MA241-012, Fall 2008Final exam summaryDisclaimer: “This review sheet is only intended to give you some guidance, and thus should not be taken asan exhaustive listing of possible final exam question topics.”Calculators which are not capable of symbolic manipulation are allowed on the final (i.e., no TI-89s, TI-92s,or equivalent). For those of you with a TI-89/92, you’ll need to either borrow another calculator or see meto make other arrangements.In addition to a calculator, you are also allowed to have a formula sheet(s) with up to 10 “items” onit. The following are a couple of examples of what I mean by an “item”:• A formula or theorem which is boxed in red or yellow in the book. Exceptions to this would besomething like the red box on page 612, which would count as 4 items (if you’re wondering if a boxwill count as more than one item, just send me an e-mail with the page number).• A fully worked out problem, as long as you don’t include a general statement of the tests or formulasthat you used in the example For instance, if you had an example showing a series which converged bythe ratio test, you could state at the end that “the limit is < 1 and thus the series converges by theratio test”, but then saying what would have happened if the limit was > 1 or = 1 would count as asecond item. See the test solutions for a rough guide to how much detail you’re allowed to have.• If you want to bring the summary sheet of all the series convergence tests that I passed out in class(without any additions, notes, or changes other than the form of the geometric series), I will countthat as 7 items.You may use as many pages and space things out as much as you like. You must turn your formulasheet(s) in with t he final. Formula sheets that have too many items on them will result inpoints off on the final! If you want me to look over your sheet to be sure you’re not over your limit,you can either scan and e-mail it to me (please keep the file size around 1MB), or if you come to the Q&Asession on Tuesday I can look it over then.The final is c omprehensive (some exceptions are noted below). In general, anything that was or couldhave been asked on a test could show up on the final. See the assigned homework to get a betteridea of specific problem types that may show up.§§5.5-5.8You will not have any questions directly from these sections on the final. However, I expect you to know andbe able to use the integration techniques from these sections if they were needed within another problem.§5.9-5.10An approximation problem with the trapezoid rule and/or Simpson’s rule could show up (the mid-point ruleshould have been review from calculus 1, so it won’t be on the final). Improper integrals of all types thatwe talked about in class, as well as the comparison theorem for improper integrals could be on the final.1§§6.1-6.4These sections cover the following applications of definite integrals: Area between curves, volumes of revolu-tion, arc length, and average value of a function. Any of these types of problems are possibilities. You willhave one or two volume of revolution problems, and at least one problem of the other types.§§6.5This section contains more applications of definite integrals. The four types of problems we looked at inthis section were: the work done in stretching/compressing a spring, the work done in pumping water outof a tank, the work done lifting a chain or rope (with or without an object attached to it), and findingthe hydrostatic force on a submerged vertical plate. Two of those types of problems were on your secondtest, but all are possibilities for the final (you will have a least one problem from this section on the final).Moments and centers of mass will not be on the final.§§7.1-7.2You may have to verify that a given function is a solution to a DE (with or without initial conditions). Givena direction field, you need to be able to sketch the solution corresponding to a given initial condition (seetest 2). You may have to use Euler’s method to approximate a solution to an IVP of the type we saw inclass (y0= F (x, y), y(a) = c).§§7.3-7.4These sections covered solving separable DEs and applications. The three types of applications we lookedat from these s ections were: orthogonal trajectories, m ixing problems, and Newton’s Law of Cooling. Youwill have one Newton’s Law of Cooling problem, and possibly one of the other application problems.§§7.7-7.9 (supplement)You should know how to solve second order linear constant coefficient DEs, both homogeneous and non-homogeneous. You are also e xpec ted to know how to set up and solve the spring-with-a-mass type ofapplication problems. You do not need to know the electrical circuit application problems.§8.1You should know how to determine the convergence or divergence of a sequence (and how/why this is differentfrom a series converging or diverging).§§8.2-8.4You need to know all of the series convergence tests discussed in these sections (we talked about all ofthem in class): nth-term test for divergence, harmonic series, telescopic series, p-series, integral test, directcomparison test, limit comparison test, alternating series test, absolute convergence, and the ratio test. Youalso need to be able to identify which test you need to use, and also remember that sometimes you needtests within tests: in both comparison tests, you need to know about the convergence or divergence of theseries you’re using for the comparison and for absolute convergence, you need to be able to determine theconvergence or divergence of the absolute value of the terms.2§8.5Given a power seriesP∞n=0cn(x− a)n, you need to be able to both the interval of convergence and the radiusof convergence .§8.6Be able to use the power series expansion of11−xto find power series expansions for other functions bysubstitutions, multiplying by constants or powers of x (not by other power series), differentiating, integrating,or some combination of these four techniques. You may also be asked to find the interval of convergence andthe radius of convergence for the resulting power series.§8.7You need to know the general formula for Taylor and Maclaurin polynomials/series and be able to computethem for a given function. If you are asked to find a Taylor or Maclaurin polynomial, then the degree willbe specified. If I ask you to find a Taylor or Maclaurin series, then I also expect you to find the patternin the terms