For this week I have another twofer, two in-class activities dealing with complex numbers, power series, and Taylor series.
Complex Numbers and Power Series - Students were introduced to some of the properties of complex numbers, and the first page of this activity has them explore operations and graphs on the complex plane. The second page has students identify the radius of convergence of given power series, with a little analysis of the coefficients on the third page. The last two pages have students graph partial sums of power series, with a hint as to what they approximate.
Taylor Series - In this activity students are given two different functions and are asked to analyze their power series about different points, and look at a very special relationship. The first function is a simple rational function, the second being the exponential function. Students are asked to numerically compare Taylor Polynomial Approximations of the rational function centered at different points, and are asked to provide a basic comparison of the approximations of the exponential function as well. The last page walks students through a derivation of Euler's Identity. For the most part students were pretty impressed with the identity, with the exception of the student asking "Is this going to be on the test?" right after the derivation.