I have to do a teaching demo on the following:
Please treat the committee as students in your Calculus II class. Please take 12-15 minutes to introduce your lesson on the Taylor Series and its applications. Take the final two minutes to indicate how you would finish your lesson for the day
Here's my plan:
Begin by stating that a Taylor Series is an expansion of a function into an infinite sum of terms.
Present the Taylor expansion for $\cos x=1-x^2/2!+x^4/4!+...$ as an example
Use Desmos (a computer will be provided) to show how the more sequence of terms that we have, the better the approximation of the $\cos x$ graph. Example:
The red line is the cos x graph and the blue line is the approximation.
Talk quickly about why the formula works (getting $c_0, c_1, c_2$) using $f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3+... $ to obtain $f(x) = f(a) + f'(a)/1! (x-a) + f''(a)/2!(x-a)^2 + f'''(a)/3!(x-a)^3 + ...$
Use this formula to obtain the Taylor series expansion of $cos x$
Have a student centered formative assessment last 2-3 minutes and find the Taylor expansion for sin x
My only concern is: Is this too much for a 12-15 minute demo?