I had an email last week from a teacher asking, how come I can use a substitution to find a power series for , and for , but not for ? The answer is that you can. Substituting (2x) in to the cosine’s series give you a Taylor series centered at x = 0, a…
The last BC question on the exams usually concerns sequences and series. The question usually asks students to write a Taylor or Maclaurin series and to answer questions about it and its interval of convergence, or about a related series found by differentiating or integrating. The topics may appear in other free-response questions and in…
The posts for the next several weeks will be on topics tested only on the BC Calculus exams. Continuing with some posts on introducing power series (the Taylor and Maclaurin series) Introducing Power Series 1 Two examples to lead off with. Introducing Power Series 2 Looking at the graph of a power series foreshadows the…
Today, for some summer fun, let’s look at synthetic division a/k/a synthetic substitution. I’ll assume you all know how to do that since it is a pretty common pre-calculus topic and even comes up again in calculus. Why Does Synthetic Division Work? An example: consider the polynomial . This can be written in nested form…
Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section, students may be asked to say if a sequence or series converges or which of several series converge. The Ratio test is used most often to determine the radius of convergence and the other tests to determine the…
One of the great things – at least I like it – about Taylor series is that they are unique. There is only one Taylor series for any function centered at a given point, What that means is that any way you get it, it’s right. Faced with writing the power series for, say, ,…
The seventh in the Graphing Calculator / Technology series Here are some hints for graphing Taylor polynomials using technology. (The illustrations are made using a TI-8x calculator. The ideas are the same on other graphing calculators; the syntax may be slightly different.) Each successive term of a Taylor polynomial consists of all the previous terms…
