Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Find two or three classmates and a few feet of chalkboard Introduce yourself to your new friends and write all of your names at the top of the chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises some are very hard Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own are from the mathematical folklore or are independently marked Here s a hint drawing pictures e g sketching graphs of functions will always make the problem easier Approximate Integration R b To approximate a f x dx let x b a n xi a i x and x i xi 1 xi 2 Then define the following approximations we use different number from Stewart for Simpson s Rule Sn Ln f x0 f xn 1 x Rn f x1 f xn x Tn f x0 2f x1 2f xn 1 f xn x 2 Mn f x 1 f x n x Sn f x0 4f x 1 2f x1 4f x 2 2f x2 2f xn 1 4f x n f xn x 6 These have the following errors EL sup f 0 x x a b ET sup f 00 x x a b b a 2 2n ER sup f 0 x x a b a 3 b 12n2 EM sup f 00 x x a b ES sup f 4 x x a b b a 2 2n b a 3 24n2 b a 5 2780n4 The word sup is short for supremum the symbol supx a b g x means the largest value of g x as x ranges over a b In practice it suffices to replace the suprema with some easy tocompute numbers which are even bigger 1 Explain each of the above approximation techniques when n 1 2 Let f x be a positive increasing function with negative second derivative on a b Place the Rb following five numbers in increasing order Ln Rn Tn Mn and a f x dx 3 If you were to evaluate each of the following integrals using each of the Midpoint Trapezoid and Simpson Rules with 5 subintervals what would be your expected error How many 1 subintervals would you need to ensure an error less than 0 00001 Z 4q Z 1 Z z a 1 x dx b z e dz c 0 0 6 ln x3 2 dx 4 4 a Show that Ln Rn 2 Tn b Show that Tn Mn 2 T2n c Show that Tn 2Mn 3 Sn 5 By explicit calculation show that Simpson s rule calculates the area under a cubic curve exactly What are the highest degree polynomials the rest of the approximation rules calculate exactly 6 By explicit calculation show that the errors for Ln and Rn are exact when f x is a linear function and that the errors for Tn and Mn are exact when f x is a quadratic function 7 Make sense of the following proof from Proofs without Words Exercises in Visual Thinking by Roger B Nelsen 1993 2
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