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# Note for MATH 132 at UMass(2)

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COURSE
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TYPE
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PAGES
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WORDS
KARMA
25 ?

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 22 views.

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Date Created: 02/06/15
The Fundamental Theorem of Caluculus7 Part 1 Adam Gamzon ln class7 we showed why the statement about di erentiability is true Now we will prove the continuity part of the statement First7 we restate the theorem Theorem 1 ff is continuous on 11 then the function 9 de ned by W m ftdt is continuous on 11 and di erentiable on 11 with gx for all m 6 11 Proof of continuity We want to show that g is continuous on 11 Since 9 is differentiable on 11 we know that g is also continuous on 11 Therefore7 we just need to show that gz is continuous from the right at z a and that gz is continuous from the left at z b To show gz is continuous from the right at z at7 we want to prove that 13299079a 133 a rode ftdt 332 a W 0 Note that7 by the extreme value theorem7 for any x in 11 there exist points u and v in aw such that u 2 ft 2 u for all t in 0 m since 1 is continuous Thus mm 7 a 2 w ftdt 2 WW 7 a so as x a of we have 0 lim mama lim mftdt limfvzia0 want want zaa That is7 limmHaJr ftdt 0 so gz is continuous from the right at z a To get continuity from the left at z b7 note that for a g x b7 17 m gltbgtegltzgt rode ftdt bftdt arlttgtdt b ftdt Then use the extreme value theorem to apply the same argument as in the case of z a 1

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