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Universities » Utah State University (USU) » MATH - Mathematics » 4200 - Foundations of Analysis » Flash Cards

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Class:MATH 4200 - Foundations of Analysis
Subject:Mathematics
University:Utah State University
Term:Spring 2011
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Extreme Value Theorem If f is continuous on closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b].
Mean Value Theorem If f is a differentiable function on the interval [a,b], then there exists a number C between a and b such that f'(c)=(f(b)-f(b))/(b-a)
Intermediate Value Theorem Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) does not equal f(b). Then there exists a number c in (a,b) such that f(c)=N
Fundamental Theorem of Calculus Suppose f is continuous on [a,b]: 1. If g(x)= the integral from a to x f(t) dt, then g'(x)=f(x) 2. the integral from a to b of f(x) dx = F(b)-F(a) where F is any antiderivative f, that is F'=f
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sin(x) x-X^3/3!+x^5/5!-....+(-1)^nX^2n+1/(2n+1)!+....
cos(x) 1-x^2/2!+x^4/4!+....+(-1)x^2n/(2n!)
e^x 1+x+x^2/2!+x^3/3!+....+x^n/n!
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 Extreme Value TheoremIf f is continuous on closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b].
 Mean Value TheoremIf f is a differentiable function on the interval [a,b], then there exists a number C between a and b such that f'(c)=(f(b)-f(b))/(b-a)
 Intermediate Value TheoremSuppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) does not equal f(b). Then there exists a number c in (a,b) such that f(c)=N
 Fundamental Theorem of CalculusSuppose f is continuous on [a,b]:
1. If g(x)= the integral from a to x f(t) dt, then g'(x)=f(x)
2. the integral from a to b of f(x) dx = F(b)-F(a) where F is any antiderivative f, that is F'=f
 sin(x)x-X^3/3!+x^5/5!-....+(-1)^nX^2n+1/(2n+1)!+....
 cos(x)1-x^2/2!+x^4/4!+....+(-1)x^2n/(2n!)
 e^x1+x+x^2/2!+x^3/3!+....+x^n/n!
  Definition
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