Math 142 Honors
Final Exam
Fall 1997

  1. Find the derivative of

    By the product rule,

  2. Integrate:

    Using parts,

  3. Integrate:

    By partial fractions,     .
    So,

  4. Evaluate:



  5. Integrate:


  6. Integrate:




  7. (a). = _____________ (Your answer should be an integer.)
    Hint: Convert all logs into terms of ln.


    (c). Sketch the graph of     , and indicate the domain and range.

    Domain is -1 £ x £ 1 and range is     .

  8. Determine where is increasing. Answer:

    So, since     x > 0,


  9. Integrate:


  10. Evaluate:


    Since,



  11. Evaluate the series . (Your answer should be an integer.)
    This is a geometric series with first term     and common ratio .
    The sum of the series is     .

  12. Find the derivative of .

    So,

  13. Express as a sum of partial fractions solve for the values of the constants                



    Hence, clearing of fractions, gives


    Let     x = 1 and you get .



    So,


  14. Show how you would use the integral test to determine if the series converges. (show all integration and evaluation of limits.)


    However,
    Hence the series converges by the integral test.

  15. Find the 3rd degree Taylor Polynomial around a = 0 for .



    So,


  16. (a). Does the series diverge or converge?
    (Explain your answer using appropriate convergence tests.)

    This converges since it is an alternating series whose terms decrease to zero.



    (b). Does this series converge absolutely? (Explain using convergence tests.)

    This series does not converge absolutely.
    The series of absolute values is     . Using a limit comparison test with we get .
    Thus, the two series behave the same and so since     is a p-series (p=1, i.e., the Harmonic Series), it diverges and so does the original series.

  17. (a). Express q in terms of x.


    Call the lower angle     g. Then



    (b). Express     as a sum of partial fractions, but do not solve for the constants.

  18. Determine the Interval of Convergence for .

    Using the ratio test on the series of absolute values,
    .

    So, the series converges if     and diverges if .
    Now check     x = -3 and x = 3 separately.
    - converges by Alternating Series Test.

     - the Harmonic Series - diverges.


    The interval of convergence is     .

  19. Express the polar curve in rectangular coordinates.


     - A Circle.

  20. (a). Find the power series for up to terms of degree 3..



    =

    (b).Given the power series
    Find the power series for     up to terms of degree 4.