Math 20B Final Exam Review Outline Winter 2020 The final exam is cumulative and will test your mastery of all of the course material, however, there will be a slight emphasis on the material that we covered after the second midterm exam; in particular, around 40% of the problems come from the material in sections 10.3, 10.4, 10.5, 10.6, and 10.8. Below is a summary of the topics/skills you need to master from each of the sections. We suggest that you start by solving the examples from your lecture notes, and then moving on to re-doing enough of the assigned homework problems from each section in order to feel confident about the main concepts. When you are solving the aforementioned problems, it is of the utmost importance that you attempt to solve them without looking at our solutions or solutions/hints on Webassign. This is the only way to really make the solutions your own. If you are able to understand and solve all of the lecture examples and the homework problems on your own and without any assistance, then you should do very well on this exam. • Section 5.6: Given a velocity function, know how to use definite integrals to calculate displacement and total distance travelled. • Section 5.7: Know how to evaluate integrals using u-substitution. If you are using u-substitution to evaluate a definite integral, don't forget to change the limits of integration. • Section 6.1: Know how to find the area of a plane region, e.g. the area between two curves, by setting up and solving a definite integral. It is usually helpful to sketch the region first; see the list of prerequisite skills a the end of this review outline. • Section 6.2: Know how to find the volume of a solid with a given base and given cross-sections by setting up and solving a definite integral. Know how to calculate the average value of a given function over a specified interval. • Section 6.3: Know how to find the volume of a solid of revolution by setting up and solving a definite integral. Remember that solids can be generated by rotating a plane region about the x-axis, the y-axis, or any other vertical or horizontal line, so practice all of the possible variations. • Section 7.1: Know how to evaluate integrals using integration by parts. Remember that for some problems you need to apply integration by parts more than once and/or combine integration by parts with u-substitution. • Section 11.3: Given the Cartesian coordinates of a point, know how to find its polar coordinates and vice versa. Know how to plot points using their polar coordinates. (You won't have an exam question that resembles the homework from section 11.3, but you need to understand the 11.3 material to do the area problems from 11.4, and you may have a exam question that resembles the homework from section 11.4). • Section 11.4: Know how to find the area of a region described by polar equations. Note that we will not ask you to sketch the graph of any polar equation; if you are asked to find such an area, the graph would be provided for you. • Supplement 1: { Know how to write a complex number a + bi in the polar form r(cos θ + i sin θ), and vice versa. { Know how to find the nth power of a given complex number using DeMoivre's Theorem. { Know how to find all of the nth roots of a given complex number. • Supplement 2: Know how to write a complex number a + bi in the form reiθ, and vice versa. Know how to integrate a complex exponential function, e.g. R e2ix dx. • Supplement 3: Know how to solve integrals such as R cos(2x) sin(5x) dx or R e7ix cos(4x) dx using eix + e−ix eix − e−ix the formulas cos x = and sin x = . 2 2i 1 2 • Section 7.2: Know how to solve integrals such as R sin3(x) cos4(x) dx and R sec4(x) tan5(x) dx using a combination of a u-sub and the trig identities: sin2 x + cos2 x = 1 or sec2 x − tan2 x = 1 • Section 7.3: Know how to solve integrals using trig substitutions. There are three types of trig subs that you can use, so you need to be able to pick the correct one: x = a sin θ or x = a tan θ or x = a sec θ. Remember that at the end of most of these problems, you need to use your little triangle to write your final antiderivative in terms of x. • Section 7.5 and Supplement 5: Know how to find the partial fraction expansion (PFE) of a given rational function and then use the PFE to integrate the function. Remember that if the degree of the numerator is equal to or greater than the degree of the denominator, then you must do long division before the PFE. • Section 7.7: { Know how to use the definition of an improper integral as a limit to determine whether it converges or diverges. Know how to handle both types of improper integrals: continuous function but infinite interval of integration OR finite interval of integration but integrand is discontinuous at one of the endpoints. On the finite interval of integration problems, you should write a one-sided limit, not a two-sided limit. { Know how to determine convergence or divergence using the Comparison Test for Improper Integrals. • Section 10.1: { Know how to write out the first few terms of a sequence. { Know how to determine if a given sequence converges or diverges (often we need to use Math 20A techniques such as L'Hopital's Rule). { If a sequence converges, know how to find its limit. • Section 10.2: { Know how to find the first few partial sums of a given series. { Given a geometric series in shorthand summation notation P arn or in longhand notation a+ar+ar2 +ar3 +··· , be able to find its common ratio and identify if it converges or diverges. { Be able to find the sum of a convergent geometric series. { Be able to use a partial fraction expansion to find a formula for the N th partial sum of a telescoping series and then find the sum of the series (e.g. like problem 4 in 10.2 online homework). 1 X th { Remember: If lim an 6= 0, then the series an diverges (the N Term Test for Divergence). n=1 • Section 10.3: { Be able to use the Integral Test to determine if a series converges or diverges. { Be able to use the Comparison Test to determine if a series converges or diverges. { Be able to use the Limit Comparison Test to determine if a series converges or diverges. Note that you can only use the 10.3 tests on series with all positive terms. Also note that on your final, you will not be instructed as to which tests to use, so you may need to try more than one test in order to understand which tests will actually be helpful. When you are using p-series or geometric series in the context of the Comparison Test or the Limit Comparison Test, you are free to state (without any proof) that p-series converge if p > 1 and diverge if p ≤ 1 and that a geometric series with common ratio r converges if jrj < 1 and diverges if jrj ≥ 1. 3 • Section 10.4: Given a series with some negative terms, be able to determine if it converges absolutely OR converges conditionally OR diverges. Remember: { The N th Term Test for Divergence can still be used on series which contain some negative terms. { The Alternating Series Test (also called the Leibniz Test) can only be used to show that an alternating series converges, but it cannot be used to show divergence or absolute convergence. • Section 10.5: Be able to use the Ratio Test and the Root Test to determine if a series converges absolutely or diverges. • Section 10.6: { Know how to find the radius of convergence of a given power series. { Know how to find the interval of convergence of a given power series; don't forget to check whether or not the series converges at the endpoints of the interval of convergence. { Know how to use the geometric series formula together with substitution to write a function as a power series with center c = 0 and find its interval of convergence, e.g. you can apply this 2 method to find the power series representation of f(x) = . 9 − x2 • Section 10.8: { Given a function f, know how to find its Taylor series with center c by taking the derivatives of f and then using the formula for a Taylor series, namely: f 00(c) f 000(c) f (4)(c) f(x) = f(c) + f 0(c)(x − c) + (x − c)2 + (x − c)3 + (x − c)4 + ··· 2! 3! 4! Note that if f is a polynomial, the resulting Taylor series will be a polynomial of the same degree as f. { Know how to find a Taylor series by using an already known Taylor series together with substitution, e.g. given the MacLaurin series for cos(x), find the MacLaurin series for cos(3x) Note that on problems which ask you for a Taylor series, you can either write your answer in shorthand (summation) notation, or you can write your answer longhand.