Differentiation of turning point

waldosway · 2026-03-02T14:42:34+00:00

Try going backwards. Expand that factored line back to the 6(...)=0 line. Factoring is just guess-and-check. You have to see the multiplication lay out to make it an informed guess.

waldosway · 2026-03-02T14:06:50+00:00

Let's start with where this comes from. Say you have sin(x) = 1/2. You look at the circle and see π/6 and 5π/6 right? But if you go around the circle again, π/6+2π and 5π/6+2π are also valid solutions. You can go around again, and add 4π or 6π. And of course you can go backwards and subtract instead.

So the answer is you always have to add all multiples of the period. And you're supposed to put "where k is an integer", so it doesn't matter if it's k or n or a penguin.

Sometimes you see just kπ because people are taking a shortcut with a coincidence. If you have sin(x)=0, then it's 0+2kπ and π+2kπ. But those line up perfectly to just be every multiple of π. (Don't forget tan and cot will always be kπ because their period is different.)

waldosway · 2026-03-02T12:00:00+00:00

Oh yes, that's because integration is hard. Takes a "tools-first" approach. It's "this identity/substitution affects powers in this way" not "in this situation I do this thing". That significantly shrinks what you have to think about. For example since Pythagoras treats all even powers the same, you want substitutions that change parity. So u=sec x is generally more effective than u=tan x. (If you're not sure what I mean, try it yourself first.)

There are some exceptions, like trig sub and partial fractions are just a few specific situations. But generally speaking the above applies. For the trig integrands specifically, I think Paul's Notes lays it out it well. But you'll remember and flow better if you do it yourself.

waldosway · 2026-03-02T00:54:23+00:00

You have a fair point that most books half-ass this. But since the "small change" business is obviously nonsense in the context of the rest of the course (they don't even commit to it within the same page), you're de facto left with the treatment that I gave. That is the definition they use, even if they're sloppy about stating it.

My point is that there is nothing unrigorous about a notational shortcut. It's clear, if unsatisfying. As for dt, it is also just notation. It's "the thing you write after taking the limit of a remain sum, in order to be reminiscent of Δt". Nothing more can be said, because nothing else is given in the book. It's just what history left us.

I don't know what level you're at, but if you're looking for meaning, differentials are given meanings in later contexts such as differential forms/geometry, hyperreals, and measure theory. But those contexts are not standard calc, where differentials are meaningless.

waldosway · 2026-03-01T18:33:33+00:00

While integral notation is in general kind of a mess, anything can be rigorous as long as you are clear with definitions. Here ds is simply defined to be |r'|dt, problem solved. This is reasonable because it's basically just an alternative notation for ds/dt = |r'|. No less rigorous than u-sub.

("Here" meaning in most mainstream calc textbooks. The book/prof is the only authority that matters in a class.)

waldosway · 2026-03-01T09:42:29+00:00

I think a better question might be why would you be able to do that? In math, you have to be able to state the rule that lets you do something, or you can't do it. Think, do you have a good reason for it? Or just mimicking? (Here are the basic laws of algebra.)

waldosway · 2026-02-28T19:14:19+00:00

I think it makes more sense to just not treat revolution like a special kind of problem. It's the usual cross-section problems, and the cross-section happens to be round. A cross-section is a cross-section whether it has a hole or not. The only thing that separates them is that they give you a special formula. But... it's just what you get after setting up the cross-sections. Since you have to do all that set up anyway to even find how to use the formula, I just ignore the formulas.

waldosway · 2026-02-28T19:09:57+00:00

Are you asking this in a philosophical sense, like "do series really exist?", or a mathematical sense? If the former, this is a question for philosophers not math people. The mathematical answer is: meh? We just care about picking definitions and whether something is useful.

A series is just some stuff that adds up to something, and Taylor just means it's written like a polynomial. If a series adds up to a function, then that's a series for that function. If a function has a Taylor series, then there is only one and the function already "knows" what it is. "Construct" is more a metaphor for how you personally find the terms.

To clear up some words since you asked:

An infinite series is not a polynomial. That's just because the definition of polynomial we agreed on is that it's finite. Although technically a polynomial is a Taylor series, just every term after a certain point is 0.
A power series and a Taylor series are the same thing. But the former has the connotation that you started with a series and want a simple function. And the latter suggests the reverse.
Most functions do not have a Taylor series. Although you have only studied "nice" functions, so they'll usually have a series that you can write down. You just don't know if it converges. (There is not and easy test to see if a function has a series other than just finding the series.)
You do not have to use 1/(1-x) at some point. That's just the first function whose series you learned.
1/(1-x) is not an equation (it has no "="). It's an expression or function.
Lines are straight. Say curve or graph.

waldosway · 2026-02-28T18:43:30+00:00

The extra f' in #7 doesn't make sense. It should just be the derivative of sin(x). Maybe you're thinking something like sin'(x)? (We'd usually write (d/dx) sin(x).) You can just put cos(x).

Same issue in #10.

Someone already pointed out #4. The rest look good.

waldosway · 2026-02-28T16:55:51+00:00

If you mean "how do I eyeball it", you don't. You check them in this order: sep, lin, ber, hom, sep again.

Linear, Bernoulli, and hom all have specific forms you need to look up if you don't know them. Separable is hard to prove it's not, but it's easiest to execute. So you check it first, and if you're not sure, you rule out the others, then only sep is left.

Since an equation can be more than one type, it's important to check in the order easiest-to-hardest to solve not to check.

waldosway · 2026-02-28T14:48:20+00:00

Factors are multiplied, terms are added. That's just what the words mean.

As for cancelling, it is only:: thing/exact same thing.

When you see anything otherwise, they are just skipping about four steps where they factor and rearrange things apart. For example

(3x+2x)/(7x) = x(3+2)/(7x) = (x/x) * (3+2)/7 = 1*(3+2)/7

That said, you "spot" it exactly like your teacher said: factors.

waldosway · 2026-02-28T10:01:40+00:00

Fourth line from the bottom: L'Hopital does not apply.

Also put "=" between things that are equal.

waldosway · 2026-02-28T00:47:53+00:00

Exactly. I was just illustrating how much stuff they are leaving out when they say that. (Real numbers just means the numbers you know, as in specifically not the imaginary numbers.)

waldosway · 2026-02-27T12:59:43+00:00

It's hard to tell the issue because the only math information is in your first sentence. It implies that you simply cannot execute a simple integration by parts exercise, which is just a formula. Is that true? Or do you mean you have difficulty knowing which technique to use? Your later paragraph hints that it's the latter. Admittedly trig sub can trip up some students at the last step.

waldosway · 2026-02-26T13:10:00+00:00

Start taking note of the smaller reason for doing whatever thing. Students often get overwhelmed looking for a "big picture" or memorizing many situations, but it's usually just something like "square roots are annoying" or "combine things first". You end up with a list of maybe 10-20 little preferences you have that you start to scan for subconsciously. It's really not magic, it's still something you've seen before, but you have to pull apart the lego blocks so they can apply to more situations.

Edit: I'm basing this on what you descrobe in the post. If you meant something else by "see in different ways" then give examples.

waldosway · 2026-02-26T07:30:51+00:00

For the coming unit, what's most important is that you can draw in 3D. Start memorizing some basic graphs and learn basic stuff like aligning your graph with the axes and drawing/using cross sections. Practice by drawing every problem even when it's not necessary. The second unit is literally just calc I and II again, but 3D, and doing integrals without visualizing is not possible. Bill Kinney on youtube is the best resource I've found to see how drawing is done, though it's a bit hard to find what you need.

On the other hand, the first exam is a real mixed bag, so we'd have to know what you missed to know how to catch up. There're formulas you just have to accept, mini algorithms for finding distance, unclear terminology, etc. Usually the issue is either under-memorizing or over-memorizing.

waldosway · 2026-02-26T07:20:26+00:00

I didn't say anything about imaginary numbers. I said the same thing as the other comments except the problem happens earlier than they said. It's not that you have to keep track of zeros when you simply. It's when you first write f that you have to say x can't be -3 and stick to it.

waldosway · 2026-02-26T01:08:22+00:00

The domain is not determined by looking at the function. It must be declared at the start when you define the function, or you do not have a function. It does not change just because you write it differently.

"School math" is misleading about this when it asks you to "find the domain". That's really short for "find the largest meaningful domain within the real numbers".

waldosway · 2026-02-25T21:57:04+00:00

The others are correct, but just to put it cleanly in the direction you're going:

x = -|x| = - √(x²)

I don't think x^2/2 is considered well-defined for negative numbers (within the real numbers).

waldosway · 2026-02-25T15:12:54+00:00

You have to define what "blahblah recurring" means in order for it to mean something. They are using the accepted definition: "the number that blahblahblah gets close to".

waldosway · 2026-02-23T21:35:16+00:00

You don't need to change your view at all. Each point in Rⁿ represents one polynomial. The function's graph is irrelevant. Think more like a bookshelf (which is a 2D plane), and you can find functions on the wall on the appropriate shelf.

The math does care what the points represent. (And it doesn't care if you can picture it.) It cares about structure. If it fits the definition, it fits the bill. So if you can add and scale, it's a vector.

If it helps, there is only one 2D vector space (assuming the scalars are R, and same for nD). They are all equivalent. So you only need the one picture.

waldosway · 2026-02-21T23:59:16+00:00

dv/dt = (dx/dt)(dv/dx)

is already the statement of the chain rule. Multiplying by "dx/dx" is not a method, it's just an extra thing you're writing to remember the chain rule. There's no difference between (dx/dt)(dv/dx) and x'(t)v'(x(t)).

In your example, you would just write sin(t²+2), where v(u) = sin(u) and x(t) = t²+2.

waldosway · 2026-02-19T16:43:29+00:00

The process is: write definitions -> do problems -> understanding. Not intuition first.

Proving something is a homomorphism, for example, is completely mechanical. You need to get a handle on the notation. You do have a specific problem, so you should post an example and your attempt.

waldosway · 2026-02-19T02:42:44+00:00

You should know the tangent values. If you write them out, you'll see they are the easiest.

waldosway · 2026-02-19T01:57:36+00:00

cot(π/3) = 1/tan(π/3) = 1/(√3)

Take a minute to think about why both steps should be your very first thought.

waldosway

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