CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265

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|                          FINAL EXAM                           |
|                  THURSDAY, DECEMBER 14, 2000                  |
|                      4:30 pm till 6:30 pm                     |
|                       ROOM  CARVER  268                       |
|         Notice the test for 265 is given in room 268          |
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Thursday, December 7:  Practice Test 

Practice test IV

1.  Find the sum of the infinite series:

               Infinity
                ____        (2n+1)
                \          2
                 \       ---------------
                 /            (2n)
                /___         5
                n = 1

     INF                  4/25
=  2 SUM (4/25)^n    = 2 ---------   = 8/21
     n=1                  1-4/25

                                           x
2.  Find the power series for   f(x) =  ---------  and  give a formula 
                                          2 
                                        3x  - 2

for its nth term and give its radius of convergence.

                  1
f(x) = -x/2 (-------------- )
               1 -(3x^2 /2)   


     = -x/2 ( 1 + (3x^2/2) + (3x^2/2)^2 + ... + (3x^2/2)^n + ...  )

     Infinity             n   2n+1
     = SUM      -1/2 (3/2)   x
       n=0
 
Its radius of convergence is   Sqrt[2/3].

3.  Write the power series for    Cos[ Sqrt[x] ] and give a formula for

its nth term and its radius of convergence.

                                                   n  n
Cos[Sqrt[x]] = 1 - x/2 + x^2/4! - x^3/6! + ...+(-1)  x / (2n)!

Its radius of convergence is Infinity.


4.  Using the alternating series error formula, how many terms are 

                                    x = 2
needed to evaluate the integral    Integral f(x)  so that the error 
                                    x = 0

will be less than 0.001?  The function f(x) is given by the series.

               Infinity
                ____          n  2n
                \         (-1)  x 
     f(x) =      \       ---------------
                 /           n! 
                /___         
                n = 0

           1 + 2 n
          2
f[n]  = ------------
        (1 + 2 n) n!

 1 2.666666667
 2 3.2
 3 3.047619048
 4 2.37037037
 5 1.551515152
 6 0.8752136752
 7 0.4334391534
 8 0.1912231559
 9 0.07604195674
10 0.02751994625   
11 0.009137057253
12 0.002802030891   
13 0.0007982993991  <---- n = 13 has the n th term < 0.001
14 0.0002123555052                 Therefore n = 12 so the
15 0.00005297470667                Taylor polynomial is of
16 0.0000124410296                 degree 12 and it actually
                                   has 13 terms.


5.  Using the Taylor error formula, how many terms are necessary to 
                    x
approximate f(x) = e   within 0.0001 on the interval [-2,2]? 

         n  2
        2  E
f[n]  = -----
         n!

Do[ Print[n," ",N[ f[n],10 ]],{n,0,15}]
 0 7.389056099
 1 14.7781122
 2 14.7781122
 3  9.852074799
 4  4.926037399
 5  1.97041496
 6  0.6568049866
 7  0.1876585676
 8  0.0469146419
 9  0.01042547598
10  0.002085095195
11  0.0003791082174
12  0.00006318470289  <-- at n=12 the error is small enough.  Therefore
                  -6      the Taylor polynomial of degree 11 works and
13  9.720723522 10        it has actually 12 terms.