** 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 {a_{n}}, 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 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.