Calculus 1C: Coordinate Systems & Infinite Series

How did Newton describe the orbits of the planets? To do this, he created calculus. But he used a different coordinate system more appropriate for planetary motion. We will learn to shift our perspective to do calculus with parameterized curves and polar coordinates. And then we will dive deep into exploring the infinite to gain a deeper understanding and powerful descriptions of functions.

How does a computer make accurate computations? Absolute precision does not exist in the real world, and computers cannot handle infinitesimals or infinity. Fortunately, just as we approximate numbers using the decimal system, we can approximate functions using series of much simpler functions. These approximations provide a powerful framework for scientific computing and still give highly accurate results. They allow us to solve all sorts of engineering problems based on models of our world represented in the language of calculus.

Calculus is a series of the following three modules:

• 18.01.1x: Calculus 1A: Differentiation
• 18.01.2x: Calculus 2B: Integration
• 18.01.3x: Calculus 3C: Coordinate Systems & Infinite Series

18.01x Syllabus:

Unit 1: Coordinate systems

1. Parametric curves
2. Introduction to polar coordinates
3. Calculus in polar coordinates

Unit 2: Understanding infinity

1. Improper integrals
2. Singularities and improper integrals
3. Introduction to infinite series

Unit 3: Infinite series

1. Advanced convergence tests for series
2. Power series and Taylor series
3. Manipulating power series

This course, in combination with Parts 1 and 2, covers the AP Calculus BC curriculum.