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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.

  Printable Key

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.

              Solution

 

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) 

              Solution

 

PROBLEM 3  (35 Points)  Determine the Maclaurin series for the function

                          3
        f(x)  =                      
                      2x - 4                        

Solution

PROBLEM 4 (35 Points)  Find the interval of convergence of

       

  Solution

 

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.)

  Solution

 

PROBLEM 6 (35 Points)  Find a unit vector that is perpendicular to the two vectors

        3i + 4j - k        and        2i + j + 2k

Solution

 

 

PROBLEM 7  (35 Points)   Consider the paraboloid  z  =  x2 + y2 

  1. Convert this equation to cylindrical coordinates.  
    Solution

  2. 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 .

  Solution

 

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).

  Solution

 

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.

  Solution