Calculus: Single Variable

This course provides a brisk, challenging, and dynamic treatment of differential and integral calculus, with an emphasis on conceptual understanding and applications to the engineering, physical, and social sciences.

About the Course

Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include:

• the introduction and use of Taylor series and approximations from the beginning;
• a novel synthesis of discrete and continuous forms of Calculus;
• an emphasis on the conceptual over the computational; and
• a clear, dynamic, unified approach.

Recommended Background

Students are expected to have prior exposure to Calculus at the high-school (e.g., AP Calculus AB) level. It will be assumed that students:

• are familiar with transcendental functions (exp, ln, sin, cos, tan, etc.);
• are able to compute very simple limits, derivatives, and integrals;
• have seen slope and area interpretations of derivatives and integrals respectively.

This material will be reviewed; however, it is important to begin the course with some background. A diagnostic exam will be made available to help you gauge your preparedness. 

The course will serve equally well as a first university-level course in Calculus or as a review from a novel perspective.

If you've never seen Calculus before, this is likely not the course for you. Please see, e.g., the more introductory course from Ohio State University:  https://www.coursera.org/course/calc1

If you are looking for the background needed to begin a study of Calculus, please see, e.g., the pre-Calculus course by UC Irvine: https://www.coursera.org/course/precalculus

The course is divided into five "chapters":

CHAPTER 1: Functions
After a brief review of the basics, we will dive into Taylor series as a way of working with and approximating complicated functions. The chapter will use a series-based approach to understanding limits and asymptotics.

CHAPTER 2: Differentiation
Though you already know how to differentiate some functions, you may not know what differentiation means. This chapter will emphasize conceptual understanding and applications of derivatives.

CHAPTER 3: Integration
We will use the indefinite integral (an anti-derivative) as a motivation to look at differential equations in applications ranging from population models to linguistics to coupled oscillators. Techniques of integration up to and including computer-assisted methods will lead to Riemann sums and the definite integral.

CHAPTER 4: Applications
We will get busy in this chapter with applications of the definite integral to problems in geometry, physics, economics, biology, probability, and more. You will learn how to solve a wide array of problems using a consistent conceptual approach.

CHAPTER 5: Discretization
Having covered Calculus for functions with a single real input and a single real output, we turn to functions with a discrete input and a real output: sequences. We will re-develop all of Calculus (limits, derivatives, integrals, differential equations) in this new context, and return to the beginning of the course with a deeper consideration of Taylor series.