Honours Integral Calculus
“Some of the best ideas in mathematics have been very simple ideas”
Text: Robert A. Adams and Christopher Essex: Calculus: Single Variable, 8th Edition, Pearson, Toronto, 2013
Prof: Dr. Young-Heon Kim
Professor Kim is a boss. He challenges us. He aims to make us understand the mathematical thinking behind a proof. He also promised us that if we worked hard and understood the basics, we should get an A, which was probably realized for most people who worked hard at the end of the term. He answers questions generally patiently and tries to sense if the class is understanding what he is doing, instead of just droning on. Highlights of the class:
On the first class. “I looked at the textbook we are using for this course. It was very disappointing. The questions are too easy.”
“Why are you guys looking so dumb today?”
“Let this by a probability distribution of the midterm. As you can see, it isn’t very high.”
Kim: “Should we do this problem this way or that way? It’s a matter of taste.” Student: “So it doesn’t matter?” Kim: “No. TASTE MATTERS A LOT!”
“Hey! doesn’t this parametric curve look like Picasso?”
Near the end of the course. “I just realized that you guys have been working very hard for this course. Please don’t ignore your other subjects. How come nobody complained? Whenever anyone comes to speak to me, it seems like you are saying “please give me more work”.
Difficulty
This course was significantly harder than Math 120. For example. the average on the first homework was 6/15, but it improved dramatically over the course of the term. Midterms were conceptually challenging but only trivial computations. The prof loved to mix multiple concepts into one question. There was generally one proof on every exam. There were often proofs on the weekly homework, and these were significantly more challenging than the ones in the exam. The questions used to nag me for days. The Final exam was 40% of Math 101 stuff and the average was scaled to 75-78. So if you work throughout the course, and make sure you understand the basics well, you should get an A. 16/29 students achieved an A.
Key Concepts
The (Riemann) Integral
Sequences and Series
The Fundamental Theorem of Calculus
Convergence
Hard Concepts
Riemann Integral: Defining integration in terms of epsilons and partitions. Similar to limit definition, but the hard part is the mathematics, not just the logic.
Probability: Some of the harder questions are difficult to wrap one’s head around, especially on the homework, where we had to find the probability of a series converging. One way is to think of probability as mass.
Functions defined by series: Last question on final was a function defined by a series of functions defined by a sequence. Can get pretty meta. One way is to look at how output function is changing with a small change in x, to understand.
Polar Co-ordinates/Parametric: Drawing, finding derivatives etc can get pretty computationally challenging and technical. But there is a step-by-step process one can follow. Also, to find parametric function it is a good idea to divide motion into motion of the centre of mass and motion relative to the centre of mass.
Centre-of-Mass: Some pretty crazy mass distributions. Make sure to define axis etc clearly, and not to change it during the course of the calculation.
Volumes by slicing: Can get pretty hard to imagine. Might require good 3D geometry imagination skills.
Conclusion
Great course. I wasted a lot of time studying for the first two midterms by practicing too many questions from the book. It was better for me to just try to understand the concepts. The homework problems were challenging but rewarding. The Webwork was a bore, but that is in Math 101 anyway, so you can’t escape it. Some aspects of the course were a bit rushed, but that is to be expected with an honors course. I would recommend it to anyone who wants to really improve their understanding of mathematics.
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