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 Indefinite Integral1. Let f(x) be a continuous function. What is the derivativeRbaf(x) dx0? What isRf(x) dx0?How does this illustrate the difference between definite and indefinite integrals?2. What is wrong with the following:Z31x dx =x2231=92−12=82= 4 + C3. § Find the general indefinite integral:(a)Z√x3+3√x2dx(b)Zy3+ 1.8y2− 2.4ydy(c)Zvv2+ 22dv(d)Zx2+ 1 +1x2+ 1dx(e)Zcsc2t − 2etdt(f)Zsec t (sec t + tan t) dt(g)Zsin 2xsin xdx(h)Zcos x +12xdx(i)Zex− 2x2dx4. § Evaluate the definite integral:(a)Z311 + 2x − 4x3dx(b)Z0−2u5− u3+ u2du(c)Z90√2t dt(d)Z50(2ex+ 4 cos x) dx(e)Zπ/3π/4sec θ tan θ dθ(f)Zπ/30sin θ + sin θ tan2θsec2θdθ(g)Z10−102exsinh x + cosh xdx(h)Z1/√30t2− 1t4− 1dt(i)Z2−1x − 2|x|dx5. § Draw the curve x = 2y − y2, and explain why the area between the curve and the y-axis isR20(2y − y2)dy. Evaluate this integral.The Net Change Theorem6. § If w0(t) is the rate of growth of a certain child in pounds per year, what doesR105w0(t) dtrepresent?7. § Recall that the marginal revenue function R0(x) is defined as the derivative of the revenuefunction R(x), which returns the total revenue received if a company sells x units of a givencommodity. What doesR50001000R0(x) dx represent?8. § If the units for x are feet and the units for a(x) are pounds per foot, what are the units forda/dx? What aboutR82a(x) dx?9. § A particle starts with velocity v(0) = −4 and has acceleration a(t) = 2t + 3, where t ismeasured in seconds and a(t) is in meters per second squared. For far does the particle travelin the first three seconds?10. § The marginal cost of manufacturing x yards of a certain fabric is C0(x) = 3 − 0.01 x +0.000 006 x2(in dollars per yard). Find the increase in cost if the production level is raisedfrom 2000 yards to 4000 yards.11. § Find the interval [a, b] for which the value of the intervalRba(2 + x − x2) dx is a
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