MATH 75 = CALCULUS I HOMEWORK #2, DUE MONDAY 16 SEPT 2002
----------------WARREN D. SMITH--------------------------
A. COW.
I have assigned over 30 COW problems.  They are mostly pretty
simple but there are a lot of them.
They have to be done by Wednesday 18 sept, but this written homework 
is due Monday 16 sept.
Connect to COW on the internet (e.g. via my web page).
Make yourself a password if first time via PASSWORD EXCHANGE.
Do not forget your password.  LOGIN with your name and password. 
HOMEWORK button (& VIEW ASSIGNMENT) will display
a list of the assigned problems from COW's "calculus book I".
Print out that list (or just keep it in another screen "window").
Then go to COW's MAIN page, hit the book-I button, and navigate thru
the menus to reach the assigned problems, then do them.  Keep doing them
until get them right... if you want credit.  It will do the grading,
it will instantly tell you if got it right, etc.  Do not use browser "back" 
buttons, use only buttons COW provides, when navigating.
You may need to consult your textbook to help, while COWing! That is
fine! You can come back and do more COWIng later & it will not forget your
previous work!  If you ever solve a COW problem, it remembers you
solved it forever after.

B. Reading. Try to read chapter 1 of the book. The Math 75 course is 
supposed to cover ch 1-5 in a little under 15 weeks.
So you should know ch 1, in particular the slope, function, 
interval, domain, range, and limit concepts, plus whatever precalculus
knowledge is needed (plots, lines, triangles, trig functions, logs, 
solving linear systems, solving quadratics, method of induction,
functions, polynomials, rational functions, range, domain, slope, 
angles, distances) by week 3, about when we will have a quiz on it...

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1. To demonstrate your mastery of manipulating fraction formulas,
           a/b + c/d         polynomial
simplify   ---------  to  ----------------,  i.e. to "rational" form.
           e/f - g/h      other polynomial
Here by "polynomial" I mean an expression involving +,- and multiplication, 
but not division.

2.  Consider these 3 equations in 3 unknowns x,y,z:
          x + 5y + 7z = 10
         3x + 2y - 3z = 3
         4x -  y      = 2
What is the only possible value for z 
(as an exact fraction, such as 99/872, please)?

3. The "addition formula" for cos is
   cos(a+b) = cos(a) cos(b) - sin(a) sin(b).
Write down the "addition formula" for sin(a+b) 
in terms of sin(a), sin(b), cos(a), cos(b).  
Use these two to derive a formula
for  tan(a+b)  in terms of tan(a) and tan(b).
(Remember definition of tan(x) = sin(x)/cos(x).  
You might want to try some sanity checking...)

4. Use the addition formula for sin(a+b)
to derive a formula for  sin(q/2)  in terms of  sin(q).
Now, we will USE this formula:

If the area A of a regular N-sided polygon, all of
whose side lengths are equal, and all of whose vertices
lie on the perimeter of the unit circle, is A=F(N), then
what is F(N), as a formula involving N and trig functions?
(For example, F(4) is the area of a square, all 4 of whose
corners lie on the unit circle, i.e. all of these corners
are at distance=1 from (0,0).)

Now what is the area F(2N) of a regular 2N-sided polygon of this sort?

Now express  F(2N)  as a function of A,
[or equivalently as a function of F(N), since this is A].  
This function should involve only A,N,+,-,multiply,divide, 
and square roots, it should NOT involve trig functions or 
unknown functions.   (So we now know how to compute the area of
the 20-gon IF we knew the area of the 10-gon.)

Can you now see a way to get arbitrarily good approximations to pi,
which can be found WITHOUT ever computing a trig function?
Hint: what happens if you keep doubling N?
[This is how Archimedes estimated pi in about 340 BC.]