Sometime early this week , I finished listening to the excellent video lectures from MIT OCW’s Calculus Revisited course . I had been meaning to listen to MIT OCW’s Single Variable Calculus course for quite sometime as my background in Calculus is a bit flaky. My interests are in machine learning, data mining and AI where Calculus has a nasty habit of making surprise entries 🙂
I somehow finished my Master’s using my old Calculus knowledge. I took a course on Numerical Methods which kind of exposed my weaknesses. I kept getting confused with the error approximations which used ideas from infinite series. Other advanced ideas like multivariable optimization were also problematic to me. Once that course was over, I swore myself to refresh my Calculus stuff and also learn multivariable calculus.
I started listening to MIT OCW’s Single Variable Calculus lecture videos and felt two things – The course was a bit slow for my pace and the course jumped right away into the mechanics without spending much time on the intuitive explanations of the Calculus. In other words, I felt 18.01 was more focused on the analytic part which emphasized proofs and derivations whereas for my purposes an intuitive explanation of the concept would have sufficed. In fact, I remembered almost all of the Calculus formulas from undergrad – My only problem was the lack of “sense” in how to apply it to the problem I faced (say in machine learning or some optimization).
Then I found the Calculus Revisited course from MIT OCW. It consists of a series of lectures on Calculus but also assumes that students have had prior exposure to it. This assumption had some interesting consequences and I fit the bill perfectly. I downloaded the set of videos and started listening to them. Interestingly, all the lectures were between 20-40 minutes which allowed for maximum focus and also allowed you to listen to multiple lectures in the same day. In fact, Arlington had a heavy snow this week and my university had to be closed for the entire week. I completed around 16 lectures in 3 days and was able to finish it ahead of my target date of Feb 15.
The course starts with the absolute basic ideas of sets, functions, induction and other stuff. If you are from CS and had taken discrete math, you can feel free to skip the first section. But I would suggest you to still take a look as it , in a sense, sets the stage for the entire course. Do take some time to listen to the lecture on limits. (Part 1 , lecture 4). Here, the discussion of limits effortlessly leads to the derivation of the formula for instantaneous speed and hence differentiation.
Part 2 forms the crux of the course and covers differentiation. Professor Herbert Gross had a beautiful way of teaching stuff about derivatives. In particular, he extensively used the idea of geometric proofs or visualizations to expound basic ideas. The way he brought out the tight relation between analysis (as in Math) and geometry was enthralling. He had a huge emphasis on geometric intuition which helped me to “grasp” the key concepts.
Part 3 had some nice discussion on Circular functions. He joked about how teachers don’t provide good motivation for learning trigonometry which I felt very true to me. He also explained some concepts that were new to me – that you do not really need triangles to define cosine and sine. Previously, I was aware of the radian concept but never put it all together. He also explained how sine and cosine tend to come up in unexpected places – like as the solution of the differential equation for harmonic motion 🙂 He also masterfully showed the close relation between circular and hyperbolic functions with a playful title of ‘What a difference a sign makes’ (in Part 5).
Part 4 discussed about integration and how it can be used to calculate 2 and 3 dimensional areas (and volumes). This part also had a great discussion on how differential and integral calculus are related. That being said, I was a bit dissatisfied with the discussion on the two fundamental theorems of Calculus. The discussion on Mean Value Theorem also felt a bit rushed. I got a bit lost on the discussion on 1 dimensional arc length calculations. May be I should revisit the lecture notes for the same when I get some free time.
Part 6 was my favorite part for two reasons – This had a discussion of infinite series and my favorite quip of the course. When discussing about the non intuitiveness and the intellectual challenges posed by infinity , professor Herbert Gross playfully quips (which goes something like this)– ‘ of course, one thing to do is to not study it. I can call it as the right wing conservative educational philosophy’ – Ouch 🙂 I think I mostly understood the idea of infinite series even though there was not much explanation of “why” it works that way. I also felt the topic of Uniform Convergence to be way beyond my comprehension level.
Overall, it is a great course and acted as a fast paced refresher for those who had already taken Calculus. The course slowly starts from basic pre-calculus ideas and rapidly gains speed and covers a huge list of calculus topics. I felt few of important Calculus topics were not covered or rushed up – First and second fundamental theorem of Calculus, Mean Value theorem, Taylor series, L’Hospital rule, discussion of exponents and logarithms etc.
But that being said, I feel the course more than makes it up for the way the basic ideas were covered. I had fun learning the ideas of limits, infinitesimals , intuitive ideas of differentiation/integration, geometric explanation of differentiation/integration, how the concept of inverse functions pervades Calculus etc. Prof. Herbert Gross had a jovial air around him and occasionally delved into philosophical discussions which made listening to the lectures more interesting.
He also had an extensive set of supplementary notes and huge amount of problems with solutions. I had to skip the problems part to conserve time. But if you have some time do spend some on it.
Lastly, I found that one of the lectures in the series was missing. Lecture 5 in Part 2 on Implicit Differentiation was the same as the one on Lecture 4. I had sent a mail to MIT OCW about this and got an reply saying they will fix it soon. Hopefully, it will be fixed soon.
In conclusion, this is a great , fast paced course on Calculus that emphasizes geometric intuition of the major ideas in Calculus. Listen to it if you already know Calculus and want a fast refresher ! I am currently listening to the lectures on Multi Variable calculus. I do intend to listen to Single variable Calculus again , may be in Summer. I will put out another post on how it went 🙂