Here is the description of the topics of questions that can appear on the final exam.

Not all of these questions will appear, in particular, the questions won't have all the parts given here. There will only be 7 or 8 questions on the test.

Some of the questions may be organized differently or include ideas from different parts of the course, but the topics below cover everything you need to know. Below, they are listed in a fairly random order (from more computational to more conceptual and to applications.)

Topic 1. Compute definite and indefinite integrals. You will need to use a variety of methods (antiderivatives,substitution, integration by parts, partial fractions). You will need to decide which method to use for each integral; sometimes you need more than one (e.g. integration by parts followed by substitution).

Topic 2. Solve the following differential equations. Find the general solution and/or solution satisfying a given initial condition. Equations you can solve can be of two types: separable 1st order equations or linear 2nd order equations (as studied in the supplementary notes). You will have to decide which method to use.

Topic 3. Given a sequence {an}, does it converge or diverge? Find its limit. Is this sequence increasing? decreasing? bounded above? bounded below?

Topic 4. Determine whether given numerical series converge or diverge. For convergent series, determine whether the convergence is absolute of conditional. You may need to use a variety of tests, and to decide which test works best for each series. The tests include the ratio test, comparison test, integral test, alternating test. You may need to compute a sum of a series (typically related to geometric series) or to work with partial sums directly. If you need to consider partial sums, the question will tell you what to do, but you must know what partial sums are.

Topic 5. Determine whether a given improper integral converges or diverges, either by direct computation of integrals and limits, or by comparison to another improper integral. Find the value (in case of convergent integrals that can be computed directly).

Topic 6. Qualitative analysis of differential equations: directions fields, properties of solutions (such as increasing/decreasing) that can be detected without solving the equation

Topic 7. Power series; Taylor series.

Given a power series, find its radius of convergence and interval of convergence. (For the interval of convergence, don't forget to check endpoints.)

Expand a given function as a Taylor series at a given point. There are two ways to do this, you must be able to use both (and to decide which one works better in the given case):
(i) compute derivatives, and
(ii) use formulas for "standard" functions together with algebra and/or differentiation and integration

Find the derivative or antiderivative of a function given by a power series.

Use Taylor series to compute limits.

Use Taylor series for approximations; estimate the error.

Topic 8. Use techniques of approximate integration: left and right Riemann sums, trapezoidal rule, midpoint rule, Simpson rule. You need to remember what Riemann sums are. The approximate integration formulas and he formulas for the error bounds will also be given. Since calculators are NOT allowed on the test, you will not be asked to compute any actual approximations. You should be able to write expressions to approximate a given integral, evaluate these expressions, illustrate the approximations, decide whether they give an over- or under-estimate, estimate the errors, and evaluate the number of steps that yields a given precision.

Topic 9. Applications of integration: computing areas, volumes, arc length, average values, and work.

Compute volume of a solid by using integration; compute the area of a figure in the plane. You will need to set up the integral. The solid may be obtained by rotation or it may be just some geometric shape. You may use any of the techniques (washers, cylindrical shells, or other type of slicing; you will have to decide which method works best.)

Compute the length of a given curve. Curve may be given as a graph or parametrically.

Compute the average value of a given function.

Set up integration to compute work performed in a particular (simple) situation. Compute the integral.

Topic 10. Applications of diffrential equations, modeling (sections 7.4, 7.5) Write and solve a differential equation for a given word problem. You may also be asked questions about the behavior of solutions.