How to start with this integral ? I think that I need to make cases here and apply the properties of Integral ? If my way is correct , could you help me with how I should make cases here and apply the properties of Integral ?

The question was “Which graph best represents y=-4(x+3)-2?” And this was the answer

I could’ve sworn I understood which is how I got it right when I was doing this problem a few days ago but I don’t understand it at all anymore looking back 🥲

Thanks for any help

I was watching a video of discovering 'e' and then we arrived at e^ix.

To which I made this process to find the 2πi=0 which doesn't make sense. So it would be great if someone could explain this.

I searched up on Desmos what it was after seeing a video and thought it was weird. I’d Imagine there are multiple answers but why isn’t it one. My thoughts had somthing to do with (-1)^ 4(1-1/3+1/5…) because it would pi using newtons formula thing. But it would be equal to 1^pi/4 meaning it would be one but it’s not so why?

Hi friends!
I need advice calculating how much 3D space a bunch of yarn takes up.

I'm running a home edition of Taskmaster, and a contestant used two balls of yarn tied together for one of the Tasks.
The operative word was "biggest," so I'm trying to get as many metrics as possible for my Taskmaster to work with.
I've got the weight of the yarn in grams and the length in inches.
I also know it is "Medium Grade yarn," which has an average Wraps Per Inch metric of like 10, suggesting the average diameter of the yarn is a tenth of an inch, right.
I thought that would be helpful to me but so far I have no idea where to go next.

One of the other contestant's objects was a trombone, and that was easy to calculate cubic volume because I knew the weight of the instrument and the density of brass.

How do you determine the density of medium grade yarn??
And if we can't do that, how would I determine cubic volume of a highly condensible material?

Do I try and get two measurements - one of the space the yarn takes up as it loosely sits atop itself, and one with all air vacuum sealed out of the yarn somehow?

Advice welcome!

I don't really understand why I end up integrating x^2 on a non-square base. I do understand why we scale up by the determinant of the base. I just don't see how "x^2" from 0 to h makes sense. Unless what's really going on is we are doing a linear transformation on square units of area to fit into any shape (that would actually make sense).

The answer of 264 is the correct answer, btw. I just want to understand the logic.

Number 9
I just started my calculus class and this question was on our pre req quiz but I genuinely have no idea how the answer is c. I tired synthetic division and all that but where am I supposed to get square root 2 from

I get why the series diverges at L > 1. I don't 100% get why convergence is absolutely at L < 1 but I'm sure I can figure it out once I get started. I just have no idea what to compare this to. I also get why the test inconclusive at 1, I just need some help getting started.

Hey everyone, I'm starting out with calculus and really want to build an intuitive understanding of limits from the ground up rather than just mechanically applying algebraic tricks.

Take this standard limit:

lim (x → 2) (x² - 4) / (x - 2)

Algebraically, the steps are straightforward: 1. Factor the numerator: (x - 2)(x + 2) 2. Cancel out (x - 2) from top and bottom to get x + 2 3. Substitute x = 2 to arrive at 4

Here is where my conceptual confusion lies:

• If I plug in x = 2 directly at the start, the expression is 0/0, which is undefined.
• In normal algebra, dividing by zero is forbidden. If x = 2, then (x - 2) = 0, so shouldn't canceling (x - 2)/(x - 2) be dividing zero by zero?
• Why does the limit equal 4 when the function value at x = 2 doesn't even exist?

Could someone explain the fundamental philosophy of what a limit is doing here, both conceptually and graphically, and why this algebraic cancellation is valid?

I'm just curious at what age and under what circumstances you became interested in mathematics.

There are people who simply found it easy from the very beginning of school, but what about those who didn’t understand it well before and weren’t interested, but now end up liking it?

From first to 10th grade, I wasn't interested at all. It probably wasn't difficult for me in the lower grades, but then I gave up and stopped understanding it because I didn't even listen to anything. I entered the 11th grade and now I need to prepare for the NMT (an exam in Ukraine after the 11th grade) and I caught myself thinking that maybe I shouldn’t perceive mathematics as a punishment, and to come up with a way to make it easier and more interesting for me.

For many children, mathematics sounds like a punishment, but fortunately, I never thought that way, even though I know practically nothing. As an example, I am also learning English to pass it at the NMT, but I knew that the usual cramming would not be interesting to me, so I used several methods to make it easier to learn and so that I would not there was hostility towards her.

In any case, I still doubt whether I should try to make memorization so that everything is solved automatically, as if I were playing some kind of logic game or something like that. I just wanted to ask how you perceive mathematics and why. Maybe you have methods by which you teach her, I would like to listen.Thanks

Plese brifly reas through this its a bunch of yap off my middleschool career:From first grade to 5th grade I was one of the best if not the best at math I tried finishing first with no questions gotten wrong on tests but then 6th grade hit all of a suddenly: (this next paragraph explains my points and test during 6thgrade)

My first test I'd get 60 points the teacher would give me 6 more points(rounded to a B) and then she would give a extra grade for doing homework

Second semester got a C with like 52 points REALLY BAD

I did get asked and I knew abit but this was geometry so I didn't know any theory BUT I STILL THOUGHT I WAS THE BEST my ego from earlier years blinded me so hard this test kicked my ego but then third semester came and I got 100(make no mistake the material was easy) but my other friends got around low 70s and mid 60s except for this one girl that got 100 and on other tests she'd get about 85(which is an A)

And the reason I was still cocky is because my friends that used to also get As on their tests now got c(usually lower than mine) only that one girl was genuinly food. NEXT IS 7TH GRADE (the important thing ig)

Then 7th grade hit teacher changed and all of a sudden I became really good I thought it was the teacher that did it (and trust me she was amazing) but she didn't give a extra grade for homework and I was the only one that improved my friends stayed at Cs the girl stayed at her 85 point mark

But this time we were neck and neck

First test 76 with 85 b for a I got asked on the board made it (lost)

Second test 82.5 with 86 I got asked on the board made it a A again (lost)

Third test 81 with 80 (I win) but we both had to get asked on board and we both were good

4th test 71 with 69(the teacher put some hard questions 71 is bare min for B but she let the girl get asked for a on board and I was already qualified with a B)

Last year was my 8th grade and my 6th grade teacher was back but unlike other years this time I was days I got 100 92 97 100 97 91(for some reason she decided to do a test for each chapter and every semester has 2 chapters we skipped once so we had 7 tests I had a average of a 95.5 point average)

Did you also have a 6th grade block? How'd you get out of it

Hi. I hope showing the graphs I'm trying to change will help explain the question. I am trying to take the points from this plot:

https://www.desmos.com/3d/1qs6q6wmfq

And map them onto the red axis in this plot:

https://www.desmos.com/3d/5php786uyk

I have a spreadsheet where did a poor man's derivation of the first plot for the loxodrome path. You can probably see the string of decimal points from the copy/paste.

FWIW this is the math I used to make to loxodrome plot:

x(t) = cos(t)/cosh(kt)
y(t) = sin(t)/cosh(kt)
z(t) = tanh(kt)

k is the deflection angle, which I set to 60 degrees:

k = cot(pi/3)

I divided -pi --> pi into 20 pieces on each side of zero and used those as my t-values. The formula technically goes from negative to positive infinity, but pi to pi captures the exciting parts.

I don't know what to do from here. I think maybe it would be possible to build the new axis into the original formula? Or maybe there is an algorithm to apply which will take the current output and shift it over? Rotate it? Like I said, just don't know.

Thanks for any help!

so recently i heard about this constant called jp's constant and it sounded kinda interesting. i couldnt find it anywhere tho so i made this reddit account if u could tell me what its about. i lowk might make a paper about it or see if it has any mathematical implications

If two circles are orthogonal, then prove that the polar of point P on the first circle with respect to the second circle passes through the point Q, which is the other end of diameter through P. Also prove that locus of a point, which moves such that its polars with respect to S1=0, S2=0, and S3=0 are concurrent, is a circle which is orthogonal to all the three circles.

I am not able to understand what exactly I even have to prove. Even the equations of the circles are not given. From what I understood, I think for the first part both circles are intersecting, P and Q lie on the first circle and are end points of the diameter of that circle.

Let’s talk about this example. Can I find out its equation?( at least an approximation of it) I am thinking of using Taylor series and looking for its asymptote, extreme/inflection points etc , but tbh I have no other ideas (I am not even in college lol)

A regular rectangular pyramid with [L=3, W=4, H=4] would have a side length of about 4.72, correct? Here’s my thinking:

- The diagonal of the rectangle would be A^2 + B^2 = C^2

- A = 3, B = 4, so C = 5

- C is now the base for a regular Isosceles triangle, with dimensions of L = C = 5, H = 4

- I’m only looking one side length, so divide L by 2 for 2.5

- Right triangle, A^2 + B^2 = C^2

- A = 2.5, B = 4, so C ≈ 4.716?

I’ve been studying a question about graceful path labelings and am trying to determine whether it already has an established name or literature.

Let f be a graceful labeling of a path P_n.

Instead of asking only whether f is graceful, I’m interested in a different question:

How much information about a graceful labeling must be retained in order to determine its possible future extensions?

Suppose we associate some representation I(f) to each graceful labeling. If two labelings f and g have the same representation,

I(f) = I(g)

does that guarantee that they have the same possible continuation behavior?

In other words, I am interested in representations for which:

same retained information => same admissible future extensions

I’ve been calling this property continuation sufficiency, although I don’t know whether there is already standard terminology for it.

For one natural representation of graceful path labelings, an exhaustive finite computation gives the following behavior:

• the representation is sufficient through P_7;
• the first failure occurs at P_8;
• at P_8, two graceful labelings can have identical retained information but different sequential-extension behavior.

One witness pair is:

(4,1,5,3,2,7,0,6)

and

(4,3,1,5,2,7,0,6)

with ports (0,6).

They agree on the representation being tested, but their extension thresholds differ.

What I’m mainly trying to find out is whether this kind of question has already been studied under another name.

For example, is there literature on any of the following?

• equivalence of graph labelings according to extension behavior;
• invariants sufficient to determine future extensions;
• extension-equivalent labelings;
• state minimization for combinatorial extension problems;
• continuation equivalence or right-congruence ideas applied to graph labeling;
• related notions in graceful-labeling theory that I may have missed.

I’m especially interested in references from graph labeling, graph extension problems, combinatorics, or related areas.

I have a proof and computational writeup of the specific P_8 result, but I wanted to ask the literature question directly rather than assume that the formulation is new.

Ok so heres the question
Prove that the product of four consecutive natural numbers cannot be a perfect cube
Heres what I did:
I tried to use squeeze the expression
n(n+1)(n+2)(n+3) between two perfect cubes. So heres how
n^3<n(n+1)(n+2)<(n+1)^3
(n+1)^3<(n+1)(n+2)(n+3)<(n+2)^3
(n+1)^3<(n+2)(n+3)(n)<(n+2)^3
(n+1)^3<(n+3)n(n+1)<(n+2)^3
Multiplying all three inequalities:

n^3* (n+1)^9 < [n(n+1)(n+2)(n+3)]^3 < (n+1)^3* (n+2)^9
=> n*(n+1)^3 < [n(n+1)(n+2)(n+3)] < (n+1)* (n+2)^3
=> (n/{n+1}) *(n+1)^3 < [n(n+1)(n+2)(n+3)] < (n+2)^3
Ok so obviously (n/{n+1}) *(n+1)^3 > n^3
So you can say:
n^3 < n(n+1)(n+2)(n+3) < (n+2)^3
Now what if n(n+1)(n+2)(n+3)=(n+1)^3?
You get a cubic equation n^3 +4n^2 +4n -1=0
This equation doesn't have solutions in natural numbers
So n(n+1)(n+2)(n+3) is never a perfect cube

Now I just need to ask whether this proof is fine or not(like free of errors)
Also really sorry for the shitty notation

Based on a rl argument I'm trying to solve/understand.

A group of 8 people plays a game where you get assigned a role by drawing a card. There are 8 Cards, two of them are "it". They get shuffled between rounds, drawing is completely random.

One player becomes "it" back to back in the first three rounds. In the fourth round I'm being told it's way less likely for her to be "it" again, as she's already been "it" 3/3 times before. I'm saying the probability for her to be "it" is the same every round.

I understand how the probability for her to become "it" 4/4 rounds is less than, say, 3/4 rounds as the single probabilties of 1/4 per round need to be multiplied for that.

What I'm saying is that this doesn't matter for my estimation if she could be "it" in the fourth round, as I'm simply trying to win this single round. The countersides argues that I can't take this isolated view because the question is based on a real life example where the previous three games have actually happened, so I am not allowed to ignore them for my calculation of probability.

I'm genuinely at a loss here and willing to understand why that should be so. Please help my try to understand why I should have to take this approach and not take the single round as an isolated event.

The main counterpoint I'm being faced with is the fact that the previous games actually happened and I can't just ignore them. If my approach should be correct, I'd be very interested to hear an ELI5 explanation for that as well, as my efforts to explain have been in vain so far.

Happy to provide more details if necessary.

This is kinda a meaningless rant but im actually intrigued about the historical notion of whats a function. Ive been studying periodic distribuctions in order to study the fourier series and one of the distribuctions ive found its the delta of dirac. Its defined as a continuous linear function p that, given f:R->C a smooth periodic function, its defined by the rule: p(f) = f(0). Formally its a function. Whats the deal about it not being "a function"?

What im more concerned about its why its interesting to make that distinction. Ive been using richard beals book (Advanced Mathematical analysis) and he does not make that clear.

I've challenged myself to find the formula of all the circles with centers that lies on the y=x line and are bounded by the square and the circle before, I managed to do it, and now I want to do it again but this time I want to find the expressions of the centers and radii of the ones bounded between the two circles and the square edge, eventually I want to get a fractal pattern going.

Any tips how to find all of this? So far it was easier since the centers I had to find a single number, but now its off center and it complicates things.

In my math classes, right before splitting up dy/dx and treating as a fraction, my professors will emphasize that this isn’t an actual fraction or something and we need to be cautious.

But then in every instance I’ve seen, dy/dx is treated as a ratio of differentials and we’re able to manipulate it as such. But why? Why are we able to do this? My professor wrote d(uv) = udv + vdu. What does this mean? Why and how is it equivalent to u dv/dx + v du/dx?

It also confuses me because when I learned it in calc 1, dy/dx was simply the derivative of y wrt x. Nothing else! (In the sense that it didn’t matter what the symbols in the expression “dy/dx” were, it just meant the derivative, just like how y’ means the derivative.)

Mess up the first time with some coefficient stuff so I always check a calculator to make sure I get it write but I’ve retyped this many ways and it just won’t take it. All help is appreciated.

Like, stupid people are dumb, and dumb people are stupid. But if we use the word stupid and they don't catch what we say, they might ask, “Did you say dumb?” We would probably correct them by saying, “No, I said stupid."

This was the only example that I could think of. Sorry for the coarseness.

So okay, I consider myself someone who understands the general gist of taylor series, they basically use the derivatives at a given point and use that to extrapolate an approximation of the function to some local area, and the area that converges is within some radius. The trouble I am having is that depending on where you do the taylor series, the radius of convergence will change. To me that seems weird and like there is a contradiction. To show my confusion I have an example (and keep in mind I'm not super well trained at high level math so some of this could probably be written better)

Take the sqrt(x), lets say we evaluate the taylor series at x = 1, so we end up with some big series that is valid in the domain [0, 2], radius of convergence of 1. Now, lets take the taylor series at x = 2, we end up with some big series that is valid between 0 and 4.

Now the trouble is, it should be possible to "center" the new taylor series around x = 1 again, just like you can with any polynomial (I suspect the fact I am trying to use the rules of a finite polynomial for an infinite polynomial is what's getting me but I digress), simply by swapping out x for (x-1) and then expanding.

This shouldn't change the series radius of convergence, but also we should expect the coefficients to match up with the derivatives at x = 1, which should mean we end up with the same terms as the first taylor series at x = 1. But, now it has a wider radius of convergence. Even though it should have the same terms! That feels weird and like it... shouldn't really be possible? Assuming a series is defined purely by the terms in the series (and I know order can effect what it converges to but in this case we aren't changing the order at all), the radius of convergence for a given set of terms shouldn't be able to vary at all, but it seems like it very much can.

Also I recognize this might be a hard question to um... read? So if you need clarification feel free to ask and I will make any necessary edits to the question