Theyre tracings of high heels. Need to calculate the area in order to find the pressure exerted when someone wears them. So I have no equation to find the integral of. Would Reimann sums be my best option?

I want to know why we use radians in calculus?

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Why not degrees?

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Is it possible to redefine and derive calculus using only degrees?

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How will the answers be different compared to now?

I am taking Calculus 2 and this question is blowing my mind right now. I used the Direct Comparison Test and got it right, but it doesn't make any sense to me. It's not clicking in my brain.

Consider the limit of x^2/x^3 as x approaches infinity. I've been taught that we say that the value of that limit is zero because the infinity on the bottom is so much bigger than the infinity on the top that it's completely negated. Ofc we can L'Hopital it and get 1/x where the one essentially becomes nothing in comparison.

My initial thoughts would be that this series converges because even though ln(n) is raised to a power of 6, EVENTUALLY it will sort of flatten out, but n on the bottom will continue to go on at a constant rate. Meaning in my head that although they both approach infinity, the number on the top will become nothing in comparison to the number on the bottom, just as before. That to me says it converges, but it doesnt as the test says.

I do not understand why. The divergence tests tell me how to get the answer, but not why. Or maybe they do and I don't understand it. If possible, explain like I'm not a math major, because I am not.

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 ?

I'm going to university to major in mathematics, and have already completed Calculus 1-3, and differential equations; so when I found this 'masterpiece' on Instagram, I wanted to try solving it to refresh myself on some double integrals.

The issue is that it's nonsensical; if you substitute e_ee for some variable 'x', and e_e for some variable 'y', you can clean up some of it, but the notation and usage of symbols literally makes no sense. For example, in the first term, there is e^e (which is fine) followed by ((e_e)^(e-*e))_e. The second term simplifies into x^2 (if you ignore the random monus thrown in there). The third term is more nonsense with out-of-place subscripts and exponents. Luckily, the 4th term simplifies into y^-e, though.

Am I missing something? Is this real, or just another mathslop post? I would like to know if I'm crazy, so some confirmation on whether or not this is even possible to solve would help!

If you treat something as close enough to zero and cancel out this term as it is zero, it is still not zero, so how can this be exact solution instead very very good approximation? I never really understand this concept. Can someone explain this?

one example: https://youtu.be/M6g-sR-lRCU?si=ldYBbiJENQKCa4OG&t=420

How can we treat a number as non-zero to divide by it, and then instantly treat it as zero to erase it?

"He pointed out, methods inconsistently treated infinitely small quantities (increments) as non-zero at the beginning of a calculation to find a rate of change, and then magically set them to zero at the end of the calculation.."

https://math.stackexchange.com/questions/2007909/whats-the-difference-between-x-approaches-0-and-equals-zero

I saw this problem in a multivariable textbook.

I think f is continuous on all points in its domain. But it clearly jumps "around" sqrt(2). Is there a point there where its not continuous?

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I was thinking about this today, and i even managed to back this theory up: the lower the decimal you divide by, the higher the number.

(I'm not a mathematician in any way and I'm still in school. was just curious)

i have a question that's been bothering me in the field of differential calculus regarding the linguistic logic behind the definition of the derivative. I find it illogical, because based on the limit definition it studies the behavior of the slope(derivative) as h approaches zero.

m(h)=(f(x+h)-f(x))/h

Though, This approach suffers from the same problem as Newton and Leibniz's works and is subjected to the same criticism introduced by George Berkeley. cause, even with a limit, we algebraically reduce the function to eliminate h in the denominator, and then we study the behavior of the derivative as h approaches zero. This literally makes you treat h initially as a non-zero number, and then suddenly make it zero.

My problem is with the linguistic logic of this approach, and certainly i believe, though I'm not certain, that Cauchy-Weierstrass dealt with the derivative as a two-step algorithm. First, he calculates the derivative for any value of h that is not equal to zero, for example.

f(x)=x^2

f'(x)=((x-h)^2 -x^2)/h where h is R/{0}

f'(x)=2x+h

Then, after evaluating and reducing the derivative, he redefines it as a function of h where h can be zero.

f'(x)=2x+h where h is R/{0}

g(h)=2x+h where h is R

then taking the limit of this new function as h approach 0.

lim(h->0)(g(h))=lim(h->0)(2x+h)

g_l(h)=2x

This is because the limit of the function before algebraic reduction equals the limit of the function after algebraic reduction.

lim(h->0)(g(h))=lim(h->0)(f(h))

and since the function after algebraic reduction fills the gap... At zero, we can then find the derivative in the context of instantaneous change!!

so my problem is just how you put a limit before reducing and you have the right to ghost it and just reduce h then oh its ok you can now calculate the limit where if i do not follow this where i can because there is no order operator that tells me why i get 0/0.

I need to know if there is a way to solve this without l'hopital to explain this to a calculus class i'm attending. I know the answer to this limit, but I couldn't find a way to solve it without using l'hopital

Just want to start off by apologizing if this is the wrong subreddit for this kind of question. I’m new here and I’m not entirely sure about the scope of what constitutes a math question here.

In particular I’m thinking of Stewart’s Calculus textbook right now. I’m doing Calc 2 and I haven’t taken calc 1 in a while. I just don’t understand why he doesn’t explain what he actually means, what his ACTUAL thinking is in any given example. It’s almost like he just assumes the student already knows how to do everything and he’s just recapping it when it’s the other way around?

Granted, I’m autistic and I desperately desire detailed explanations for topics like this and I just find it so frustrating. Maybe it’s not for other people. I get that. But I also used to be a high school math teacher and I am trained in trying to anticipate students confusion, questions, etc. and in explaining myself and the thinking behind an idea or example as thoroughly as possible. That may have to do with the fact I’ve got a strong academic background in humanities as well, idk.

It just seems to me that if your literal job is to teach people how to understand and do something you’d obviously explain your thinking clearly, any symbols or notation clearly, etc. And nobody does! It’s driving me insane. I genuinely enjoy doing math and learning it but I do not enjoy or appreciate being gaslit by someone telling me they’re teaching me something when they aren’t.

I understand a significant amount of responsibility here belongs to the publishers who require as few words and pages as possible to minimize printing costs. But that doesn’t explain why professors don’t explain themselves either. And I know about the “curse of expertise” and all that but it gets to a point where you’re at the inverse of Hanlon’s razor where “sufficiently advanced incompetence is indistinguishable from malice” and the level of pedagogical incompetence in undergraduate and graduate level mathematics educational skill is just astounding to me and I cannot believe these people can actually charge money for their educational services in the form of tuition and textbook prices when they suck this bad at their job (which is teaching). And I know I know their primary job might not be teaching if they’re a mathematician, but then stop having these terrible teachers teaching math and get mathematicians who wanna actually teach math well teach the classes instead! It’s almost like they don’t want anybody to actually learn it, and only want people who don’t need to be taught it to be able to do it.

Rant over.

EDIT: I will say in Stewart’s defense that it’s almost always the first few chapters sections of each chapter that give me the most issues. The chapter sections where he’s trying to “walk you into the idea before he presents the basic idea” like in chapter five how he spends the first two sections going through “here’s how you’d do this without integrals” followed by “here’s how you’d do this with limits” before finally getting to the easy part which is “here’s how we do this with integrals and antiderivatives, sorry I made you suffer with all that other crap beforehand but it’s a right of passage”.

I'm a beginner, only having learnt the basics of calculus about two weeks ago. Why does this not work? I thought that I applied the rules correctly but somehow this result is incorrect.

I was solving a bunch of series and got stuck on this one, I don't even know what test I should try for it, please don't solve just give me a hint on where I could start

How one could calculate the area of the shape between the sine and cosine function?

I just got curious and would love to know

Thanks

I understand that radians make more sense because they make certain calculations easier, but why does it mean that in calculus you are forced to use radians? If it is just an arbitrary measurement system why does calculus need you to use radians to get the correct result?

finding the circumference a circle can be done by using the radius, which can be a rational number. and then you are stuck with an irrational number for the circumference. and with triangles you get stuck with radicals that are irrational for a side length

but is it possible to have a real length that is irrational? it seems like in the physical world it would always be completely ratioed, even if you would be there for seemingly forever.

I'm asking this because somebody said at one point you would be PI years old. I'm okay with being 3.14159 years old, but there would be no continuation with "..." it would just have to end and be a perfect ratio at some point, right?

My professor did something in his notes that i thought was odd. I kind of just accepted it at first, but was wondering if someone could explain it to me.

I sometimes have trouble understanding the way he uses square roots, especially.

So my question is, is this okay to substitute sqrt(x)*sqrt(x) for x just to cross the top one out ??

And like, how would i know to think of using that method? Like what about the problem would tell me that I should do that, instead of doing something else like, idk, factoring out sqrt(x)/x. Lol

It all started with arguments about 0.(9)=1 on multiple random posts,

I assume everyone knows it already but other than the usual 3*1/3 and algebraic way, I also always backed it up with:

1. The sum: lim_{n->∞} of Σ (x=1 to n) of 9/10^(x), or 0.9+0.09+0.009+... converges/equal to 1

2. 0.(9) can be rewritten as lim_{n->∞} 1-0.1^(n)=1 with the ε-N proof of it as an overkill back up with N=log(ε^(-1))

3. ∄x ∈ ℚ that satisfies 0.(9)<x<1 ⟹ there's no difference in between 0.(9) and 1.

However, I always find a pattern from the people who disagree with those proofs in particular, their arguments are

1. Limits cannot be reached since it means approach to something therefore they're not equal, you can get infinitely close but never get there.

2. In hyperreals they're not equal.

3. It's an infinite approximation therefore not equal.

Now even though I consider myself to be familiar with/like the subject mathematics I have a bit of doubt in both of our claims, considering I'm still a high schooler and my school curriculum only covers up until basic limits, derivatives and integrals with not summation or Taylor series in the Calculus category, and I only recently self learned some Calculus courses.

But here are some of my counter arguments against them.

For the first one, limits work differently as it finds the value by looking at the behavior of the function or "neighborhood?", limits don't care what happens on that exact point based on limit definitions.

2nd, in hyperreals they're still equal by definition, it's from a simple google search I'mma be for real with you so I feel uncertain.

3rd, Infinite is not an approximation, it is equal, 0.(9) already includes infinite amount of 9's so it makes sense to use infinity and have the exact value.

Now my question is, are my counterpoints correct? In general I want to clear some minor parts of confusion about limits in order to get a better understanding of them so I would like to hear your thoughts/insights

i know the squeeze theorem proof but it uses the area of the sector being x/2 which already assumes radians. in degrees the limit comes out as pi/180 and i can't see why changing units would change the limit so much

is it just that radians are the only unit where arc length equals the angle, so the derivative of sin comes out clean? feels like i'm missing something obvious

One of my friends wrote their IGCSE exam earlier this year and encountered this question. He was unable to solve part (b) and asked us for help when the official papers (and mark schemes) were released.

Our class tried solving it and we got (a) by differentiating C with respect to v, and our answer of 11.4 km/h matched the mark scheme.

For part (b), we considered the total cost as (150/v)*C(v) and tried differentiating it. We got a cubic equation which (when graphed on desmos) gave the minimum cost as $4875.

Is this answer correct? Our teachers (and the mark scheme) give a higher minimum cost and did not hear us out.

UPDATE: They have heard us out and sent a response to CAIE

Also, if we are indeed correct, what should we do? IGCSE does not entertain students challenging their mark scheme and our administration said that the correct answer is the one in the second image.

This is a screenshot from one of my homework problems. I am not understanding how the answer is not there is a jump. f cant be continuous because for 1) there is a hole at x=-2 and 2) there is a jump at x=0. The only thing i can think of is that x_0 does not in fact mean "any particular x value".

Could anyone explain?

Which of the two is generally considered easier or less work? I know both are said to be difficult, but I’m trying to decide which one to take over the summer and which one to take during the spring semester. I’ll have more classes during the spring, so I’ll have less time to fully focus on a course.

I passed Calc 1 and Calc 2 with an A, and I’m doing pretty well so far with the vector portion of Calc 3, which I’m assuming is somewhat similar to the foundation of linear algebra. So I’d say my foundation for both courses is pretty solid. Unless linear algebra is more difficult than I’m making it out to be.

I’m required to take both courses. I’m trying to figure out which one would be more manageable alongside my other classes and which one would be better to take over the summer, when I can dedicate more of my time and attention to it.

I wanted to know of there was some way for me to determine how I should go about finding or creating curves properly so that I dont have to keep doing guess work every time I need to measure or make armholes and sleeves

Edit: apparently the appropriate shape is called a pyriform, and there seems to be a universal formula for it. I just need to find out where to start, and if anyone can get me started on how I can learn this so I can use it for armholes, I would be forever grateful

Edit #2: after looking at the Desmos link BadJimo sent it got me thinking of looking at this in terms of bezier curves, or as RailgunDE112 put it, splines. If I think of the armhole as an irregular polygon with 8 points vaguely representing a square (4 points in the corners and one midpoint between each line). I can use those points as controls for the beziers. So the armhole would look like 4 separate bezier curves (if we were to think of the armhole as a grid separated into 4 quadrants) coming together to form what would be an ovoid/elliptical shape.

I've heard the continuously compounding interest explanation for the number e, but it seems so.....artificial to me. Why should a number that describes growth so “naturally” be defined in terms of something humans made up? I don't really see what's special about it. Are there other ways of defining the number that are more intuitive?

My professor showed u-sub where he got 2arctan(sqrt(x-1)) I would have tried u-sub if the trig sub didnt work out so well. Please tell me im not going crazy.

Edit: I know my work is sloppy, it is a timed test and my attention to detail does need work. I was just wondering if the answer I got matches with my professor's answer. I got zero points on the problem and, before speaking to him, wanted to make sure my answer was right. I accept that my work is insufficient for a thorough explanation, honestly I was racing against the clock. Just wanted to know if the end result was correct. Thank you for your feedback goats.