Math 222 Lecture 1 Final Exam topics

The final exam will cover the entire semester, with only the most recent material from chapter 11 given any additional emphasis because it has not been on any prior exam. Problems assigned and the questions on previous exams are a good starting point in thinking about the kinds of questions that might be asked: Note that material we covered and which might have been on an earlier exam could well be on the final, even if there did not happen to be a question on that material on the earlier exam. Here is a list of the topics we have covered, with some comments:

Chapter 7: Most of this chapter was on techniques for finding antiderivatives, but don't overlook improper integrals! Since we also have had topics since chapter seven which use integrals, e.g. finding area in polar coordinates or an integral test, some of this material might be tested in a different context. Note that some kinds of problems could be lengthy: I could not ask you to do an integral using partial fractions unless it was simple enough to get done in a relatively short time.

Chapter 8: This divides into two parts. The sections on specific curves (circle, ellipse, parabola, hyperbola) are all alike, differing only in details, while sections 8-8 and 8-9 step back and look at the general quadratic equation without specializing as to which curve it represents. If we ask you to sketch a curve you should be sure to show the significant points for that kind of curve, and put the curve in the right place, and not draw sharp corners on a smooth curve, but we won't be grading you on artistic ability. We might ask you what angle of rotation would eliminate an x-y term, but because of time constraints we will not ask you to carry out that rotation. That would not prevent us asking what kind of curve resulted, which you could find using the discriminant.

There will be no questions on Chapter 9, hyperbolic functions, and no questions where you need to be able to manipulate hyperbolic functions. If you choose to express an answer using hyperbolic functions (and you do it correctly) that will be OK.

Chapter 10 on polar coordinates both has material appropriate for questions and serves as useful background to our recent material on cylindrical and spherical coordinates. You will not need to know by name any polar curves other than cardioids and the graphs you already knew in rectangular coordinates such as lines and circles, but you should be able to sketch the graphs of other polar equations well enough to find intersections and to set up integrals for area. You should know how to find the angle between the tangent line to a polar curve and the radius vector, and how that relates to the slope of the curve.

Chapter 18, on differential equations, and the supplementary material on undetermined coefficients, fall into several sections. You should be able to solve both differential equations in general and initial value problems using the techniques for: first order, variables separable; first order, linear; second order with constant coefficients (homogeneous or not). For non-homogeneous second order equations you can assume that the method of undetermined coefficients, as described on the class handout which is also available HERE. If you wish you could use the method of variation of parameters instead.

Chapter 16 covered both sequences and series. On the exam the emphasis will definitely be on series, but remember that sequences are still involved: The very definition of the sum of an infinite series involves a sequence, and sequences are also required to apply the ratio and root tests for convergence of series. There will probably be some problems asking whether a series of numbers converges, but more emphasis will be on power series: Finding a series, finding where it converges, estimating the remainder for a truncated series, etc.

The part of Chapter 11 we have done introduces parametric equations to describe curves, coordinates (three systems) in space, and vectors and vector operations both in the plane and in space.
Since sections 11-4 through 11-8 have not been on any previous exam, here is a little more detail about what we might ask on that material: We will not ask you to graph a surface or curve in space. We might ask you, though, to describe the set of points in space which satisfied some equations or inequalities, or to find equations or inequalities to characterize a surface we described in words or showed in a picture. You should be able to find the vector which extends from one point in space to another and to do arithmetic with vectors. You should be able to compute and to use for geometric problems the dot and cross products. You should be able to find equations for lines or planes. You should be able to describe in words the effect of vector operations, e.g. "the set of all points such that the vector from a point P to the given point is perpendicular to the vector v."