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.

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)

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 {x 0 , x 1 ,…, x n } in [ a,b ] such that a = x 0 <x 1 <…<x n = b

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 {x 0 , x 1 ,…, x n } in [ a,b ] such that a = x 0 <x 1 <…<x n = 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 = {x 0 ,…,x n } is a partition of [ a,b ]. For each i =1,…n, we let M i ( f )=sup{ f (x): x є [x i-1 , x i ]} and m i ( f )= inf { f ( x ): x є [x i-1 , x i ]} We let x i =x i -x i-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 b f ( x ) dx

Upper and Lower Sum:

Suppose that f is a bounded function defined on [ a,b ] and that P = {x 0 ,…,x n } is a partition of [ a,b ]. For each i =1,…n, we let

M i ( f )=sup{ f (x): x є [x i-1 , x i ]}

and

m i ( f )= inf { f ( x ): x є [x i-1 , x i ]}

We let x i =x i -x i-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 b f ( x ) dx

Theorems dealing with properties of Integrals kf is integrable and ∫ a b kf = k ∫ a b f. Also f+g is integrable and ∫ a b ( f+g ) = ∫ a b f + ∫ a b g. If f is integrable on both [a,c] and [c,b], then f is integrable on [a,b]. Furthermore, ∫ a b f = ∫ a c f +∫ c b f

kf is integrable and ∫ a b kf = k ∫ a b f. Also f+g is integrable and ∫ a b ( f+g ) = ∫ a b f + ∫ a b g.

If f is integrable on both [a,c] and [c,b], then f is integrable on [a,b]. Furthermore, ∫ a b f = ∫ a c f +∫ c b f

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

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:

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:

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:

If f is differentiable on [ a,b ] and f’ is integrable on [ a,b ], then:

Proof: Let P = { x 0 , x 1 ,…, x n } be any partition of [ a,b ]. We apply the Mean Value Theorem to each subinterval [ x i -1 , x i ] and obtain points t i є ( x i -1 , x i ) such that: Since m i ( f ’) ≤ f ’( t i ) ≤ M i ( 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 ’) = ∫ a b f ’ Thus, f ( b )- f ( a )=∫ a b f ’

Let P = { x 0 , x 1 ,…, x n } be any partition of [ a,b ]. We apply the Mean Value Theorem to each subinterval [ x i -1 , x i ] and obtain points t i є ( x i -1 , x i ) such that:

Since m i ( f ’) ≤ f ’( t i ) ≤ M i ( 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 ’) = ∫ a b f ’

Thus, f ( b )- f ( a )=∫ a b 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.

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.