Final Exam

Write your name and student number on the front of your blue book together with a grading table. Start each question on a new page. Make sure you SHOW ALL WORK. No points will be given for answers where no justification is given.

1. (16 points) Find for

   

2. (32 points) For each of the following evaluate the integral, if it exists.

   

3. (20 points)
   1. Calculate the eccentricity for the conic section

      

      Draw a graph and clearly label all foci, vertices, directrices, and asymptotes.

   2. Graph the curves given below for the values of the parameter t indicated. Be sure to label important points. Carefully explain why these curves have the shape you draw.

      

4. (24 points)
   1. Write the polar equation

      

      in Cartesian coordinates (i.e. x and y).

   2. Write the Cartesian equation

      

      in polar coordinates. Be sure to simplify your answer.

   3. Calculate the area inside the cardioid

      

5. (16 points) Let .
   1. Find the limit of the sequence .
   2. Find a simple expression for the n-th partial sum .
   3. Find the limit of the sequence .
   4. Find the sum of the series

      

6. (24 points) For each of the following, state whether the series converges absolutely, converges conditionally, or diverges. Be sure to name any tests or theorems you use.

   

7. (24 points) For what values of x do the following series i. converge absolutely, ii. converge conditionally, and iii. diverge.

   

8. (14 points)
   1. Use the Binomial Theorem to find the Taylor polynomial of order three for about a=0 (i.e. write out the first few terms in the series up to and including the -term).
   2. Using series multiplication and your answer from part (a), find the Taylor polynomial of order four generated by

      

      about a=0.

9. (14 points)
   1. Find the Maclaurin series for .
   2. Using integration and your answer to part (a), find the Maclaurin series for .
10. (16 points)
    1. Write down the Maclaurin series for .
    2. Using the series obtained in part (a), estimate

       

       with an error less than .

converges?