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# Mathematical Analysis I MA 425

NCS

GPA 3.93

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This 6 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 425 at North Carolina State University taught by Stephen Schecter in Fall. Since its upload, it has received 8 views. For similar materials see /class/223713/ma-425-north-carolina-state-university in Mathematics (M) at North Carolina State University.

## Reviews for Mathematical Analysis I

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Date Created: 10/15/15

Reviewing for the Second MA 425 Test Steve Schecter March 21 2006 1 Sequences 35736 a Important notions i Cauchy sequence 351 ii Cachy7s convergence criterion 355 iii Sequences that approach 00 or foo 361 b Important homework problems all assigned 2 Limits of functions chapter 4 a Important notions Cluster point 411 Limit of f at z 0 414 Sequential criterion for limits 418 Boundedness of f on a neighborhood of 0 421 V Existence of limit implies boundedness on a neighborhood 422 vi Limits of algebraic combinations of functions 424 6 6 proofs given in lecture and homework vii limmacf gt 0 implies f gt 0 on a neighborhood of 0 429 viii One sided limits 431 ix limmacf L if and only if limmacdr f L and limmac f L 433 X In nite limits 435 Xi Limits as x a 00 or z a foo 239 E 39 L b Important homework problems all assigned 3 Continuous functions 51754 a Important notions i De nition of continuity 511 ii Sequential criterion for continuity 513 iii Dirichlet7s function is not continuous anywhere 515g iv Algebraic combinations of continuous functions are continuous 521 1 S lt S viii Intermediate Value Theorem 536 x Q X X i Composition of continuous functions is continuous 526 Bounded functions 531 Boundedness Theorem 532 Max min Theorem 534 Uniform continuity 541 Lipschitz functions 544 Lipschitz implies uniform continuity 545 b Important homework problems all assigned 4 Derivative 61762 a lmportant notions ngrr The derivative 611 Differentiable implies continuous 612 Derivatives of algebraic combinations of functions 613 Chain rule 616 Mean Value Theorem 624 b lmportant homework problems all assigned Reviewing for the Second MA 425 Test Steve Schecter October 20 2005 1 Sequences 34736 a Important notions lt Subsequences 3417 342 Bolzano Weierstrass Theorem 348 Cauchy sequence 351 Cachy7s convergence criterion 355 Sequences that approach 00 or foo 361 b Important homework problems 34 147 35 47 problems assigned Sept 267 problem 1 assigned Sept 30 2 Limits of functions chapter 4 a Important notions g 39 L Cluster point 411 Limit of f at z 0 414 Sequential criterion for limits 418 Boundedness of f on a neighborhood of 0 421 Existence of limit implies boundedness on a neighborhood 422 i Limits of algebraic combinations of functions 424 6 6 proofs given in lecture and homework i limmacf gt 0 implies f gt 0 on a neighborhood of 0 429 One sided limits 431 1X limmacf L if and only if limmayr f L and limmac f L 433 X In nite limits 435 Xi Limits as x a 00 or z a foo b Important homework problems all assigned 3 Continuous functions chapter 5 a Important notions i De nition of continuity 511 1 Sequential criterion for continuity 513 Dirichlet7s function is not continuous anywhere 515g iV Algebraic combinations of continuous functions are continuous 521 V Composition of continuous functions is continuous 526 Vi Bounded functions 531 Vii Boundedness Theorem 532 Viii Max min Theorem 534 ix Intermediate Value Theorem 536 X Uniform continuity 541 Xi Lipschitz functions 544 xii Lipschitz implies uniform continuity 545 b lmportant homework problems all assigned 1 Limits 0 Reviewing for the Second MA 425 Test Steve Schecter April 3 2003 ffunctions chapter 4 a Important notions lt 4 Viii lt b lm 17 Cluster point 411 Limit of fac at 90 C 414 Sequential criterion for limits 418 Boundedness of f on a neighborhood of C 421 Existence of limit implies boundedness on a neighborhood 422 Limits of algebraic combinations of functions 424 6 6 proofs given in lecture and homework limx cf gt 0 implies f gt 0 on a neighborhood of C 429 One sided limits 431 limx cf L if and only if limQHCJr f L and limx c f L 433 ln nite limits 435 portant homework problems all assigned March 3 rst two assigned March 2 Continuous functions chapter 5 a lm portant notions De nition of continuity 511 Sequential criterion for continuity 513 Dirichlet7s function is not continuous anywhere 515g Algebraic combinations of continuous functions are continuous 521 Composition of continuous functions is continuous 526 Bounded functions 531 Boundedness Theorem 532 Max min Theorem 534 Intermediate Value Theorem 536 Uniform continuity 541 Lipschitz functions 544 xii Lipschitz implies uniform continuity 545 b Important homework problems 51 7 12 54 27 67 9 3 The derivative chapter 6 a Important notions The derivative 611 Differentiable implies continuous 612 i Derivatives of algebraic combinations of functions 613 iv Chain rule 616 v Mean Value Theorem 624 v Taylor7s Theorem 641 b lmportant homework problems 61 1b7 1c7 27 4 62 67 137 15 64 117 157 17

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