ALA K+Kf+Bt, part a, fall 2002
Program
1. Understand what is "linearization by a tangent" by starting matlab,
giving the command setpath, opening RMplus (with the command open('RMplus.fig'))
and then the "Linearization lab".
Verify that the tangent to   y = f(x) = x.^3   for
  x = .5   is   y = .125 + .75 (x - .5). 
Verify that the tangent to   y = f(x) = 1. / (x.^2 + 1)   for
  x = 1   is   y = .5 - .5 (x - 1). 
Find the tangent to   y = f(x) = 1. / (x.^2 + 1)   for   x = -.5.  Zoom
in to make sure you really found the tangent. Compare to the tangent found by
symbolic derivation.
What is the tangent to   y = f(x) = sqrt(x.^2 + 1)   for   x = 1 ?
2. Understand what is the "inverse" of a given function   f(x)  (it is NOT   1 / f(x) !!),
by opening the "General lab".
Study the given functions   f(x) = x.^2   (in blue) and   g(y) = y.^3   (in red) by varying   x   and   y, 
respectively. Note, in particular, that we may consider   y   to be the independent variable
and   x   to be the dependent varaible, as in   x = g(y).  Then erase the given function   g(y) = y.^3  
by writing nothing in the corresponding edit box, and giving a return. If you now vary  
y,   the interpretation of the code is that you like to find the   x   for which   f(x) = y,   denoted  
f^(-1)(y),   if there is no more than one such   x.   Test this! Can you change   D_f   so that there is a unique  
x = f^(-1)(y)   for all   y   in   V_f ? How do you think   x   is found?
3. Test the following program for computing succesively better and better values
of the derivative of   f(x) = sqrt(x)   for   x = 1:
for i = 1:15
x = 1 + 1/10^i;
disp( (sqrt(x) - 1) / (x - 1) )
end
Explain the idea, and what you see! What is the surprise?
4. Now open the   f : R x R -> R   lab and seek the tangent plane to   z = g(x,y) = -x.^2 - y.^2  
for   x = -.5,  y = 0,  by selecting   f(x,y)   of the form   m + kx + ly   and varying   m, k   and   l.
5. Go back to complete the lab work suggested earlier, in case you haven't had time to
do this before.
/Kenneth
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