Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch 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: in particular, the last few exercises may be 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, or are independently marked.The Definite Integral1. § Estimate the area under the graph f(x) = cos x from x = 0 to x = π/2 using four rectanglesand either left or right endpoints. Determine whether your estimate is an underestimate oran overestimate.2. § Speedometer readings for a motorcycle at 12-second intervals are given in the table below.Estimate the distance traveled by the motorcycle during the first minute of driving. Can youdetermine whether the left or right estimates are over- or underestimates?t (s) 0 12 24 36 48 60v (ft/s) 30 28 25 22 24 273. § Find an expression for the area under the graph of f as a limit. Do not evaluate the limit.(a) f(x) =4√x, 1 ≤ x ≤ 16(b) f(x) = x cos x, 0 ≤ x ≤ π/24. § Determine a region whose area is equal to the given limit. Do not evaluate the limit.(a) limn→∞nXi=1π4ntaniπ4n(b) limn→∞n−1Xi=0ln1 +inn + i5. (a) Find an expression for the area under the curve y = x2from 0 to 2, using right endpoints.(b) Use the fact that12+ 22+ 32+ ··· + (n − 1)2+ n2=nXi=1i2=n(n + 1)(2n + 1)6to evaluate your expression in part (a).(c) What happens if you use left endpoints?6. § Evaluate the Riemann sum for f(x) = 3 −12x, 2 ≤ x ≤ 14, with six sub intervals, takingthe sample points to be the left endpoints. What does the sum represent?7. § Express the limit as a definite integral on the given interval:(a) limn→∞nXi=1xiln(1 + x2i) ∆x, [2, 6] (b) limn→∞nXi=1q2x∗i+ (x∗i)2∆x, [1, 8]8. § Using only the following facts, and the linearity (Sum, Difference, and Constant Multiplerules) ofPandR, evaluate the integrals below. The facts:nXi=11 =n1,nXi=1i =n(n + 1)2,nXi=1i2=n(n +12)(n + 1)3,nXn=1i3=n2(n + 1)24The integrals:(a)Z5−1(1 + 3x) dx (b)Z20(2 − x2) dx (c)Z50(1 + 2x2) dx (d)Z21x3dx9. Using the facts in the previous problem, prove the following formulas:(a)Zbax dx =b2− a22(b)Zbax2dx =b3− a33(c)Zbax3dx =b4− a4410. Based on the formulas from the previous problem, guess the formula forRbaxndx. Check thatyour formula works for n = 0. What happens when n = −1? For which n are you confidentthat your formula is correct?11. § Evaluate each of the following integrals by interpreting them in terms of areas.(a)Z3012x − 1dx(b)Z3−1(3 − 2x) dx(c)Z2−2p4 − x2dx(d)Z0−31 +p9 − x2dx(e)Z2−1|x|dx(f)Z1−1x5cos x dx12. (a) Use the fact that cos x = sin(π4− x) to prove thatRπ/40cos2x dx =Rπ/40sin2x dx.(b) Use the previous problem and the Pythagorean theorem to findRπ/40sin2x.13. § IfR51f(x) dx = 12 andR54f(x) dx = 3.6, findR41f(x) dx.14. § FindR50f(x) dx if f (x) =3, x < 3x, x ≥
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