MTH 436 Spring 06 -- Tips for Exam 1 Exam 1 is scheduled for Friday, March 3, 3-5, in Tyler 106. Exam 1 covers classes 26-34 and homework assignments 1-4. You are expected to know and be able to state all definitions, propositions, and theorems given in class and in homework assignments as well as their proofs. You should know all of the important examples studied in class. You are expected to know solutions to all homework problems. Some problems on the exam will ask you to state and prove a theorem proved in class; some will ask you to state a definiton and give an example; some will be similar (or identical) to homework problems. While I was browsing my notes and homework assignments, the following questions jumped at me as good candidates for exam questions. (I am indicating below theorems, propositions, examples studied in class without stating precisely all the assumptions. Consult you notes for that.) -- State the definition of a Lipschtizian function f on an interval I. -- Prove that if the derivative f '(x) is bounded on I, then f is Lipschitzian on I (or the converse). -- Show that a Lipschitzian function is uniformly continuous. -- Give an example of a function on an interval [a,b] which is uniformly continuous but not Lipschitzian. -- Show that 1/x on (0,1] is not Lipschitzian. -- State the theorem about discontinuities of a monotone function. Prove the theorem. (Only one case as in Th 27.1.) -- State the definiton of the Riemann integral of f on [a,b]. Show that if the integral of f on [a,b] exits, then it is unique. -- State the Cauchy condition for integrablity. Prove the easy part: if f integrable on [a,b], the condition holds. -- Define the oscillation of f in [a,b]. State the characterization of integrability in terms of oscillations. -- Prove if f is continuous in [a,b], f is integrable in [a,b]. -- Prove if f is monotone in [a,b], f is integrable in [a,b]. -- Prove any part of Th. 29.1. -- Prove if f is integrable in [a,b], then |f| is integrable in [a,b]. Give an example that the converse is not true. -- Prove Th.32.1. -- Prove Th.32.2 (b). -- State and prove the 1st FTC. -- State the 2nd FTC. Prove (a) of it. (The running integral is Lipschtzian.) Of homework problems, the following seem particularly good: H1 #6, #7; H2 #2; H3 #6; H4 #2,3,4,5. Consider it a practice exam (Of course, the actual exam will be much shorter). Some of the problems listed above will appear, some will not. A problem or two not listed above may appear as well. Who knows. I hope you all get As!