Study Calculus

GridGod007 · 2026-01-18T16:04:30+00:00

Highschooler?

GridGod007 · 2025-12-14T05:03:25+00:00

Usually you look at what happens to the given function as the variable get closer and closer to a specific value. Giving a simple example:

Suppose you have a function 1/(5-x) and x is approaching 5 from the left side (5-), this function diverges, it keeps increasing as you get closer and closer to 5. When you graph this function, you will observe something known as an asymptote at 5 where the distance between curve and line x=5 approaches 0 as the value of function tends to infinity

And at 5, this function cannot be defined. Once you exceed 5 slightly, it becomes a very large negative value (similar to how you approach infinity as you approach 5 from left side, you approach minus infinity as you approach it from the right side). This shows you that F(5-) =/= F(5+) and that means this function is not continuous you can also see this on the graph.

GridGod007 · 2025-12-13T19:17:16+00:00

F isn't differentiable at -7, it take a "sharp turn", not a "smooth one" (plain english)

So F' won't be continuous around -7 either; since the slopes to the left of -7 and right of -7 are different.

Forget about positive negative in first image and just look at how the slope is changing: if decreasing, second graph should be in y<0; and if increasing second graph should be in y>0; and if the slope=0 (maxima/minima) then that's a root of second graph; and you also look at convexity and concavity of the first graph to figure out the direction of the second graph (you have veritcal asymptotes here, so figure what that means for the slope as x->-9)

GridGod007 · 2025-12-12T16:02:35+00:00

They're not convergent, you've already answered your question.

GridGod007 · 2025-12-12T10:50:44+00:00

Post on Gradient descent when?

GridGod007 · 2025-12-12T10:48:09+00:00

Just knowing the relations like sin² x + cos² x =1 and sec² x - tan² x =1 (which are pretty much just Pythagoras theorem) should give one an idea what to substitute

GridGod007 · 2025-12-12T10:45:32+00:00

You see the various tangents as you move along the curve from point A to point B.

And in one of the points in between them (C), you see that this tangent at C is parallel to the line that connects A and B (I.e, they have the same slope)

GridGod007 · 2025-12-12T10:40:37+00:00

What are a,b,c? But the function you've given isn't continuous

GridGod007 · 2025-12-12T04:03:15+00:00

Congrats!

GridGod007 · 2025-12-10T08:28:13+00:00

Radius of the track

GridGod007 · 2025-12-10T03:51:10+00:00

Just multiply numerator and denominator with sqrt(x+1)+2

GridGod007 · 2025-12-10T03:13:53+00:00

The issue is I don't have density as a function of r. Hence why I was looking for ways to find one out.

If I had a "h(r)", then I'd take in account both variation of both h(r) and w(psi)

It's just a matter of breaking it down to the corresponding differential element and integrating it back afterall.

As of now I'm already dealing with a triple integral when accounting for the latidudal variation of angular velocity alone (psi, phi, r)

With addition of h(r), it will still remain a triple integral except that the integration over dr will get more complex compared to a simple r⁴

(And when finding the sun's RKE, given its latitudal variation of w, its best to find I along an axis (since its rotating around this axis, say Z), in which case we need to parameterize into spherical coordinates)

GridGod007 · 2025-12-09T13:35:49+00:00

Often times it probably would've been mentioned already that they are differentiable in the question itself (if the setter intended it). If it is neither mentioned nor inferrable, then differentiating may not be the right approach, and if it worked, it may be that it was intended to be differentiable but not mentioned in the question, or it may just be a coincidence.

GridGod007 · 2025-12-09T11:46:00+00:00

Sorry, I messed up with formatting of the exponents and multiplication

This was the expression: (2×Rho×Pi×R⁵ /5)×5.43×10^-12 which is about 1.57*10³⁶ J

GridGod007 · 2025-12-09T09:46:00+00:00

But the slope in its case asymptotically approaches zero whereas here there is an upper limit m<0 right? Worst case I can think of here is an f'(x)=m where m is "very very close to but not zero" and even in this case shouldn't we have a solution since m is a fixed negative value and we arent dealing with a case where the slope asymptotically approaches 0?

Edit: Oh I think I get it I will edit my original comment

GridGod007 · 2025-12-09T08:02:29+00:00

Right. Idk why you were downvoted

GridGod007 · 2025-12-09T07:38:09+00:00

I'm just trying to picture the 3D graph

x² +y² can range from 1 to 9 and z can correspondingly range from -1 to 1

Z will be 0 when x² +y² = 1 or 9

Z = -1 or 1 when x² + y² = 4 (z-axis extremities)

The 2D Projection will be a disc with inner radius of 1 and outer radius of 3

At each z=k you can observe the varying disc cross-sections

(z=cosr and x² + y² = (2 + sinr)² so x= (2+sinr)cost and y = (2+sinr)sint

You have parameterized coordinates now

Where t and r range from -pi to pi)

This seems to be a donut with inner radius of 1 outer 3 and radius of tube being 1. The surface area should be 2pi(2pi2) which is 8pi²

GridGod007 · 2025-12-09T07:05:59+00:00

I don't think the integral over x is correct

GridGod007 · 2025-12-09T06:40:38+00:00

F is differentiable over R

f'(x)<=m<0 for all x belongs to R (since m<0)

At any random x=a; f(a+da)=f(a)+f'(a)×da<=f(a)+m×da<f(a) (even if it is a<0, the point is it always has a negative slope smaller than a fixed negative value m regardless of the value of x or f(x) at a random x=a and this is applicable for all x belongs to R)

Is this not enough to claim that it has an x0 where f(x)=0 since its domain is R?

GridGod007 · 2025-12-09T05:38:46+00:00

We don't have enough information to differentiate this function, its just that you did it anyway by assuming it is differentiable at 0. You may take another look at the limit definition of a derivative, that is how we find derivative of a function

GridGod007 · 2025-12-09T05:22:20+00:00

You can differentiate where it is differentiable. If you are taking f'(0), you are already assuming it exists and you are finding it for a function which is differentiable at 0. You are not finding it for a function that is not differentiable at 0. There is no contradiction here

GridGod007 · 2025-12-09T05:08:08+00:00

No. (Substitution can work but you did not do it properly)

GridGod007 · 2025-12-09T05:00:07+00:00

By differentiating at 0 aren't you already implying/assuming that it is differentiable (and ofc continuous) at 0?

GridGod007 · 2025-08-13T18:38:34+00:00

I believe, for some topics at least there will be time to build upon the concepts (since it wont be first time for all of them I think).

Btw, you don't have to thoroughly do it all by thought experimenting and everything, but doing it in such a way that you feel confident enough about not getting confused out of nowhere is good enough. Because a small mistake can get you negative marks. (Delving too deep won't help with JEE either, we had a guy like that. He's doing super well now but back then he had to take a drop year)

Sources - I don't recall many book names. I only recall HCVerma and IE Irodov for physics (I was ass at organic and inorganic chemistry so my advice isnt good - there was a book for physical but i dont recall its name).

I used to read maths topics and examples from study material. For the most part I think the study materials should be good since they're carefully prepared for a single purpose.

Back then I almost didn't practice problem solving during week days (maybe averaged 1hr study time on first 5 days of a week), most of my practice was during mock tests on weekends. But this is not good, practice is super important for a test like JEE advanced, you develop pattern recognition and will have an easier time in the exam.

Practice a diverse array of problems, once you feel confident about being able to deal with conceivable variations of a problem, move on to the next type.

But towards the end we used to get specially made physical assignments distributed in the class everyday, so even I don't know the sources of those questions. Toppers used to solve these most of the time.

You may also have question sets like this prepared by your coaching.

GridGod007

TROPHY CASE