MATH 75 = CALCULUS I HOMEWORK #8, DUE MONDAY 18 NOV 2002
----------------WARREN D. SMITH--------------------------
A) Reading. Basically we are done - ch 1-5, appendix on Newton method,
ch 10.1-10.4 on Taylor series, ready reference and frontspiece and
endspiece of book are the works. Rest of course is review
and filling-in.

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                        2
1. Write the area (=pi r  / 2)  of a semicircle
as a definite integral (the area under the curve of
the semicircle).    (Do not actually try to do the integral.)

2. Find the Taylor series (based at x=0) of arctan(x). That includes
an infinite number of terms - you need to describe them all!
Hint:  It may help, as an tool, first to find a formula
for the derivative (d/dx) of 

                           A
                          x
                        -----------
                               2  B
                        ( 1 + x  )
for any nonnegative integers A,B.

3. How could this series from exercise 2 be used to compute pi?   Hint:
think of some special value of x, such that pi could
be easily found if you knew arctan(x).  (Not allowed to use the extra
credit answer below, must think of something else.)

4. What is the area under the curve  y=1/x  and above the x-axis,
from x=1 to x=10?  What about from x=1 to x=100?  

OPTIONAL EXTRA CREDIT:
Can you prove that
Pi/4 = 4arctan(1/5) - arctan(1/239) ?
Does this formula have advantages - in what way is it a "better" way
to compute pi than the formula you found in exercise 3?