) 4. (5 pts). (5 pts) 5. (5 pts). (5 pts) 6. (5 pts). (5 pts) Total Points: 1

2 urface Integral: If R is the shadow region of a surface defined by the equation f(x, y, z) = c, and g is a continuous function defined at the points of, then the integral of g over is the integral f g(x, y, z) dσ = g(x, y, z) f p da, where p is a unit vector normal to R and f p 0. R Green s Theorem: P dx + Q dy = R ( Q x P ) da y where is a positively oriented simple closed curve enclosing region R, and P, Q have continuous partial derivatives. Divergence Theorem: F d = F n dσ = D F dv where D is a simple solid region with boundary given outward orientation, and component functions of F have continuous partial derivatives. tokes Theorem: F dr = F n dσ where, given counterclockwise direction, is the boundary of oriented surface, n is the surface s unit normal vector and component functions of F have continuous partial derivatives.

3 1. The arclength of the curve r(t) = 3 t3/ i + 3 ( t)3/ j + (t 1) k for 1 4 t 1 is: A. /4 B. 3/4. / D. 3/ E. 1/. Find the directional derivative of the function f(x, y, z) = x y z 6 at the point (1, 1, 1) in the direction of the vector, 1,. A. 6 B.. 0 D. E The function f(x, y) = 3x + 1y x 3 y 3 has A. no critical point B. exactly one saddle point. two saddle points D. two local minimum points E. two local maximum points 4. The function f(x, y) = x 3 + y 3 3xy has how many critical points? A. None B. One. Two D. Three E. More than three 3

4 5. The max and min values of f(x, y, z) = xyz on the surface x + y + z = are A. ± 3 B. ± 6. ± D. ± E. ± 3 6. Find the maximum value of x +y subject to the constraint x x+y 4y = 0. A. 0 B.. 4 D. 16 E Find the parametric equations for the line passing through P = (, 1, 1), and normal to the tangent plane of at P. A. x = t + 4, y = t, z = t B. x = 4t +, y = t + 1, z = 3t 1. x 4 D. x 4 = y 1 = y 3 = z 1 3 = 3 1 E. x = 4t, y = t 1, z = 3t + 1 4x + y + z 3 = 8 8. One vector perpendicular to the plane that is tangent to the surface x + xy + z 3 = at the point ( 1, 1, 1) is: A. 3 i j + 3 k B. + j + k. +5 k D. j + k E. 5 i + j + 3 k 4

5 . uppose z = f(x, y), where x = e t and y = t + 3t +. Given that z x = xy y and z y = x y x, find dz when t = 0. dt A. 3 B D. E Find the equation in spherical coordinates for x + y = x. A. ρ = sin φ cos θ B. ρ sin φ = sin φ cos θ. ρ = sin φ cos φ D. ρ = ρ cos φ E. ρ sin φ = ρ sin φ cos θ 11. Let : x = u v, y = uv, z = u + v. If (0, b, 5) is a point on the tangent plane to at (0, 1, ) on, then b = A. 3 B. 1. D. 0 E. 1. Find the area of the region bounded by x = y y and x + y = 0 A. 1/3 B. /.3. 1 D. 4/3 E. 5/3 5

6 13. Find the area in the plane that lies inside the curve r = 1 + cos θ and outside the circle r = 1. A. π/ B. 1 + π/. 1 + π/4 D. + π/ E. + π/4 14. A sheet of metal occupies the region bounded by the x axis and the parabola y = 1 x. At each point, the density is equal to the distance from the y axis. Find the mass of the sheet. A. 1/4 B. 1/3. 1/ D. /3 E Evaluate A. 1 ydx + xdy + zdz, where : F (t) = t(t 1)e t i + sin( π t ) j + t t + 1 k, 0 t 1. B D. 0 E Let be the boundary of the triangle with vertices (0, 0), (1, 0), (1, 1) oriented counterclockwise. Then ydx xdy = A. 1 B D. 1 E. 6

7 17. Let F = f, f = x + y. If is any smooth curve joining the points (1, 1), (, ), then F d r = A. B. 1. D. 1 E. 18. Let D be the solid region bounded by the surfaces x + z = 4, y = 1, y = 0, and be the boundary of D. If F (x, y, z) = 1 3 (x3 i + y 3 j + z 3 k), then with n being the unit outward normal, evaluate F ndσ. A. 8π 8 B. 3 π. 8π D. 10π E Find a, b in the following formula which connect the triple integral from rectangular coordinates to spherical coordinate 3 x 0 0 x +y A. a = 0, b = ρ sin ϕ B. a = π/4, b = ρ 3 sin ϕ sin θ. a = π/4, b = ρ 3 sin ϕ sin θ D. a = π, b = 3 ρ3 sin ϕ sin θ E. a = π/, b = ρ 3 sin ϕ 0 ydzdydx = π/ π/ 3 csc ϕ 0 a 0 bdρdϕdθ. 0. F = xy i + (x + 3y ) j is a conservative vector field, i.e., F = f. If f(0, 0) = 0, then f(1, 1) = A. 1 B.. 3 D. E. 4 7

8 1. Evaluate octant. A. 1 6 B D. 5 5 E. 4 yd, where is the part of the plane x + y + z = 1 in the 1st. If F (x, y, z) = xz i + xyz j y k, then curl F evaluated at (1, 1, 1) equals A. 3 i j + k B. 3 i + j k. i + j k D. 3 i + j + k E. i j + k 3. Evaluate 0 x e y dydx. A. (e 4 1) B. e 4 1. e4 D. e 4 1 E. e

9 4. Let R be the region in the xy plane bounded by y = x, y = x and y = 4 x. Evaluate the integral yda. A. 8 3 B D. 8 3 E. 4 R 5. Find the surface area of the part of the surface z = x + y below the plane z =. A. π 4 (3 3 1) B. π 4 (3 3 ). π 6 (373/ 1) D. π 6 (3/ 1) E. π 6 (y3/ 1) 6. Find a, b such that A. a = 3, b = x B. a = z, b = 3. a = 3, b = y D. a = z, b = 3 E. a = 3, b = x 3 x z xdzdydx = a b z x dxdydz.

10 7. If F (x, y, z) = (x sin x + y) i + xy j + (yz + x) k, then curl F evaluated at (π, 0, ) equals A. π i j + k B. i j k. i π j + k D. i j + π k E. i + j + k 8. Evaluate (x + yz)dx + (y + xz)dy + xydz ( π ) where c: r(t) = t (1 + t) i + cos t j + t + 1 t k, 0 t 1. A. 1 B.. 3 D. 4 E. 5. Evaluate (x + y + z )d where is the upper hemisphere of x + y + z =. A. 1π B. 8π. 6π D. 4π E. 3π 10

11 30. Evaluate y x dx + x + y x + y counterclockwise. dy where is the circle x + y = 1 oriented A. π B. 4π, No to Green s theorem because the function is not continues at origin. 0 D. 4π E. π 31. alculate the surface integral F n d where is the sphere x + y + z = oriented by the outward normal and F (x, y, z) = 5x 3 i + 5y 3 j + 5z 3 k. A. 48 π B. 16π. 4π D. 5 π E. 0π 3. What is the spherical coordinates (ρ, ϕ, θ) = and the cylindircal coordinates (r, θ, z) = for the point (x, y, z) = (1, 1, 1)? Answer: (ρ, ϕ, θ) = ( 3, cos 1 ( 1 3 ), π 4 ) Answer: (r, θ, z) = (, π 4, 1) 11