Name                                     MATH 107 PRACTICE FINAL   Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work. PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why. A.       (11 Points)  Suppose that    is defined at x = 3 then f '(x) is also defined at x = 3 . Solution B.       (12 Points)  Let x = x(t), y = y(t) be parametric equations for a differentiable curve such that x''(-1) = y''(-1) = 3 , then the curve is concave up at the point (x(-1),y(-1)) . Solution C.       (12 Points)  If f(x,y) is a differentiable function at the point P, then Dgradf(P)(P)  the directional derivative in the direction of gradf(P) cannot be negative.   PROBLEM 2  Test the following series for convergence.   If applicable, determine if the series converges absolutely or conditionally.   A.      (17 Points)  Solution B.       (18 Points)    PROBLEM 3  (35 Points)  Determine the Maclaurin series for the function                           3         f(x)  =                                             2x - 4                         PROBLEM 4 (35 Points)  Find the interval of convergence of           PROBLEM 5    (35 Points)  Find the length of one of the petals of the graph of the curve         r  =  4 sin(3q) (You may use a calculator to perform the integration.)   PROBLEM 6 (35 Points)  Find a unit vector that is perpendicular to the two vectors         3i + 4j - k        and        2i + j + 2k     PROBLEM 7  (35 Points)   Consider the paraboloid  z  =  x2 + y2  Convert this equation to cylindrical coordinates.   Solution Convert this equation to spherical coordinates.   Solution     PROBLEM 8  (35 Points)  Let f(x,y)  =  x cos(y2) .  Use the chain rule to determine the angular rate of change of f given that x  =  r cos q,    y  =  r sin q .   PROBLEM 9  (35 Points)  Find the equation of the tangent plane to the surface         x2                 + y2 - z2  =  1         4   at the point (2,1,1).   PROBLEM 10  (35 Points)  The temperature of a room in your factory can be modeled by the equation f(x,y)  =  exy .  There is a round table of radius  centered at the origin.  Use the method of Lagrange multipliers to determine the hottest points on the table.