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This 3 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH360 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 51 views. For similar materials see /class/215400/math360-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.
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Date Created: 09/28/15
Review Sheet for Second Exam Mathematics 360 November 18 2002 The exam will be held in class Books and notes will not be allowed but you may compile a one page list of de nitions theorems and examples You may not put any proofs on this page and it will be collected with the test In all cases if you are just asked to prove something you can use any theorem we have proven in class or that is proven in the textbook If you are asked to prove something directly from the de nition then you may not use anything other than the de nition Know the de nitions 0 convergence of a series 221 an of real numbers 0 absolute and conditional convergence of a series of real numbers 0 compactness of a set 0 sequential compactness of a set 0 connectedness of a set 0 path connectedness of a set continuity at a point 0 uniform continuity on an interval di erentiability at a point 0 integrability on an interval Be able to prove a series converges by the comparison test the ratio test the root test or the alternating series test a set in R is compact by showing it is closed and bounded using the Heine Borel theorem a set is connected by showing it is path connected a function f R a R is continuous at a point using the 6 6 de nition a function attains a maximum or a minimum by proving it is continuous on a compact set a function attains an intermediate value between two points for example a function has a zero by proving it is continuous on a connected set a function is uniformly continuous by showing it is continuous on a compact set a function is uniformly continuous by showing its derivative is bounded a function is integrable by showing it is increasing or decreasing a function is integrable by showing it is continuous except at nitely many points Know the proofs Theorem 294 iv the ratio test Theorem 294 viii the alternating series test Theorem 421 continuous image of a compact set is compact Theorem 422 continuous image of a connected set is connected Theorem 431 composition of continuous functions is continuous Theorem 475 iii the product rule Theorem 4710 Rolle7s theorem Theorem 488 fundamental theorem of calculus Know the key examples 00 n 1 Zr 17quot n0 sum if M lt 1 be able to prove this using the explicit formula for the partial 1 A lt71 d b t En 1verges u g n The topologist7s sine curve7 f sin Be able to prove that it is discontinuous at z 0 understand why the graph of x fz7 together with the vertical segment 0 gtlt 7117 is connected but not path connected be able to prove that x2f is differentiable everywhere7 but the derivative is not continuous converges Let f be the characteristic function of the irrational numbers 0 z E Q f 96 1 z 9 Q Be able to prove that it is discontinuous at every point7 and it is not integrable on any interval that z gt gt fx is continuous at z 0 but not any other point that z gt gt x2fx is differentiable at s 0 but discontinuous everywhere else Be able to construct examples of functions which do not satisfy the maximumminimum theorem eg de ned on noncompact sets or the intermediate value theorem eg de ned on disconnected sets Know examples of functions that are continuous but not uniformly continuous uni formly continuous but not Lipschitz Lipschitz but not differentiable