Let’s begin with a pair of rabbits. Assume that that rabbits begin to reproduce two months after their own birth. This is, after reaching the age of two months each pair produces one male and one female, and another pair each month thereafter. Also assume that no rabbit dies. This is the sequence of numbers of rabbits generated by the original pair of rabbits:
Find the Taylor expansion for a given trascendental function, such as e^{x}, sinx, cosx.
Find a Taylor polynomial by using substitution. Ex: e^{2x}, sin2x^{}, cos(x/2), etc. Note: Taylor series can be added, subtracted, and multiplied by constants and powers of x, and the resulting series will also be Taylor series. Example: To find a series for cos 2x, simply substitute 2x in for each x in the Taylor series for cos x.
Find the expansion of e^{ix}, compare the result to the expansion of sin x and cos x and relate the result to: e^{ix} = cos x + isin x
Find Taylor/Maclaurin series expansions by integration or differentiation of power series. Ex: ln(1-x), arctan x etc.
The Binomial Series expansion.
Background: Integration by partial fractions, Improper integrals, Geometric Series.
Strategies for testing Series, by Stewart: test the following series for convergence or divergence, #1 to #34. Solutions included.
Feb 13-17:
Vector valued functions.
Motion along a curve: problems involving position, velocity and acceleration of a particule on a 2-Dimensional space.
Feb 20- and Beyond: REVIEW.
Discussion of previous Calculus BC Exams,
Worksheet on integration: int by substitution, by parts, partial fractions.
Area between curves, volumen of a solid of revolutions.
Ex 1 thru 6, also 17 & 18 p 454. Page 465 ex 1 thru 10
and page 474 1 tru 4, and 17, 18, 27, 28.
April 2nd thru May 9: Discussion of previous exams.