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Calculus: Single Variable Part 1 - Functions

Coursera · University of Pennsylvania · 11 HN comments

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Course Description

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: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.

In this first part--part one of five--you will extend your understanding of Taylor series, review limits, learn the *why* behind l'Hopital's rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O.

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This course is offered by University of Pennsylvania on the Coursera platform.
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Hacker News Stories and Comments

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I took the (Penn) Calculus sequence by Robert Ghrist and really enjoyed it as a review of single variable calculus. He uses Taylor series as the basis of his explanations which I felt was really clean way to provide intuition for some of the more complicated theorems and formulae. The lecture videos have high quality animations and are broken into digestible chunks (around 10-15 minutes each). The downside is that only a few example problems are shown being worked out, but I only found this to be an issue for a few of the lectures (mainly in the applied calculus section of the course).

The course also covers some interesting, non-standard topics. In particular, I liked the lecture on a discrete version of calculus ( which can be used to find easy solutions to series and recurrence relations (e.g. the "discrete anti-derivative" can be used to provide quick closed-form solutions to sums of the form "n^k from n=1 to K" - an example occurs at the 5:28 mark of the linked lecture, but some background from earlier in the video will be necessary to follow along).

The lecture videos are available on Youtube (, but I would recommend working through the problems on Coursera (especially the challenge problems) as well. I would also recommend that viewers watch the videos as 1.5x speed or faster. Dr. Ghrist speaks so slowly in these videos that I found it distracting.

For those who have some knowledge of the standard intro calculus textbooks, the level of rigor and difficulty in this course is above the Stewart book that many universities use, but below the Spivak/Apostol/Courant type of book that an honors course may use.

This used to be a single course, but Coursera split it up into 5 pieces, with somewhat unhelpful names. The sequence is "Part 1 - Functions"[1], "Part 2 - Differentiation"[2], "Part 3 - Integration"[3], "Part 4 - Applications"[4], and "Part 5 - Discrete Calculus"[5]. The first four parts names are reflected in their Coursera titles, but the "Discrete Calculus" course is titled "Single Variable Calculus" instead since it contains the final exam for the overall sequence.

It's also worth mentioning that Dr. Ghrist also has other video lectures available on Youtube ( for other math courses including a sequence on multivariable calculus called "Calculus Blue."






University of Pennsylvania has an amazing course on single variable calculus.

Don't let the idea of doing 'basic' calculus turn you away as it is an incredibly tough course. The reason it can be so challenging and the reason I find it so incredible is that it teaches Calculus through the lenses of Taylor Series. Very different to other Calculus courses and as someone who hated my first year university maths course it's helped me really come to appreciate the beauty of it!

Here's the link to the first course of 5:

I strongly recommend Robert Ghrist's other courses as well, they're fantastic.

* Multivariable calculus (a linear algebra based approach, and a very nice intro to differential forms)

* Applied dynamical systems (ongoing, started recently)

That’s fantastic! Thanks for sharing!
Ran through this series a number of years ago while I was trying my best to self-study my way to a CS bachelor's equivalent; highly recommended. I'd taken calculus in various forms before and always loved it but this one made me think about the material in a completely different light.
The calculus course appears twice (same eventual URL), with different HN citations.

> Calculus: Single Variable Part 1 - Functions Coursera · University of Pennsylvania

The Linear Algebra OCW is great if you do all the exercise and read the book as you study. For Calculus it's not so great, though the ODE course is fine.

This kind of courses from UPenn are really good if you need to refresh single variable, though I did an older version were all the courses were given together than I can't find in the current Coursera:

I'm going to plug Calculus: Single Variable from the University of Pennsylvania on Coursera (

This was the best Calculus course I've taken online.

without a doubt: Calculus in a single variable with Robert Ghrist

for lots of people here it'll revisit some material you learnt at school but it does go further and the materials are fantastic and the exam at the end is no pushover either.

Jul 07, 2015 · ddellacosta on Pure to Applied
I've been fumbling in slow motion through Professor Ghrist's Calculus MOOC on Coursera (, and while I find it challenging it's quite enjoyable. His presentation style is quirky and his manner of speech...unique, but he presents concepts in a very compelling and lucid fashion. It's obvious he and his team have put a tremendous amount of effort into the class, and it's especially apparent in the videos.

I enjoyed reading about what he does the rest of the time: unsurprisingly he is doing interesting things in applied math. Thanks for posting this kjak, I wouldn't have discovered it if it hadn't come up on HN!

I will try to list resources in a linear fashion, in a way that one naturally adds onto the previous (in terms of knowledge)


First things first, I assume you went to a highschool, so you don't have a need for a full pre-calculus course. This would assume you, at least intuitively, understand what a function is; you know what a polynomial is; what rational, imaginary, real and complex numbers are; you can solve any quadratic equation; you know the equation of a line (and of a circle) and you can find the point that intersects two lines; you know the perimiter, area and volume formulas for common geometrical shapes/bodies and you know trigonometry in a context of a triangle. Khan Academy website (or simple googling) is good to fill any gaps in this.


You would obviously start with calculus. Jim Fowlers Calculus 1 is an excellent first start if you don't know anything about the topic. Calculus: Single Variable is the more advanced version which I would strongly suggest, as it requires very little prerequisites and goes into some deeper practical issues.

By far the best resource for Linear Algebra is the MIT course taught by Gilbert Strang If you prefer to learn through programming, might be better for you, though this is a somewhat lightweight course.


After this point you'd might want to review single variable calculus though a more analytical approach on MIT OCW as well as take your venture into multivariable calculus

Excellent book for single variable calculus (though in reality its a book in mathematical analysis) is Spivaks "Calculus" (depending on where you are, legally or illegally obtainable here (as are the other books mentioned in this post)). A quick and dirty run through multivariable analysis is Spivaks "Calculus on Manifolds".

Another exellent book (that covers both single and multivar analysis) is Walter Rudins "Principles of Mathematical Analysis" (commonly referred to as "baby rudin" by mathematicians), though be warned, this is an advanced book. The author wont cradle you with superfluous explanations and you may encounter many examples of "magical math" (you are presented with a difficult problem and the solution is a clever idea that somebody magically pulled out of their ass in a strike of pure genius, making you feel like you would have never thought of it yourself and you should probably give up math forever. (Obviously don't, this is common in mathematics. Through time proofs get perfected until they reach a very elegant form, and are only presented that way, obscuring the decades/centuries of work that went into the making of that solution))

At this point you have all the necessery knowledge to start studying Differential Equations

Alternativelly you can go into Probability and Statistics


If you have gone through the above, you already have all the knowledge you need to study the areas you mentioned in your post. However, if you are interested in further mathematics you can go through the following:

The actual first principles of mathematics are prepositional and first order logic. It would, however, (imo) not be natural to start your study of maths with it. Good resource is and possibly

For Abstract algebra and Complex analysis (two separate subjects) you could go through Saylors courses (sorry, I didn't study these in english).

You would also want to find some resource to study Galois theory which would be a nice bridge between algebra and number theory. For number theory I recommend the book by G. H. Hardy

At some point in life you'd also want to go through Partial Differential Equations, and perhaps Numerical Analysis. I guess check them out on Saylor

Topology by Munkres (its a book)

Rudin's Functional Analysis (this is the "big/adult rudin")

Hatcher's Algebraic Topology


It is, I guess, natural for mathematicians to branch out into:

[Computer/Data Science]

There are, literally, hundreds of courses on edX, Coursera and Udacity so take your pick. These are some of my favorites:

Artificial Intelligence

Machine Learning

The 2+2 Princeton and Stanford Algorithms classes on Coursera

Discrete Optimization

Convex Optimization


Some I can recommend that are still available on Coursera:

- Introduction to mathematical thinking [1]

- Introduction to Mathematical Philosophy [2]

- Machine Learning (actually a CS course, but involves linear algebra and some calculus) [3]

- Calculus: Single Variable [4]





Many thanks!
Personally looking forward to the ebook coming out! I was actually just this morning going through and signing up to a bunch of mathematics related Coursera courses. For those interested, quite a few are starting soon:

[Jan 7th] Calculus: Single Variable -

[Jan 7th] Calculus One -

[Jan 28th] Algebra -

[Jan 28th] Pre-Calculus -

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