course mth174
Please feel free to add as much information you'd want to these...
Nested Interval theorem:When considering a set of nested intervals we know that it's a collection of real numbers L(sub)n= [a(sub)n, b(sub)n],closed and bounded, with an
Intermediate value theorem:
if f(x)=y is continuous on the intervals (a,b), then there is a number between (a,b), we can call it g, then f(c)=g.
if we have integral(f(t) from x to x+ d'x= f(c)
then F(x+d'x) - F(x) = f(c) d'x
f(c)= F(x+d'x)- F(x) / d'x
then take the limit as d'x -> 0
completeness axiom:
If a nonempty set, S, of real numbers has an upper bound M (for all x in S), then S has a least upper bound b.
We know that the fundamental theorem (I) is for a continuous function bounded by the intervals (a,b) so F(x)= integral( f(t)) from a to b
We know that for a function f(x) from (a,b) the integral of f(x) from a to b will be equal to the area under the defined set of intervals
Go to
http://www.wiley.com/college/bcs/0471408263/instructor/instructor.html
and look at the document Theory 1.
Theory 3 is also relevant to the integral.