FTC1 - + m = x . or the same 1 reason, lim + M = x ....

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The Fundamental Theorem of Calculus I Statement Suppose that f is continuous on [ a,b ]. If we deFne g ( x )= ± x a f ( t ) dt , then g ± ( x )= f ( x ) for all a < x < b . Proof 1. ±ix x in the interval ( a,b ). 2. DeFne the function h ( ± )= g ( x + ± ) - g ( x ) ± , for nonzero values of ± everywhere that g ( x + ± ) is deFned. [We do this because g ± ( x ) is then just the limit lim ± 0 h ( ± ), which is easier to work with.] 3. Substituting the expression for g , we get h ( ± )= ± x + ± a f ( t ) dt - ± x a f ( t ) dt ± = ± x + ± x f ( t ) dt ± 4. Consider ± > 0. 5. We know that f is continuous on the closed interval [ x,x + ± ]. By the Extreme Value Theorem (see page 272) we know that f attains an absolute minimum value f ( m ± ) and an absolute maximum value f ( M ± ) on [ x,x + ± ] for some m ± and M ± lying in [ x,x + ± ]. [We write m ± and M ± because these values depend on our choice of ± .] 6. This means that f ( m ± ) f ( t ) f ( M ± ) for all t lying in the interval [ x,x + ± ]. 7. By Comparison Property 8 on page 375, it follows that f ( m ± ) ± ± x + ± x f ( t ) dt f ( M ± ) ± . 8. Combining this with Step 3 , this means that f ( m ± ) h ( ± ) f ( M ± ). 9. We have x m ± x + ± from Step 5 . Taking limits preserves nonstrict inequalities so upon taking the limit ± 0 + and using the Squeeze Theorem , it follows that lim
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Unformatted text preview: + m = x . or the same 1 reason, lim + M = x . [Remember that were still only considering positive so we can only talk about right limits.] 10. By the continuity of f , this means that lim + f ( m ) = lim + f ( M ) = f ( x ). 11. Taking limits preserves nonstrict inequalities and we want to take the limit of the expression f ( m ) h ( ) f ( M ) from Step 8 as + . 12. From Step 10 , we know that f ( m ) f ( x ) and f ( M ) f ( x ). By the Squeeze Theorem , this means that lim + h ( ) = f ( x ). 13. We can similarly perform Steps 4-12 for the case where < 0 and we get lim -h ( ) = f ( x ). 14. The previous two steps imply that lim h ( ) = f ( x ). 15. Since g ( x ) = lim h ( ) by the way we have dened h , the previous step shows that g ( x ) = f ( x ). 2...
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This note was uploaded on 06/26/2010 for the course MATH 1A taught by Professor Wilkening during the Fall '08 term at University of California, Berkeley.

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FTC1 - + m = x . or the same 1 reason, lim + M = x ....

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