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 Fundamental Theorem of Calculus, part 21. § Use the Fundamental Theorem of Calculus to evaluate the following integrals:(a)Z5−26 dx(b)Z101 +12u4−25u9du(c)Z2−1x3− 2xdx(d)Z415 − 2t + 3t2dt(e)Z10x4/5dx(f)Z213t4dt(g)Z20x2 + x5dx(h)Z91x − 1√xdx(i)Z20(y − 1)(2y + 1) dy(j)Zπ/40sec2t dt(k)Z1−1e2udu(l)Z√3/21/26√1 − t2dt2. § What’s wrong with the following equations?Z1−2x−4dx =x−331−2= −38Zππ/3sec θ tan θ dθ = sec θ]ππ/3= −33. What’s wrong with the following equations?Zπ/4π/61cos tdt =1sin tπ/4π/6=√2 − 2Z20√3 − x dx =r3x −x22#20= 24. (a) What is the derivative of g(x) = x ln x?(b) Use your answer to part (a) to find an antiderivative for f(x) = ln x.(c) Use your answer to part (b) to evaluateR21ln x dx.5. § Evaluate limn→∞nXi=1i3n4by first recognizing the sum as a Rieman sum for a function definedon [0, 1], and then evaluating the corresponding integral using the Fundamental Theorem ofCalculus.6. (a) § Prove that cos x2≥ cos x for 0 ≤ x ≤ 1. Hint: What’s the relationship between x andx2?(b) § Deduce thatRπ/60cos x2dx ≥12.7. § Prove that 0 ≤Z105x2x4+ x2+ 1dx ≤ 0.1 by comparing the integrand to a simpler function.8. § If f is continuous on [a, b], prove that:2Zbaf(x)f0(x) dx = [f(b)]2− [f(a)]29. § A manufacturing company owns a major piece of equipment that depreciates at the (contin-uous) rate f = f(t), where t is the time measured in months since its last overhaul. Because afixed cost A is incurred each time the machine is overhauled, the company wants to determinethe optimal time T (in months) between overhauls.(a) Explain whyRt0f(s) ds represents the loss in value of the machine over a period of timet since the last overhaul.(b) Let C = C(t) be given byC(t) =1tA +Zt0f(s) dsWhat does C represent and why would the company want to minimize C?(c) Show that C has a minimum value at the numbers t = T where C(T ) = f(T ).10. § If f is a differentiable function on (0, ∞) such that f (x) is never 0 andRx0f(t) dt = [f(x)]2for all x > 0, find f.11. § Evaluate limx→01xZx0(1 − tan 2t)1/tdt.12. § Evaluate the following limit by interpreting it as a Riemann sum for a continuous function onthe interval [1, 2], and evaluating the integral using by the Fundamental Theorem of Calculus:limn→∞ 1√n√n + 1+1√n√n + 2+1√n√n + 3+ ··· +1√npn + (n − 1)+1√n√n +
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