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4

A lot depends on what you plan to use the bachelor's for. If you plan to go to grad school, than go crazy and take a bunch of hard upper div classes. I really caution AGAINST grad school in pure math though (for you, based on your evident aptitude level). Even for stars, this can be quite a daunting pyramid--how many of the people here are tenured profs ...


3

Anecdotally, my father's generation (1950s-60s, UK) all learned logarithms in primary school and used them extensively for calculations. All children had their own slide rule which they carried with them and used in classes other than maths classes. There were no electronic calculators. My father became a radar engineer. My generation (1980s-90s, UK) only ...


1

If you teach the chain rule with Leibniz notation I recommend this suggestion of Steven Gubkin. It makes computations more explicit and straightforward, and students pick it up fairly well in my experience. For the remainder I'll address some of the subtleties involved with derivative notation, the function concept and how that relates to the chain rule. ...


7

The intuition here is basically that of Ben Crowell's answer, and that kind of intuitive explanation might be worth going through first. What I want to show is the kind of activity you can explore with students to investigate how it works in a "less than completely obvious" situation, when at least one of the rates of change itself keeps changing. One way ...


10

What is difficult about the chain rule is the function concept, more specifically the composition of functions. Notation that hides or leaves implicit the composition of functions causes a great deal of confusion for students. However, the fundamental issues are not the notation used (all choices are messy to some extent), but what the use of the notation ...


28

I just start with constant rates of change, where it's pretty blazingly obvious that the chain rule works. E.g., Jane hikes 3 kilometers in an hour, and hiking burns 70 calories per kilometer. At what rate does she burn calories? Our class is generally much more comfortable with the $f'(x)$ notation, and as a result I stayed away from the $\frac{dy}{dx}=\...


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