(last modified on 3/13/09)

REVIEW TOPICS FOR THE FINAL EXAM

You can find here some of the the more important topics.
Many of the problems and questions in the final exam will be
from this list. But questions from other topics studied may appear as well.

You may be asked to state (clearly, completely and correctly) some definitions
and/or give some proofs done in class. These are explicitely mentioned below
and marked with Q.

TOPICS

Order of growth of functions at infinity.

Improper integrals.
Q: State the definition of the convergent improper integral: Integral from a to infinity of f(x) dx.

Sequences and their limits
Q: State the definition of a convergent sequence (with epsilon and N).

Limits: Indeterminate forms, l"Hospital's rule.

Series
Q: State the definition of a convergent series.
Geometric series, telescopic series

Series with non negative terms
(Tests: limit comparison test, the integral test, the ratio test, root test)
Q: State the integral test. Prove it.

Alternating series (test)

Conditional versus absolute convergence.

Power series: Chapter 14 (do not miss: radius of convergence, interval of convergence, ratio and root tests, formulas for the power series at x=0 for the exponential, sin, cos, ln(1+x), integration and differentiation of power series, Taylor polynomials and series.

Conic sections: completing the squares, graphs.

Polar coordinates: convert from polar to rectangular, or rectangular to polar.
Graph curves given in polar coordinates. Areas, surfaces.
Q: Define the angle psi (in words, and show it on a picture).
Q: Deduce the formula for tan psi.

Vector algebra (and geometry) in the plane and in space.
The velocity is has direction tangent to the path.
The magnitude of the velocity is the speed.
Q: Define the curvature of a curve in the plane.
Finding the tangential and normal components of the acceleration (both their magnitude and as vectors).

Dot product, cross product, lines, planes, quadric surfaces, cylindrical and spherical coordinates.