Section 9.4 -- Exponential Growth and Decay ----------- Technically this section could be included in the previous one; the entire section is built on the differential equation dy/dt = ky, which is a separable differential equation. It would probably be a good review for you to solve it on a piece of scratch paper, and then compare your work to the author's on the top of page 604. This is such a well known equation that unless they specifically tell you to solve it while showing your steps, you can just skip straight to the solution, given in box 2 on p604. Note that to find y_0 (or P_0 or C or whatever your multipicative constant is) all you need to know is the initial value, i.e. y(0). If you don't see why this is true, stick t=0 into the right hand side and work it out. To find the constant k in e^(kt) you first need to solve for the initial value constant, and then have another value, like "y(10)=15,000" or something. These might be a breeze for you; if not, make sure you read through the examples, particularly the population growth and radioactive decay ones. Section 9.5 -- Logistic Equation ----------- We didn't spend as much time on this as on other differential equations, and usually that means it won't be as important on the test. However, a lot of you had difficulty with this section on the homework, and I'd hate to see you lose 25 points just in case there _is_ a logistic equation on the exam. Definitely work through #1 on the first review sheet, and maybe work out the general solution shown on pp616-7. You should also look at some of the examples to see how you figure out what the different constants are. You should know what the constants mean, too -- i.e. what is the "carrying capacity" ??? Even so, I think I'd concentrate on sections 9.6 and 9.4 before spending hours on 9.5. Section 9.6 -- Linear Equations ----------- Ok, right off the bat: you'll have a first order linear differential equation problem on your exam. It'll probably be a mixing problem like #33 or #34, or number 3 on Review I. I can't say it any clearer: know how to do these problems. Be completely at ease with integrating factors. (Remember: the integrating factor is e raised to the integral of p(x), not the integral of p(x) by itself.) Woe to the person who doesn't know how to do these and neglects to review this section! Section 9.7 -- Predator/Prey ----------- Not covered! Section 10.1 -- Curves Defined by Parametric Equations ------------ You should be able to sketch the graph of a curve given by parametric equations, and know the definition of an initial point and a terminal point. You should practice eliminating the parameter and converting parametric equations into xy form, like in problems 7-15. You might want to read your manual and/or talk to me to figure out how to graph parametric equations on your calculator -- it might be a good help on the exam. It's conceivable that you might have to do some sort of mix/match problem like #22 on the exam. If you had lots of trouble with that one I'd recommend stopping by my office door and reading the answer key for that problem. Section 10.2 -- Tangents and Areas ------------ You need to be able to find tangent lines to curves given in parametric form. Know the chain rule formula used in the box on p649. Anything like example 2, or any of the homework problems, or any of these types of problems that we did in class are fair game. I'd bet money you'll get one of these. Areas with parametric equations weren't stressed at all in class, and you only had a few homework problems on this, so it's probably not as important. You might want to know the formula, and practice one or two simple ones like example 3, but not at the expense of area with polar equations. Section 10.3 -- Arc Lenth and Surface Area ------------ If I recall correctly you didn't have to do any arc length / surface area stuff on the last midterm, so it's a good bet to show up now. You should know those formulas! You might only have to set them up, but just in case you should be comfortable working them out, too. Remember the standard trick for arc length problems -- you add up (dx/dt)^2 and (dy/dt)^2 and you end up with a perfect square, which cancels out with the square root. That same trick sometime works with surface area integrals; sometimes, though you can do a u-substitution with surface area integrals. Section 10.4 -- Polar Coordinates ------------ Of course you need to be comfortable with polar coordinates, but you probably won't get 25 points for writing down 4 different "names" for the point (1,0) in polar form. You *might* get 5 or 10 points for that, though, as part of one problem. In any case, you should read through this section not because there are lots of possible test questions here, but because if you don't understand this stuff, you stand no chance of understanding 10.5. You might want to know the equations for finding the tangents to polar curves, but these weren't mentioned at all in class so I don't imagine they'll play a big part on the midterm. I certainly wouldn't spend time on this until I knew the tangents to parametric curves stuff down pat. Again, you might want to make sure you can graph polar stuff on your calculator. It's a great help for the next section. By the way, be careful not to confuse polar and parametric curves. Sometimes a parametric curve is given in terms of theta, but it's still parametric. A parametric curve has equations for x and y, like x = cos (theta) y = sin (theta) Whereas a polar equation only has r's and thetas: r=cos(theta). You shold be able to switch back and forth between polar and xy form. Remember the "Bible" for doing this: x^2 + y^2 = r^2 x = r cos (theta) y = r sin (theta) tan(theta) = y/x Section 10.5 - Areas and Length in Polar Coordinates ------------ You need to know how to do these. I'd bet *lots* of money that you'll get one. You should be absolutely comfortable with the formulas in the boxes on p671 and the examples that follow. Read the caution on p672 and the example that follows -- it's very similar to what we did in class last Thursday. This is where graphing polar equations on your calculator can be helpful, because otherwise you might miss intersection points. For example r=cos(theta) and r=sin(theta) intersect at the point (1, pi/4). The graphs also intersect at the origin, but there isn't any specific value of theta that makes both sin and cos equal to zero. (Think about this!) My best bet would be you'd have a problem like example 2 -- notice the problem on your review sheet is kind of like that. Also look at the ones we did last week in class. You could know the formula for Arc Length in polar form, but it was more or less neglected, so I don't think it will be important. You certainly shouldn't spend time on it until you know the arc length problems in parametric form. To summarize: Arc length: parametric form is more important Surface Area: parametric form Areas: polar form is more important Conic Sections -------------- This section was more or less skipped, but you're responsible for anything that was said in lecture. The homework is probably a good guide for what kind of problems might be possible. It's conceivable that you'll be given something like x=5cos(t), y=7sin(t), asked to eliminate the parameter, and then graph the resulting conic section. (An ellipse in this case.) It might be worth looking at pp679-80 if you're not sure how to draw a hyperbola. Section 10.7 -- Conic Sections in Polar Coordinates ------------ Not covered. Section 11.1 -- Sequences ------------ Repeat after me: A sequence is a list of numbers, nothing more. Read through this section and try to understand everything. There are technical definitions of limits, but I don't believe they were stressed in lecture so you can probably use your intuition: if the numbers in the list tend towards zero, then the limit of the sequence is zero, and so on. Be especially careful about applying L'Hopital's Rule to a limit where everything is written in terms of n. Technically you need to apply it to the related function f(x); see example 4. The boxes on page 699 are worth noticing. Box 7 is important for the next section, and Box 8 has some important definitions. Notice that Monotonic means _either_ increasing _or_ decreasing. Sometimes we don't care if a sequence is going up or going down -- the important thing is that it's just going in one direction and never "turns around." That's why we bother with the term monotonic. Look at p700 to see why we might care. Section 11.2 -- Series ------------ Repeat after me: A sequence is a list of numbers, nothing more. A series is the sum of a list of numbers! A series has two associated sequences. The first is a_n, the numbers being added up together. The second is s_n, the series of partial sums. s_n is the sum of the first n terms in the sequence a_n. This is confusing, I know. But read through what I just wrote and pp704-5 until you get it, because it's very important. In particular a _series_ converges if its associate _sequence_ of partial sums converges. The box on page 706 about the geometric series is very important! Geometric series in general are important, whether or not they show up on the exam! One possible question would be something like example 4, or #3 on the sheet we did in class today. (Both have to do with decimal expansions and fractions.) Both of the theorems on page 709 are important. We talked about them in class today, and I showed you that the converse is not necessarily true. This is very important! The example is this: if a_n = 1/n, then the limit of a_n is definitely zero, but if you add them all up, 1 + (1/2) + (1/3) + (1/4) + ... you get infinity. Section 11.3 -- The Integral Test and Estimates of Sums ------------ Not covered, *Except* that you have to know the theorem in box 1 on p716 about p-series. Section 11.4 -- The Comparison Tests ------------ You need to know the comparison test on p722. The Limit comparison test on p723 is not part of the course, and in fact you probably will not get any credit if you try to use it on the test. If you get a comparison test problem, it will probably be a "baby" one, so if you read pp721-23 and understand it you'll be in good shape. Other than reviewing, Thursday will be spent on the comparison test, so that will probably help. Happy studying!