The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guidline, and may not explicitly mention everything that you need to study.The exam will focus on chapters covered since the last midterm, but some of these chapters do rely on older material, so it is important that you remember the material from earlier chapters as well. It is also still important that you are comfortable with the basics of differention and integration (as covered in Calculus I).
Please review all homework, quiz and workshop problems for the chapters given below, and make sure you can solve all of them. You can also look at these problems for additional practice for each chapter.
8.4: Know the formula for the nth Taylor polynomial of a function, and remember that a Maclaurin polynomial is a Taylor polynomial centered at 0. You should know how to approximate the function value near the center using Taylor polynomials, and you should be able to find a bound for the error in the approximation using the error formula. You do not need to memorize the error bound formula, but you should know how to apply it if it is given to you. Make sure you know how to find the Taylor polynomials of the elementary functions listed at the end of this section.
10.1: Know what it means for a sequence to converge or diverge, and how to determine which case holds for a given sequence. You should also know how to find the limit of a given sequence using the methods described in this chapter.
10.2: Make sure you understand what a series is, especially how it is different from a sequence. You should know what the sequence of partial sums is, and what it means for a series to converge or diverge. You should know how to find the sum of a series using the methods of this section, especially for a geometric series.
10.3: You should understand all the theorems of this section and how to apply them in order to show whether a series converges or diverges. Make sure you go over all of the examples in this section. Also think about how you would use the integeral test or a comparison with a p-series or geometric series to find an error bound for a partial sum approximation for the series.
10.4: You should know how to determine whether a series converges absolutely, converges conditionally or diverges. You should know how to use the Leibniz Test, and how to find the error when using a partial sum to approximate an alternating series with terms decreasing to 0.
10.5: The ratio test is one of the most useful tests we have for determining whether a series converges or diverges. Know how to use both the ratio test and the root test, and practice lots of problems from this section.
10.6: Given a power series, you should know how to find its center, its radius of convergence, and its interval of convergence. You should also be able to find the values of x for which it converges absolutely, and for which it converges conditionally. Learn how to find the power series of functions of the form a/(b+cxn) by comparing them to the sum of a geometric series, and how to find other power series by integrating or differentiating a series you already have (term by term). You do not need to know the subsection on solving differential equations using power series.
10.7: You should know how to find the Taylor series for a function. Methods for doing this include using the formula for the co-efficients in the series, substituting terms in a series you already have, multiplying series by powers of x or by other series, comparing with the sum of a geometric series, integrating or differentiating a series you already have term by term, or by using the binomial theorem.
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