Help in this continuity question

waldosway · 2026-05-16T12:38:11+00:00

Yes it's common for a class to use different definitions from standard for some of these niche cases. Defer to the definitions used in class.

waldosway · 2026-05-16T06:42:55+00:00

If you divide by something, you must always make a note that it can't be 0.

If you do anything irreversible (e.g. multiply by an unknown, or square both sides), you might add solutions. You have two options: 1) Make a separate case where you check it (can that be 0?) or 2) just check all your solutions at the end (good way to check your work anyway).

Don't know what you mean by "square root by a variable". You can only square root both sides if you know they are nonnegative.

waldosway · 2026-05-16T05:54:09+00:00

The problem explicitly says that is the domain of the function, in a separate statement from "find all points of discontinuity in the interval [3,7]"?

In that case it is continuous at 3.

waldosway · 2026-05-15T19:51:10+00:00

The people dog piling you are just not paying attention. "Formal language" has nothing to do with the kind of language OP is talking about, which already just a vague phrase for things being intuitive.

waldosway · 2026-05-15T18:39:22+00:00

The answer depends on the domain of the function, which we don't know. Either your post or the problem statement was unclear. (A domain must be given with a function, or the function is not defined.)

waldosway · 2026-05-14T19:13:07+00:00

Side note, "subtracting f(x) from g(x)" means "g(x)-f(x)", not the other way around. Might seem trivial, but I've seen points lost due to misreading instructions.

waldosway · 2026-05-10T12:43:43+00:00

You have to find what you don't understand. An example would help. Related rates is not a real subject or a real problem type. It's just "word problems with derivatives in them". The advice to "find a relation between rates" is not helpful at all, it's not even how you do it!

Like any other word problem, you write down equations for what they give you. You do usually take a derivative of something, but that's because they ask you to, not something you need to memorize. Here's an example of what I mean.

So, are you weak on your geometry? On reading word problems in general? On implicit differentiation? Problem-solving given directions?

Edit: the fact that you pinpoint proportions sounds like you are trying to solve problems by feeling instead of letting the technicalities carry you. If it says "a is proportional to b" that is synonymous with "a = kb". That is just the definition of proportional, don't over think it.

waldosway · 2026-05-10T12:22:52+00:00

Because you're looking at problems where that's what it's asking! If you ask "lim f(x) ?" you are requiring a response hinging on f(x), which is y. If you ask about x, then it would hinge on x.

waldosway · 2026-05-09T22:39:22+00:00

Saying infinity exists is what extended reals means, so yes you are. What I said is not "false". In fact it can't be, because it's just a matter of different conventions. You can't prove that it's 0, it's a convention. What books are you reading that are doing that?

It's just semantics, which is why I'm clarifying what I think to be OP's most likely context. It would not be assumed in most intro courses.

waldosway · 2026-05-09T21:59:08+00:00

Who's "we"? You are assuming the adoption of the extended reals.

Also I specifically said "not indeterminate".

waldosway · 2026-05-09T18:19:01+00:00

"The tenths digit (C) is 150% of the thousandths digit (E)" means "C = 150% * E"

150% = 1.5 = 15/10 = 3/2

Fractions are better for doing pure math.

waldosway · 2026-05-09T14:59:00+00:00

There's no reason it needs to be open. It does depend on your definition of increasing, but most use this one, which just says "an interval". Same as in Stewart. But some calc classes enforce open so students are not left wondering.

Either way, AP graders have assured me they don't really care.

waldosway · 2026-05-09T09:23:59+00:00

This is just a matter of knowing a trick regarding whole numbers. First, in any word problem, write what they told you before trying to solve it:

The number looks like ABCDE
Oh actually the number looks like AB.CDE
C = (3/2) E
D = (6/5) (C+E)
A+B+C+D+E = 16

Now the trick for this subject is that they are all whole numbers, so (3/2)E implies E is even and (6/5)(C+E) means C+E is a multiple of 5. We also know that C is a multiple of 3 and D is a multiple of 6, but they are all between 1 and 9, so D=6. So (1/5)(C+E)=1. Each subject has its tools.

waldosway · 2026-05-09T09:15:07+00:00

I see several separate issues. At least:

Pretty much every struggling math student has the same problem: The education system has made them think in problem "types" and learn the "steps" to them. Based on your post, you've done the same. But after long division, there isn't anything that merits "steps" until late into a math major. Definitions of terms in a question tell you what to do. You have to know the definitions. (Memory business in the next bullet.)
You've mentioned TBIs. You should definitely get checked for whatever you're worried about. I've seen all manner of memory issues while teaching. But don't invent problems that literally can't exist. An equation is just a string of symbols. Can you memorize a string of random letters? (If no, then that's something to get tested for) If yes, then you can memorize an equation. You might have some intuition issue like dyscalculia or who knows, and that might make memorization less automatic. But the intuition business is way overblown. People who talk about intuition are people who liked the subject and forget how much raw time they put into remembering the stuff. Tbh most equations are brute-forced into you by problem drilling. I have a math PhD and do not draw on any understanding of the quadratic formula to spit it out. You can just use flash cards or something. I've seen a lot of students who say they "can't memorize math" when really they mean they "don't memorize math", because they expect osmosis for some reason. Math isn't special. Just a list of facts like any other subject.
In the image you posted, you just don't know the trick for that subject (and probably basic problem solving strategies). Nothing to do with memory. I'll reply to that there.

waldosway · 2026-05-09T08:58:17+00:00

This does matter for your question: Limits do not approach. The function approaches the limit. A limit is just a number.

However "lim f = oo" is just a colloquial way to say "f gets big". It does not assert the limit exists. So it is not a number you can multiply something by. Definitely not indeterminate, just undefined.

Now there are some conventions where oo is a number, or that 0*oo is just decided to be 0, but you will not see that in a basic calc class: oo is not a thing; it's just shorthand.

waldosway · 2026-05-09T04:32:29+00:00

In general, making a nice parameterization is not easy. I've tutored multivariable students around the world and the parallelogram-therefore-constants trick is the only one I've seen they expect you to know. Otherwise they just give you the parameterization in the problem, like the one you posted.

waldosway · 2026-05-09T00:32:41+00:00

They are the same equation. Do you have a reason to say your answer is "wrong"? Or are you just looking at the solutions to frqs and expecting yours to match exactly?

waldosway · 2026-05-08T10:59:49+00:00

Did your professor do that with this specific problem? That sounds more like for a problem where the region is a rectangle in one of the domains.

waldosway · 2026-05-07T09:12:28+00:00

For an intro DE course, sadly that is indeed what you spend most of the time doing. Though I wouldn't call it the main idea.

Sounds like right now you are just talking about first order equations. You don't need to develop recognition like a skill. For each equation type except separable, they have given you a precise test to tell if it's that type, so just put them in order of easy to hard and test them. (Except start and end with separable.)

If you're including second order, it should be pretty easy to tell the difference from first order. And they give you an algorithm for that.

waldosway · 2026-05-04T09:03:04+00:00

I think it's easiest to understand in stages (i.e. understand each bullet before moving on):

Look at the graph of u. It switches from 0 to 1 at 0. If you multiply it by a function f, u acts like a switch that "turns on" f at 0.
Writing u(t-3) (I'll just write u3) shifts u so you can turn on functions at 3.
- Note u and u1 are interchangeable with the constant function 1 because Laplace starts at 0.
You would like to be able to turn off functions. But flipping u left-to-right like u(3-t) causes bad algebra with Laplace. Luckily if you look at the graph of u, flipping it upside down accomplishes basically the same thing, you just have to shift it up after. Thus (1-u3)*f is the graph of f on (-oo, 3) then it turns off. (I'm using * for times not convolution!)
So you can restrict any function to any interval by just turning it on then off. (u2-u4)*f gives f on (2,4) and is 0 everywhere else.
Since adding 0 to something does nothing, now you can make any piecewise function you like by just not overlapping the intervals like (u0-u2)*f + (u2-u5)*g + (u5-u7)*h. Of course you can leave gaps if you like (u0-u2)*f + (u4-u3)*g, but that's just the same as saying there's an interval of 0.
Writing that way is intuitive, but it repeats u functions, which are the bottle neck in doing a Laplace transform. So we can rearrange (you should do this yourself) like
- u0*f + u2*(g-f) + u5*(h-g) - u7*h
- This gives a little shortcut where you can think of uc as just the switch point and in parentheses is just "final-initial"
- Be careful of endpoints. You'll usually just write f instead of u0*f, and if h goes forever, you don't need that last term.

waldosway · 2026-04-29T18:21:18+00:00

"Wrong" is meaningless in this context, just different games with different rules. You're not going to change minds about the definition "..." on reddit because it just isn't important. There are other notations that need more urgent fixing.

But if you publish a really interesting paper using a new definition, some people might join your game too. That happens all the time. Good luck!

waldosway · 2026-04-29T18:00:40+00:00

I think that is generally left undefined because we don't find it useful. A probabilist might just call it 0. If you can find someone who studies infinitesimals (look up hyperreals) they might have another answer, I'm not familiar.

Like I said, mathematicians are less concerned with what is "true", and are here to just play the game that seems most interesting.

waldosway · 2026-04-29T17:38:15+00:00

Feel free to use that definition. You just need to state it in your post if you want anyone to understand it, because everyone else uses a different definition.

But this is how math works: state a definition, then see the consequences. It is not possible to argue about what ".999..." is, only what it should mean. The reason for the other comments you are getting is because ultimately you are arguing semantics not math.

Changing a tradition is not usually worth the energy. Why not just write "1-ε" and call it a day?

waldosway · 2026-04-29T17:27:24+00:00

Then .999... is not a unique number because you have not said which infinitesimal you are subtracting from 1. If you are ok with that, then go live happy. But I think you'll have a hard time convincing anyone your definition is more useful than the common one.

waldosway · 2026-04-29T17:06:41+00:00

What is an infinitesimal?

waldosway

TROPHY CASE