MATH1052, Semester 1 2001

Sample Final Exam

  1. Show that the vector function

    2.  

     

    E

    is conservative by finding the potential.

  2. Evaluate the line integral

    3.  

     

    where f = 3xyi + 2yzj + 4xzk, and C is the straight line connecting

    the point (0,0,0) with (1,1,1).

  3. Sketch contour plots for the values f(x,y) = 3, 9 for the function

    4.  

     

    Solve for x(y) and plot the shapes of the curves. Be sure to indicate the

    equations of the curves that you are plotting.

  4. Use the method of Lagrange multipliers to find the maximum and minimum values of

    5.  

     

    subject to the constraint 

  5. Find and classify the critical points of the function

    6.  

     

  6. Evaluate the derivative of  F(x,y)= sin y + x cos y  at the point  (0,0) in the direction of the vector u= 2i-j .

    7.  

     
     
     

  7. Construct the tangent approximation to the function

    8.  

     

    at the point (1,2,2).
     
     

  8. Solve the differential equations
y(0) = 0

, y(1) = 1

 9. Solve the differential equation , y(0) = 0, y(1) = 1
  1. A tank contains an amount of material y(0) = 6 at time t = 0. The material is consumed in a reaction at a rate given by a constant k times the current amount while material is added into the tank at the rate 3 + cos(t). Write an equation for y(t) and find the solution.