Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Spring1B/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, or are independently marked.Review and Preview1. (a) What is the definition of a definite integral?(b) What is the definition of an indefinite integral?(c) What is the statement of the Fundamental Theorem of Calculus (FTC)?2. (a) § Use the definition of a Riemann sum to approximateR20(x2−x)dx with four subinter-vals, taking the sample points to be right endpoints.(b) Use FTC to calculateR20(x2− x)dx exactly. What is the error in your estimation frompart (a)?(c) Repeat steps (a) and (b) with n subintervals, where n is an arbitrary number. You maywant to use the following facts:nXi=1i =n(n + 1)2nXi=1i2=n(n +12)(n + 1)3Remark: We will develop approximation methods better than the Right (or Left) Endpointmethod later in this course. You’ve met one already in Math 1A: the Midpoint approximation,which takes the sample points for calculating a Riemann sum to be the midpoints of eachsubinterval. Particularly ambitious students are invited to repeat steps (a) and (b) with theMidpoint approximation.3. (a) Use geometry to evaluate f(x) =Rx0√1 − t2dt.(b) Check the Fundamental Theorem of Calculus, by showing directly that f0(x) =√1 − x2.(c) Use the substitution t = sin θ to evaluate f(x). You will need the Pythagorean Theoremand the Double Angle Formula:1 − sin2θ = cos2θ cos2θ =12(cos 2θ + 1)Check that you get the same answer as in part (a).Remark: We will develop methods similar to the one in part (c) with which to evaluate manyintegrals that involve terms like√1 − t2.4. (a) State the Chain Rule for differentiation.1(b) Explain how the Chain Rule leads directly (via FTC) to the Substitution Formula forintegration.5. (a) What’s wrong with the following calculations?Z√1 + exexdx =u=1+exZ√u (u − 1) du =Zu3/2− u1/2du=25u5/2−23u3/2+ C =25(1 + ex)5/2−23(1 + ex)3/2+ CZπ0cos2x sin x dx =u=cos xZπ0u2du =u33π0=π33−030=π33(b) Evaluate the two integrals above correctly.6. § Use a substitution to evaluate the following integrals:(a)Z20y2py3+ 1 dy (b)Z51dt(t − 4)2(c)Z10sin(3πt) dt(d)Zsin πt cos πt dt (e)Ze√x√xdx (f)Zcos(ln x)xdx7. (a) § Use an integral to estimate the sum1000Xi=1√i.(b) Is your estimate an overestimate or an underestimate?(c) Estimate the error in your answer to part (a). When estimating errors, you shouldalways provide an overestimate rather than an underestimate.(d) Without a calculator, how could you improve the estimate of the value of the sum?Remark: In addition to developing techniques through which we can estimate the value of anintegral using a sum (as in question 2), we will develop techniques to estimate the values ofsums, including infinite sums, using integrals.8. § Sketch the curves x + y = 0 and x = y2+ 3y and find the enclosed area.9. § Sketch the curves y = x2+ 1 and y = 9 − x2, and find the volume obtained by rotating theenclosed region around the line y = −1.10. Let p(x) be a cubic function such that the curves y = p(x) and y = x2intersect when x = 0,x = a, and x = b.(a) Is the function p(x) uniquely determined by the above condition?(b) Express the above condition algebraically in terms of the function p(x) − x2.(c) § The two curves inclose two regions. If these two regions have the same area, how isb related to a? Remark: there are various cases here, depending on the order of 0, a, b.Up to switching the names of a and b and switching x 7→ −x, there are two possibilities:either 0 < a < b or a < 0 < b. Consider those two
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