MA174: Multivariable Calculus
                     Final EXAM (practice)
       NAME             Class Meeting Time:
       NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back
       of the test pages for scrap paper.
                       Points awarded
            1. (5 pts)       9. (5 pts)
            2. (5 pts)       9. (5 pts)
            3. (5 pts)       9. (5 pts)
            4. (5 pts)       9. (5 pts)
            5. (5 pts)       9. (5 pts)
            6. (5 pts)       9. (5 pts)
                    Total Points:
                          1
             Surface Integral:
             If R is the shadow region of a surface S defined by the equation f(x,y,z) = c, and
             g is a continuous function defined at the points of S, then the integral of g over
             S is the integral
                                 Z Z g(x,y,z)dσ = Z Z g(x,y,z) |∇f| dA,
                                     S                 R         |∇f · p|
             where p is a unit vector normal to R and |∇f ·p| =6   0.
             Green’s Theorem:
                                     I Pdx+Qdy=ZZ ∂Q−∂PdA
                                      C                 R  ∂x    ∂y
             where C is a positively oriented simple closed curve enclosing region R, and P,
             Qhave continuous partial derivatives.
             Divergence Theorem:
                                   ZZ F·dS=ZZ F·ndσ=ZZZ ∇·FdV
                                     S            S              D
             where D is a simple solid region with boundary S given outward orientation,
             and component functions of F have continuous partial derivatives.
             Stokes’ Theorem:            I          ZZ
                                          C F·dr =    S ∇×F·n dσ
             where C, given counterclockwise direction, is the boundary of oriented surface
             S, n is the surface’s unit normal vector and component functions of F have
             continuous partial derivatives.
                                                      2
                                                                    2   3/2    2         3/2            ~       1         1
                                                                          ~                 ~
                  1. The arclength of the curve ~r(t) = 3 t                i + 3(2 − t)     j +(t−1)k for 4 ≤ t ≤ 2 is:
                      A. √2/4
                      B.    √3/4
                      C. √2/2
                     D. 3/2
                      E. 1/2
                  2. Find the directional derivative of the function f(x,y,z) = x2y2z6 at the point
                     (1,1,1) in the direction of the vector h2,1,−2i.
                      A. −6
                      B.    −2
                      C. 0
                     D. 2
                      E. 6
                  3. The function f(x,y) = 3x+12y −x3 −y3 has
                      A. no critical point
                      B. exactly one saddle point
                      C. two saddle points
                     D. two local minimum points
                      E. two local maximum points
                  4. The function f(x,y) = x3 +y3 −3xy has how many critical points?
                      A. None
                      B. One
                      C. Two
                     D. Three
                      E. More than three
                                                                        3
            5. The max and min values of f(x,y,z) = xyz on the surface 2x2+2y2+z2 = 2 are
                    √2
               A. ± 9
                    √3
               B. ± 9
                     √6
               C. ± 9
                     √
               D. ± 2 2
                      9
                     √
               E. ± 2 3
                      9
            6. Find the maximum value of x2+y2 subject to the constraint x2−2x+y2−4y = 0.
               A. 0
               B. 2
               C. 4
               D. 16
               E.  20
            7. Find the parametric equations for the line passing through P = (2,1,−1) , and
               normal to the tangent plane of
                                            4x+y2+z3=8
               at P.
               A. x = t+4,y = t,z = −t
               B.  x = 4t+2,y = 2t+1,z = 3t−1
               C. x−2 = y−1 = z−1
                    4      2      3
               D. x−4 = y−3 = 2−3
                    2      9     −1
               E. x = 4t−2, y = 2t−1, z = −3t+1
            8. One vector perpendicular to the plane that is tangent to the surface 2x2 +
               xy2 +z3 = 2 at the point (−1,1,1) is:
                     ~   ~    ~
               A. −3i−2j+3k
                   ~~   ~
               B. −+j+k
                   ~ ~
               C. −+5k ~
                   ~~
               D. 2−j +k   ~
                   ~   ~
               E. 5i+2j+3k
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