thanks for saving me the effort to type up rolle's theorem and mean value theorem for functions
Back to integrationing:
Mean Value Theorem (integrals):
if f() is continuous on [a,b] (closed interval)
∃ a value "c" somewhere on [a,b] (inclusive) where
<sub>a</sub>∫<sup>b</sup> f(x)dx = f(c)(b-a)
Average Value:
f(c) = Avg value of f
If f is integrable on [a,b]
then avg value =
1/(b-a) <sub>a</sub>∫<sup>b</sup>f(x)dx
Second Fundamental Theorem of Calculus
If f() is continuous on (I) where a is a number in that interval
then:
d/dx[ <sub>a</sub>∫<sup>x</sup>f(t)dt] = f(x)
Rieman Sum:
<sub>i=1</sub>∑<sup>n</sup> f(c<sub>i</sub>)Δx; x<sub>i</sub> ≤ c<sub>i</sub> ≤ x<sub>i</sub>
Definite Integrals:
If f() exists on [a,b]
holy crap okay I don't think I can do this in symbol notation. Heck, i don't even undestand what I have written in my notes. but it's the limit definition if someone wants to fill in the hole
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