Calc II Module I: Approximations, Infinite Sums, and the Definite Integral
This is the first module of our Calculus II course. Prior to taking Calculus II, all students have completed a Calculus I (Differential Calculus) course. This module takes approximately 5-6 weeks to complete. In the activities in this module:
Students develop and apply a definition of series in the process of conjecturing an exact value for the area and perimeter of the Koch Curve by approximating these values from a finite number of iterations of the Koch Snowflake.
Students begin to generalize from their experience with the Koch Curve series to conjecture and justify convergence and divergence properties of more general series and some special series.
Building from their prior work in developing finite sum approximations, students investigate polynomial approximations for functions and develop and apply a definition for the Taylor polynomial.
Students develop and apply a definition of the Riemann sum and the definite integral by building from finite approximations where careful attention is paid to simplifying assumptions made.
Students are prompted to consider the differences between a Riemann Sum and partial sum approximations for series.
Students investigate typical application problems (area, distance travelled, volume, total change, average value etc) as soon as the definite integral is discovered. This encourages students to both deepen their understanding of the meaning of the definite integral and develop their own independent approaches to these types of problems.