Review sheet for the final exam
The final exam will be cumulative, but you should expect an emphasis on topics not covered yet. That means half or
slightly more than half of the exam will cover material taught since the second exam, and the rest will cover material
taught before the second exam. This review sheet, however, covers only that material taught since exam II.
Siddique's students will take the exam in Maryland 201, and Stephanie's students will be in Maryland 202.
- Taylor Series, basic notions A Taylor series is an infinite series, for each particular value of x. Taylor
series are expansions based on derivatives, and can be thought of as generalization of the tangent line approximation.
We usually form them at the point a=0, but other centers of the Taylor expansion are possible as well. A sort of
"taylor series at infinity" can be found by substituting y=1/x and finding the taylor series at y=0. Taylor series
are most useful for analyzing short term behavior, especially if the radius of convergence is small. How can you use them
to approximate values, like e^4 and sin(3)? Which ones should you memorize? The Taylor remainder is the values left
over by a particular Taylor approximation. Whenever we can prove that the remainder goes to zero we have good
convergence properties.
- Taylor Series, simple procedures. Find simple taylor series, based on differentiation, evaluation at the point x=a,
and reassembly. Find the radius of convergence via the ratio test. Analyze convergence at endpoints.
- Taylor Series, complex procedures. Find more complex taylor series by addition, subtraction, multiplication,
or composition. Multiplication seems to cause the most grief. Also by integration (arctan(x) or arcsin(x) would require
this), or differentiation. Find Taylor series by simplifying the problem, solving a simpler problem, and using composition
to write the more complex version.
- Taylor Series, reversal. Given the Taylor series, find the original function. This usually done by comparing f
to xf, but it can also involve comparing f to f', or simple recognition.
- Taylor Series, sequence analysis
- Taylor Series, Physics. Understand well the three basic examples in class -- objects in motion, relativistic
energy, and dipole attraction. Each illustrates some important concept: interpretations of at^2/2, presimplification for
easy computation of Taylor series, and expansion at infinity.
- Fourier Series Understand the basic idea of the analytic tool, know how to compute the coefficients, and
know how to interpret, especially in the case of sound analysis. But you don't have to know the musical names of the
tones, obviously.
- Taylor Series, Diff. Eq. Solve differential equations using Taylor series to amaze your friends.
See also 11.5-11.9.
Good fortune!
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