MATH 75 = CALCULUS I HOMEWORK #4, DUE MONDAY 21 OCT 2002
----------------WARREN D. SMITH--------------------------
Advanced uses of derivatives.  (Now that you are masters!)

1. Find the real root (x such that y=0) of the cubic polynomial

          3
     y = x  - x - 1

accurate to 3 decimal places (or more).  (Hint. Make a plot
to try to figure out the answer roughly.  Use that rough answer
as the initial guess for a few improvements via Newton's method.)


2. In class we've discussed ways in which your calculator (perhaps)
computes log(x) and square root of x.  In this exercise we'll
work out a way in which your calculator could DIVIDE (compute a/b)
even if all it has inside are circuits to ADD, SUBTRACT and MULTIPLY
(has no divider)!
                                            -2
A. Consider finding the root x of  y = b - x    where b is 
some known positive constant.  What is this root (as an exact formula 
in terms of b)?

B. What would Newton's update rule be for finding this root?
Can you express this updating formula in such a way that no divisions
are needed, only +, -, and multiply are needed?

C. If b>0 is some known number, and you use Newton's method
starting from some initial guess and it converges, what quantity Q
will it converge to?  And if you know Q, can you see how to compute
   a/b     and   SquareRoot(b)
WITHOUT ever doing a division or square-rooting (just +,- and multiplies)?


3. Optimum design of a conehead dunce hat!  Suppose this hat is a cone
with circular base, containing volume 1.  (Coneheads can distort their
head to any cone shape, so long as it has volume 1!)  If the height
of the cone is H and its base area B is            2
                                           B = pi r
then what is a formula for its volume V?  Since we require V=1,
what does that tell us H must be (in terms of r)?
The surface area S of the curved part of
the cone is given by                      2    2
                  S = 2 pi r SquareRoot( H  + r  )
Now. Find the optimum cone, meaning find H and r so that you
get a conehead-hat with V=1 and with MINIMUM surface area S
(so it costs the least).  (Hint: express S in terms of r ONLY -
no H - and then minimize by finding S'(r) and considering the
critical points. Can you think of some sanity checks?)


4. Read in book about "racetrack principle".  Now prove
that  sin(x) < x   for all positive x  and  x < tan(x)  for
all x in (0, Pi/2).   (These inequalities also have a geometric
interpretation in terms of lengths along lines and circular arcs...)