fundamental theorem of calculus final version
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The Fundamental Theorem of Calculus
2 December 2008
Katie Ford
Math 101
So what’s the deal?
The Fundamental Theorem of Calculus (FTC) links the two branches of calculus: differentiation and integration.
This seems completely obvious to us now, but it really wasn’t originally.
History
Isaac Barrow (1630-1677)
Isaac Newton (1642-1726)
Gottfried Leibnitz (1646-1716)
Definitions
Let f: D→R. f is uniformly continuous on D if for every ε>0 there exists a δ>0 such that |f (x)-f (y)| < ε whenever |x-y| < δ and x,y є D.
Let [a,b] be an interval in R. A partition P of [a,b] is a finite set of points {x0, x1,…, xn} in [a,b] such that a = x0 <x1 <…<xn= b
More Definitions: Upper and Lower Sums and Integrals
Upper and Lower Sum:
Suppose that f is a bounded function defined on [a,b] and that P= {x0,…,xn} is a partition of [a,b]. For each i=1,…n, we let
Mi(f)=sup{f(x): x є [xi-1, xi]}
and
mi(f)= inf {f(x): x є [xi-1, xi]}
We let xi=xi-xi-1 (i= 1,…,n), and then the upper and lower sums of f is defined with respect to P to be:
Upper integral- U(f)- inf {U(f,P): P is a partition of [a,b]} Lower integral- L(f)- sup {L(f,P): P is a partition of [a,b]} Integrable- if U(f)=L(f)=∫a
bf(x)dx
Theorems dealing with properties of Integrals
kf is integrable and ∫abkf = k∫a
bf. Also f+g is integrable and ∫a
b(f+g) = ∫abf + ∫a
bg.
If f is integrable on both [a,c] and [c,b], then f is integrable on [a,b]. Furthermore, ∫a
bf =∫acf +∫c
bf
The FIRST Fundamental Theorem of Calculus
Let f be integrable on [a,b]. For each x є [a,b], let
Then F is uniformly continuous on [a,b]. Furthermore, if f is continuous at c є [a,b], then F is differentiable at c and
Proof:
Since f is integrable on [a,b], it is bounded there. That is, there exists B>0 such that |f (x)| ≤ B for all x є [a,b]. To see that F is uniformly continuous on [a,b], let ε > 0 be given. If x,y є [a,b] with x<y and |x-y|< ε/B, then
Thus F is uniformly continuous on [a,b].
Now suppose that f is continuous at c є [a,b]. Then given any ε >0, there exists a δ>0 such that |f(t)-f(c)|< ε whenever t є [a,b] and |t-c|< δ. Since f(c) is a constant, we may write:
Proof (continued)
Then for any x є [a,b] with 0<|x-c|<δ, we have:
Since >0 was arbitrary, we conclude that:
The SECOND Fundamental Theorem of Calculus
If f is differentiable on [a,b] and f’ is integrable on [a,b], then:
Proof:
Let P = {x0, x1,…, xn} be any partition of [a,b]. We apply the Mean Value Theorem to each subinterval [xi-1,xi] and obtain points ti є (xi-1,xi) such that:
Since mi(f ’) ≤ f ’(ti) ≤ Mi(f ’) for all i, it follows that L(f ’,P) ≤ f(b)-f(a) ≤ U(f ’,P)
Since this holds for each partition P, we also have L(f’) ≤ f(b)-f(a) ≤ U(f ’)
But f’ is assumed to be integrable on [a,b], so L(f ’) = U(f ’) = ∫ab f ’
Thus, f(b)-f(a)=∫ab f ’
Bibliography
Lay, Steven R. Analysis With an Introduction to Proof. 4th ed. Upper Saddle River, NJ: Pearson: 2005.
Bardi, Jason Socrates. The Calculus Wars. New York: Thunder’s Mouth Press, 2006.
Stewart, James. Calculus, Concepts and Contexts. 2nd ed. Pacific Grove, CA: Wadsworth Group, 2001.
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