You missed the point. Why would Cal spend that much time/text explaining the derivative? A simple picture and an equation to show the difference works fine. Concept vs memorizing. This isn’t a blog about Calculus, it’s about studying. Here’s your math website -http://math.ucr.edu/home/baez/FUN.html#mathematics

Cal, great job on this post. I used most of the methods of your post already except making the list of the concepts. I’m in grad school right now and I’m feeling the “concepts flood” building a little too much in my QFT class. I just started making my list to help me out. Thanks again.

Background: I got a computer science degree, in some sense, in the early 1980s. But, to avoid some money situations (drastic change of major to computer science, taking business calculus), and fear of “the” calculus due to a bad trig teacher…. So, I had to take “the” calculus. I got a bunch of dummy books, and a TI-89 calculator. My background of doing computer code maintenance for years got me through this. Those tests I took: calculators were not permitted. But, basic trig formulas were provided. Calc II course in the summer: we were allowed the integration formulas, only, along with trig identities. Of course,during that summer, I hired a lawn mower service and house cleaning service while I was taking calculus II in the summer’s 8 week course.

Key: Get to know classmates! Work with them. Get into study groups. Without my classmates, I would have not gotten my high A’s in calculus, probably gotten low B’s to C’s, without them for sure in calculus II.

In this day and age: get dummy/idiot books (I did), and read/work those problems as well.

How can one study most effectively?…

There are very few rules for how to study most effectively because there are too many variables. Here is a basic algorithm followed by some general tips, both of which can be generalized across many settings: Repertoire: 1. List the repertoire that wil…

[...] from effective study groups to class attendance. The advice is all built around a common theme (familar to Study Hacks readers): understanding the material is everything and the only thing that [...]

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Your explanation of the ‘intuitive’ side of differentiation is not good.

Qualification: Maybe your explanation is okay for an MIT ‘plug and grind computer scientist’, but let’s set those people aside! !!

First, yes, in calculus and in nearly all of ‘mathematical analysis’ of which calculus is the most important part at first, ‘intuition’ is important. Really, for each good definition, theorem, or proof, there is a ‘picture’ (graph, diagram, etc.) that illustrates what is going on and makes much of the algebraic manipulation fairly obvious.

So, you are correct encouraging pictures.

Second, your picture omits any illustration of the role of h. So, you need to draw not just the tangent line but a few ‘secant’ lines, one for each value of h you want to illustrate.

Third, the derivative is a ‘local linear approximation’ to the function. Or, pick a value of h, remove the lim_{h \rightarrow 0}, rearrange, and get approximately

f(a + h) = f(a) + h f’(a)

which shows that as tweak h, f(a + h) gets tweaked by approximately just hf’(a) which is linear in h. Of course, the right side here has just the first two terms in the Taylor series.

That we are taking a limit here is crucial because we get the ‘chain rule’ that says, roughly, if we differentiate f(g(a)), then we get the local linear approximation of f evaluated at g(a) times the local linear approximation of g evaluated at a — when we differentiate a composition of two functions (here f composed with g), we get the product of the two linear approximations. This fact generalizes to where f and/or g is a vector valued function of a vector variable — nice since we get a product of two matrices.

This chain rule is especially nice since in the related subject of ‘calculus of finite differences’ we do not take a limit as h –> 0 and do not get a chain rule.

Here we see a secret: One good approximation to a complicated discrete situation is to ‘smooth’ the situation and use calculus. Can get some practical attacks on some NP-complete problems this way, etc.; some of these attacks are called ‘Lagrangian relaxation’.

Third, differentiation is linear: So with mild assumptions

(af(x) + bg(x))’ = af’(x) + bg’(x)

which is important. The two main pillars of analysis are continuity and linearity, and here we see linearity (again).

Fourth, what happens at a = 0 for f(a) = |a|, that is, the absolute value of a? Right: Without some ‘smoothness’ assumptions, there need not be a unique tangent to f at a. So, essentially everywhere in science and engineering where calculus gets used, we are asking for some ‘smoothness’ conditions. Such an assumption is more reasonable than we might think because every continuous function can be approximated as closely as we please by a function ‘smooth’ enough to be differentiable.

Fifth, the tangent, local linear approximation, and derivative get much of their importance from the fundamental theorem of calculus that if integrate f’ then get back f. A good illustration is to have f be distance traveled so that f’ is speed. Then if for, say, each second, multiply speed f’ by time of one second and add, get a good approximation to f. That is, at a, f’(a) tells us how rapidly we are ‘accumulating’ f. Or the speedometer tells us how fast we are eating up road on our way to our SO’s house.

Sixth, yes, in solving problems, first we have to guess at manipulations that might get us to a solution, and for good guesses intuition is crucial. But to do well on tests, still need to be able to solve the problems. One way to know that can solve lots of problems is to take a good sampling of the problems in the book and solve those.

Net, intuition does help and at times is crucial but is not sufficient.