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Berkeley MATH 1B - Math 1B - Discussion Exercises

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Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Find two or three classmates and a few feet of chalkboard. Introduce yourself to your newfriends, and write all of your names at the top of the chalkboard. As a group, try your hand atthe following exercises. Be sure to discuss how to solve the exercises — how you get the solutionis much more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: some are very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, are from the mathematicalfolklore, or are independently marked.Here’s a hint: drawing pictures — e.g. sketching graphs of functions — will always make theproblem easier.Approximate Integration — CorrectedTo approximateRbaf(x) dx, let ∆x = (b − a)/n, xi= a + i∆x, and ¯xi= (xi−1+ xi)/2. Thendefine 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)) ∆xTn= (f(x0) + 2f(x1) + ··· + 2f(xn−1) + f(xn))∆x2Mn= (f(¯x1) + ··· + f(¯xn)) ∆xSn= (f(x0) + 4f(¯x1) + 2f(x1) + 4f(¯x2) + 2f(x2) + ··· + 2f(xn−1) + 4f(¯xn) + f(xn))∆x6These have the following errors:|EL| ≤ supx∈[a,b]f0(x)(b − a)22n|ER| ≤ supx∈[a,b]f0(x)(b − a)22n|ET| ≤ supx∈[a,b]f00(x)(b − a)312n2|EM| ≤ supx∈[a,b]f00(x)(b − a)324n2|ES| ≤ supx∈[a,b]f(4)(x)(b − a)52880n4The word “sup” is short for “supremum” — the symbol “supx∈[a,b]g(x)” means “the largest valueof g(x) as x ranges over [a, b]”. In practice, it suffices to replace the suprema with some easy-to-compute 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 thefollowing five numbers in increasing order: Ln, Rn, Tn, Mn, andRbaf(x)dx.3. Of the Midpoint and Trapezoid rules, pick your favorite. If you were to evaluate each of thefollowing integrals using that rule with 5 subintervals, what would be your expected error?1How many subintervals would you need to ensure an error less than 0.00001?(a)Z41q1 +√x dx (b)Z21√z e−zdz (c)Z64ln(x3+ 2) dx4. (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 curveexactly. What are the highest degree polynomials the rest of the approximation rules calculateexactly?6. By explicit calculation, show that the errors for Lnand Rnare exact when f(x) is a linearfunction, and that the errors for Tnand Mnare exact when f(x) is a quadratic function.7. Make sense of the following proof from Proofs without Words: Exercises in Visual Thinkingby Roger B. Nelsen


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Berkeley MATH 1B - Math 1B - Discussion Exercises

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