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Partially deductive mathematics
Posted:
Oct 16, 2001 3:28 PM
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As a high school students I read many times that mathematics is a deductive
science. During university years all my courses were taught in deductive way.
My problem is that I have almost never learned mathematics in a deductive
way. One exception was Abstract Algebra, but even then only for a part of
the semester. Normally, my way of learning was to identify the most
important/useful statements and take them as if they were revealed from the
Mount of Sinai. Then build from and around them. As much as possible on my own.
I wonder how many other people learned in that way. And wouldn't that way
be the best to learn Calculus. For example, we know that
integral from a to b of f(x)dx is F(b) - F(a).
So, d/dx(integral from a to g(x) of f(x)dx) = d/dx (F(g(x)) - F(a)) =
F'(g(x)*g'(x)
Another example. Instead of proving (x^r)' = rx^(r-1) the long way, do the
following
y = x^r
lny = rlnx
(1/y)dy/dx = r/x
so, y' = x^r(r/x) = rx^(r-1)
Student who would notice that for x<0, we may still need some more
gymnastics would get 10 extra points.
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