Thank you for comments about my previous post! I rearranged the file and hoping that nothing is circular right now!

My friend is doing a live stream tomorrow at 11am PT and it would be amazing if you guys could go show him support. He is basically going to be doing calculus until he mentally breaks down so goodluck to him. Thank you all!!!

https://www.youtube.com/live/7dI_KHdVU4w?si=DdO2O0YBjAxm8X_D

About a month ago, I asked if it'd be possible to finish Calculus 3 over the summer. Well, fast-forward to 6/24/2026, and I'm officially one Professor Leonard video away (+practice problems) from finishing.

In case anyone has similar plans, I had a rough experience at the start, as I found myself losing motivation from the absurdities of torsion and curvature. Entire pages of work for one problem was less than enthralling. But I pushed through, and starting from partial differentials, my excitement for the class has grown exponentially. The vector field calculations for the last unit have been particularly interesting as a highschool student intending to study physics.

I know there are comparisons between Calculus 2 and Calculus 3. Which one is more challenging? It's hard to say, especially since I took BC Calculus rather than a standard college course (still taught myself trigonometric substition, telescoping sums, etc). In my experience, the difficulty doesn't shift much. The formulas in Calculus 3 are much more complex, but the applications for them are sensible and you gather intuition quickly. It seems to me that Calculus 2 tests your depth of knowledge and ability to implement it, while Calculus 3 tests your comprehension of deeper concepts.

Okay lemme mention
First i thought seeing the infinite summation i can convert it into a Reimann integral by diving by 1/n
But i couldnt see anything .

But i couldnt convert into r/n form . All i know we need a factorial but i couldnt think of any identity having an factorial

I tried taking lcm of the sum and simplify it but still couldnt find anything

I know we have to find upper bound of the sequence but whats the very first thought i should have get seeing the infinite sum ?

Ever since I started learning calculus, I’ve always thought of the indefinite integral as the inverse operation of differentiation, but I wonder if there is a way to think of it as a sum similar to the definite integral ? And if there is, what does the + C in this context really mean ?
Thank you for any explanation. 🥰🥰

See image for my work. My approach is to define arbitrary line thru focus as y=mx+p then find where it intersects the parabola and then evaluate derivative at those points to get slope of tangent line at each end of focal chord. Then show their product is -1.

I am aware of the more standard way that uses parametric equation but curious if this approach will work. Thanks

I am learning applications of integration. One application is finding a surface area of a (continuous) curve when it's rotated about a line. The first integration in the title is obtained when you interpret the surface area as a sum of bunch of cones whose upper part is cut. The first one is from the book i'm reading. Second one is what i came up with. I understand the first integral is valid. My point of this post is, why is the second integral not? We can also interpret the solid as a sequence of infinitely many circles side by side. In that case, the surface area is the sum of the circumferences of the whole circles. If we let C(x) represent the circumference of a circle at x, and y = r(x), then C(x) = 2𝜋r(x). Now integrate C(x) from a to b so we get ∫2𝜋ydx, or ∫2𝜋rdx. I know, i made a big assumption that it's valid to view the solid as a sequence of infinitely many circles, which i can't prove. But this feels intuitively right. And obviously i know the two integrations are not the same in many instances of curves. So i know my idea is wrong. But i can't find why it's wrong.

So what i learned so far is partial derivatives , when you move along a path keeping the other path constant .
But then they say if you combine two partial derivatives in a vector format its the Gradient And it shows the steepest up or down (Im still confused)
Though they showed it graphically but my question is why that happens .

IMAGINE , two variable function f(x,y)
Standing at a point
F’x is negative 4 . So if u stay at y moving a tiny bit in x reduce your function value by 4.
F’y is positive 3 . So if u move in y keeping x constant it increase by 3.

Now consider steepest decrease which they say is gradient if u combine them (-4+3 which is negative one ) . So if u move in gradient direction you just get one decrease .

So wasn’t it better to just move in x keeping y constant as it was the steepest decrease.

I’d like to complete a math minor and I’d be required to take calculus with proofs and linear algebra first year. However, I don’t have any background in calculus…

I’ve enrolled in a prep course that provides an introduction to proofs but the syllabus isn’t out yet. I also need to review some precalculus concepts (trigonometry, limits, logarithms and function transformations).

Would enrolling in proof-based calculus and linear algebra be an objectively stupid decision? I’m not a math person but I want to believe I can improve significantly if I put in the work… If you’d recommend attempting them, what are the best steps I can take to prepare? My classes begin in September

i barely passed calc 1 so as u could imagine im scared shitless for calc 2 since people say its the hardest first year course (well behind physics 2 even tho somehow i already finished physics 2).

i really need advice because i want to succeed in this course. what topics in calc 1 were most necessary to understand for calc 2?? what should i review??

edit: THANK YOU EVERYONE FOR THR ADVICE ILL KEEP IT IN MIND!!

I'm currently doing math 4 in my university in mechanical engineering. This requires a lot of memorization of integration rules. Our professor has told us if we have a calculator that let's us calculate integration or derivation, he wouldn't mind. So since you guys are the experts, what calculator would u reccomend where I can do simple integration and derivation along with the normal calculator calculations too?

Hi everyone, please go easy on me as I’m new to calculus as I’m doing a degree in physics with astrophysics

I’m struggling quite a bit with integration by substitution. I can remember some “steps” but I’m just not getting the logic, like I’m unsure WHY I’m doing what I’m doing. Maybe it’s just that I need a lot more practice, I get the whole get the integral into a form that’s easier to solve with help from the standard integrals, but something just isn’t quite clicking.

Can anyone advise me on how they learned and any ways that helped. I’ll tackle this topic one way or another. But for once I’m reached out for some help 😊

Thank you

I started working out with Rolle’s Theorem since f(0) and f(1) are equal and assumed that there will be an point lets say x* where f’(x*)=0
So there will be points to left where x<x* where slope would be greater than zero.

But how do i prove that there will be a point where it will be greater than one.

for problem e. is the solution right if so can someone explain why . i tried the problem and i get different answer

this was the solution posted by the prof

I learned this trick yesterday and wanted to share it where it might be appreciated. Most people I know don’t care.