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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