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University of Pennsylvania

Calculus: Single Variable

University of Pennsylvania via Coursera

This course may be unavailable.


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.


By signing up and paying a nominal fee (financial aid can be provided), you'll be eligible to earn a Course Certificate in this course, including a higher level of identity verification to your Coursera coursework. For each assignment, your identity is confirmed through your photo and unique typing pattern. If you earn a Course Certificate, you will also be given a personal URL through which your course records can be shared with employers and educational institutions.


Note: The following only applies to sessions starting on September 8th, 2014, and prior. This Calculus course has been evaluated and recommended by the American Council on Education’s College Credit Recommendation Service (ACE CREDIT) for college credit so you can get a head start on your college education. More than 2,000 higher education institutions consider ACE credit recommendations for transfer to degree programs. If you add this option to sessions starting on or prior to September 8th, 2014, towards the end of the course, you will take an online proctored exam which will be combined with your coursework to determine your eligibility for college credit recommendation.


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.

Taught by

Robert Ghrist


4.7 rating, based on 19 Class Central reviews

Start your review of Calculus: Single Variable

  • Anonymous
    I've completed several MOOC's and this was definitely the best. Another reviewer has already commented on the fact that Professor Ghrist designed this course specifically for the MOOC platform, with lots of great animations that help illustrate the concepts. I will also add that Professor Ghrist actively monitors the forums (more than any other online professor I've seen) and very frequently takes the time to respond to his students. Even if it's only to offer encouragement for students expressing frustration. This class really demonstrates the beauty of a MOOC done right.
  • Matt
    Robert Ghrist produced the most beautiful course I've seen on Coursera. It is a single variable calculus course designed for those who have already seen calculus at some level. The production quality of the video lectures is second to none. They we…
  • Anonymous
    This is a long course of currently 14 weeks, covering functions, Taylor series, differentiation, integration, limits, convergence, sums and more. By the excellent presentation of the course (course-wiki, videos and fora) I improved in all of it's to…
  • Bart
    I am in awe. Robert Ghrist mentioned in the forum he put in 18 months of solid work into this course and it shows. There are many ways to present calculus and his way is not the easiest route, but I believe it has many advantages over the other methods I've seen. Ghrist has a consistent way of explaining the material. And the lectures are a real joy. Colourful formula's dance across the screen with helpful animations. There are enough good examples. The fora are active and helpful. In most topics there are a few interesting bonus material video's which make the course less dry.

    It does take some self discipline to actually learn all the material by heart and to do enough homework to prepare for the next quiz.
  • Anonymous
    I've taken many MOOCs and this is by far the most well presented. Prof Ghrist clearly invested a huge amount of time and effort to put this together. It's also a somewhat different teaching approach so it's interesting even if you've already taken calculus before.

    Note that this course has now been split into 5 smaller courses but the content is the same.
  • Anonymous
    it is great.... the best thing is that it takes effort to explain the basic concept from depth. It is very important for people who need to learn the basic calculus in the short time. Professor G. also is really very good.. his way of teaching is very attractive and interesting. I want to thank him specially and also to University of Pennsylvania.
  • Amedeo Nannini
    The best course on Calculus Single Variable I have ever followed. Robert Ghrist is so nice and clever while explaining difficult concept. Better you have some prior knowledge of the stuff to deeply learn from this course. Not to be miss if you wish a beautiful learning experience on calculus.
  • Anonymous
    I am nearly finished with the fifth month. Prof G covers material very quickly. The quizzes near the end of the course include quiz questions that require getting 10 - 16 answers correct within a question in order to score that question correctly. To get a passing grade on one of these quizzes you may have to answer 20 questions correctly - miss one or two and you fail the quiz. I find this to be a diabolical and distinctly unhelpful way to learn the material. Still, very good course. Very challenging.
  • Anonymous
    This is definitely one of the most exciting courses around. Though accessible for beginners (like myself), it presents the material in a way that makes you curious and eager to dig deeper. And this is indeed the primary goal.
  • It is simply the best mooc course I had ever seen.
    Clear and deep.
    The right way to really understand calculus and appreciate one of the best creation of the human kind
  • Nitin
  • Anonymous
    I agree with everything that has been said above. Professor Ghrist is a great teacher. I enjoyed the course very much
  • Anonymous
  • Amar Doshi
  • Francisco Javier Jiménez

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