Project 4 due Fri 25 Sept

Now Ready.
Corrected an error in #2 on Thurs -- I mentioned this in class.

Here are some warm up examples to give you a feel for how Maple
does limits. Do them, but DON'T turn them in.

The range of Maple answers for limit questions. First miss spell limit
as lim and see what Maple answers to lim(sin(x)/x,x=0). Let Maple try
to find limit as x->0 of the functions 1/x, 1/x^2, abs(1,x)
[This is the derivative of abs(x) = |x|.] and sin(1/x). The answer
for sin(1/x) is interesting. Check and see if the answer is the same
as limit(sin(x),x->infinity) which you know does not have a limit.
Get the one-sided limits (right and left) of abs(1,x) (ie
limit(abs(1,x),x=0,left) and same with right replacing left).

Just 4 problems this time:
1. f(x)=sin(1/x) is called the topologist's sine curve. Plot is graph
for the ranges -1..1, -0.1..0.1 and -0.01..0.01. Why does the curve
look increasing choatic? For the domain=-1..1 let "x" "drag down" f(x), that
is plot x*sin(1/x) and check the limit as x->0. For the same domain
let "1/x" "blow up" f(x), that is plot sin(1/x)/x. You will need to
restrict the range y=-m..m for some sitable m for this last plot.

2. Find a bunch of limits. First do "series(sin(x),x=0,11)" and then 
repeatly find the limit as x -> 0 of
((x)-sin(x))/x^3
((x - x^3/6)-sin(x))/x^5
((x - x^3/6 + x^5/120)-sin(x))/x^7

...

((x - x^3/6 + x^5/120 ... + (-1)^n x^(2*n+1) / (2*n+1)! )-sin(x))/x^(2*n+3)
until Maple can't find the limit. [Maple echos the question if it does
not know the answer.]

Now increase "Order" to a value larger than 2*n-1 and re-try the last
limit. [So Order is like Digits only for series, and limits.]

3. Although the topologist sine curve is not a function that is likely
to mug you in a dark alley, the function f := (x,y)->x*y/(x^2+y^2) which
looks almost friendly has similar problems. As we did in class,
write g(r,theta)=f(r cos(theta),r sin(theta)). Now plot g for r=0..1
t=0..2*Pi as rectangular co-ordinates. The r=0 in the plot of g is
the "orgin" in the plot of f. Repeat for f1=xy/sqrt(x^2+y^2) and
f2=xy/(x^2+y^2)^(3/2). [For this last plot you may have to limit the
range of z, this uses the view=-m..m option to plot3d.

4. Use f = xy^2/(x^2+y^4) and g(r,t)=f(r cos t , r sin t ) and plot
g in rectanglar co-ordinates like in #4. Use the orientation option
with [145,45] so that the r=0 is in front. Plot again with a smaller
r range like 0..0.1 or 0..0.01 and see how these graphs might fool one
into thinking the limit existed and is 0. Show lim f(x,mx) is zero
but lim f(y^2,y) is not and hence the limit fails.