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Calculus II

by: Reuben Hudson DDS

Calculus II MATH 132

Reuben Hudson DDS
GPA 3.55


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This 1 page Class Notes was uploaded by Reuben Hudson DDS on Friday October 30, 2015. The Class Notes belongs to MATH 132 at University of Massachusetts taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/232223/math-132-university-of-massachusetts in Mathematics (M) at University of Massachusetts.

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Date Created: 10/30/15
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 1 W ftdt I 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 mgrggltzgt2gltagt 133 rode ftdt lim mftdt m4 er 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 ftdt 2 mm 7 a so as x a of we have 0 133 mm 7 a 2 133 m ftdt 2 133 mm 7 a 0 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 gltbgt2gltzgt 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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