Calculus: Single Variable

This course provides a brisk, entertaining 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, entertaining, unified approach.

About the Instructor(s)

• Robert Ghrist is the Andrea Mitchell University Professor of Mathematics and Electrical & Systems Engineering at the University of Pennsylvania. Prof. Ghrist is an applied mathematician whose expertise consists of finding novel applications for previously un-applied branches of Mathematics to Engineering Systems. Examples include applications of algebraic topology to sensor networks, sheaf theory to optimization and network data, CAT(0) geometry to robot motion planning, and braid theory to dynamical systems. His work has been honored by Scientific American as a "SciAm50 Top for Research Innovation" in 2007 and a Presidential Early Career Award for Scientists and Engineers (PECASE) in 2004. The S. Reid Warren, Jr. Award was granted to Prof. Ghrist by Penn students in 2009 for exceptional teaching. Prof. Ghrist is the 2013 recipient of the Chauvenet prize, the highest award for mathematical expository writing.

Prof. Ghrist will be assisted by Dr. Vidit Nanda, a post-doctoral researcher at the University of Pennsylvania. In his research, he develops algebraic and topological tools for analyzing large high-dimensional datasets as well as their transformations. Dr. Nanda began his career in Applied Mathematics (changing majors to Mathematics) shortly after taking 1st-year Calculus from Prof. Ghrist a decade ago.

Course Syllabus

The course is divided into five "chapters":
CHAPTER 1: FunctionsAfter 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: DifferentiationThough 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: IntegrationWe 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: ApplicationsWe 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: DiscretizationHaving 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.

A more detailed lecture-by-lecture syllabus can be found here.

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 the more introductory course by OSU's Jim Fowler: https://www.coursera.org/course/calc1

If you are looking for the background needed to begin a study of Calculus, please see the pre-Calculus course by UCI's Sarah Eichhorn https://www.coursera.org/course/precalculus

Suggested Readings

There is a fun picture-book available that gives the main ideas of the course:

R. Ghrist [2012], FLCT: the Funny Little Calculus Text, http://www.math.upenn.edu/~ghrist/FLCT/

You can preview the entire book for free or download a copy through google play. This is a supplemental text only and is not required for the course...but it might make you laugh.

Course Format

The class will consist of nearly 60 animated lecture videos, each about 15 minutes in length. The schedule will be approximately 5 quarter-hour lectures per week over 13 weeks. Occasional "bonus" lectures will provide more advanced off-the-syllabus perspectives. You will get to practice your skills with lots of homework problems. These will not count towards your grade for the course, but, because of this, there will be open forums for discussing how to solve the homework problems. Grading will be based on graded chapter quizzes (5), and a final exam.

FAQ

• Will I get some kind of Statement of Accomplishment after completing this class?

  Yes. Students who successfully complete the class will receive a Statement of Accomplishment signed by the instructor.

• What is the format of the class?

  The class will consist of lecture videos, usually about fifteen minutes each. There will be homework problems that are not part of video lectures. There will be approximately seventy-five minutes (5 lectures) worth of video content per week.

• Will the text of the lectures be available?

  The lectures will contain extensive text, and we are preparing lots of supplementary material. Otherwise said, you do not need to take detailed notes of the lecture -- it's already been done for you.

• Is this a hard course?

  Yes. Calculus, like the rest of Mathematics, takes time and effort to master. If you are prepared to work hard at the assignment, I'll work hard to explain the principles as clearly as possible.

• Do I need a graphing calculator or special mathematical software?

  No! This course will emphasize conceptual understanding. All the computations should be done using a pencil, eraser, paper, and your brain, though not necessarily in that order of importance.

• Does this course cover all of Calculus?

  No. It will be assumed that you've seen some of the subject, at a high-school equivalent level (e.g., at the level of the Calculus AB exam). In addition, we will cover only single-variable calculus, not multi-variable.

Provided by:

University: University of Pennsylvania

Instructor(s): Robert Ghrist