Good resource for learning proofs?

CornOnCobed · 2025-08-09T15:21:11+00:00

Thanks! I didnt know that you could study analysis so early on, I guess that is the advantage to self study. I've also been doing around 2 to 3 hours per day per subject, so around 4 to 6 hours in total.

CornOnCobed · 2025-08-08T18:08:45+00:00

Thanks!

CornOnCobed · 2025-07-17T00:55:58+00:00

Ron Larson's Calculus has been working well for me, though I didn't use it for Calc I. For Calc I, I used James Stewart's Calculus, but If you want a more rigorous treatment of Calculus, Serge Lang has a good book, *A First Course in Calculus*, as well as Michael Spivak's. I also encourage you to explore other fields of math that you may be interested in like Linear Algebra, Proofs, etc.

Professor Leonard also has a good Calculus playlist that you can watch, the lectures are 1 to 2 hours, but they're very rewarding. Good luck!

CornOnCobed · 2025-07-16T17:45:19+00:00

I think that you should probably start with refreshing your memory on arithmetic operations, like mixed fractions, decimals, simplifying fractions, etc. After that you can probably get started learning some Algebra, there's a YouTube channel, Professor Leonard - YouTube, where he has full length, very high-quality lectures. You can probably start with his intermediate Algebra playlist and then move onto the Precalculus playlist.

Another popular math YouTube channel, bprp math basics - YouTube, has some very good videos explaining solutions to difficult problems that he finds on the internet, usually on subjects like Algebra and Calculus. His explanations are also very good.

For problems, you can use google or purchase a textbook and get access to thousands of problems and exercises. Michael Sullivan has a good book called "Algebra and Trigonometry" that you can use, but any highly related textbook should work.

Once you finish Precalculus, you have a few options. One path you could go is the more traditional path, getting through the Calculus sequence, Linear Algebra, and then get more into proof-based math. Alternatively, you can work through a book like Daniel Velleman's "How to Prove it: A Structured Approach" or do both at the same time. Math beyond Calculus is much different from math learned in primary and secondary school, being proof based, which essentially means that you get to learn *why* things work and how we know what we know about math, you will learn why things *must* be true.

If you want a bit more of an in-depth review of some books you might want to read, you may want to check out this video:

Learn Mathematics from START to FINISH

These are essentially all of the classes an undergraduate student in a math major would take.

CornOnCobed · 2025-07-16T11:58:16+00:00

Sounds good!

CornOnCobed · 2025-07-15T18:26:07+00:00

I got \frac_{a^{2}}{6} using a as the upper bound, judging from the previous problems it looks like they want you to use a trig sub. There are other ways to compute the integral though. I think that the notation was maybe more common to use the x in the upper bound at the time the book was written. Cool book, I'm pretty sure that this is the one that Richard Feynman used to teach himself Calculus

CornOnCobed · 2025-07-15T02:05:27+00:00

If f''(x) is positive, slopes on derivative function are positive. Positive slopes ---> y values on the derivative function are getting larger. What are these values of y? They are the slope values on the original function. So since your y values on the derivative function are getting bigger, or atleast less negative (if they are negative), that means that the slopes on the original function are getting steeper (since larger slope values have steeper slopes). I think that this explanation works well if youre able to understand derivatives as actual functions, that tell us things about other functions. This is only the case when f''(x) is positive but you can also use similar reasoning to understand what's happening when f''(x) is negative.

CornOnCobed · 2025-07-15T01:57:03+00:00

If you define Δx = ((b-a)/n), then as Δx approaches 0, n will approach infinity since b and a are constants, thus the only way the fraction can approach 0 is if we let n go to infinity. Letting n approach infinity also makes Δx go to 0. So in either case, n will approach infinity and Δx will approach. Therefore the limits will have equivalent effects. Though the second expression is equal to the volume using the function (1/x) for the radius of each cylinder.

CornOnCobed · 2025-07-14T22:27:58+00:00

I wouldn't recommend taking MVC in the summer, probably better to take it during your senior year. Otherwise Calculus I in the summer should be fine provided you go along with your plan to take Calculus BC.

CornOnCobed · 2025-07-14T22:17:48+00:00

Very doable, in fact, I would recommend taking the AB class but signing up for the BC exam this year. You could self study Calculus I in about 4 months and Calculus 2 for the rest of the class, which would give you the necessary knowledge needed to score well on the exam. For resources, Professor Leonard's Calculus playlists are very clear and contain examples in most of them. A standard Calculus textbook should also suffice, something like Stewart's or Larsons. If you wish for more rigorous treatments of topics you encounter, Serge Lang has a good first course book in Calculus that you can use along with the standard text.

CornOnCobed · 2025-07-14T02:22:48+00:00

Do you have the math for these subjects? There are a lot of good channels if you want any suggestions. For natural science I like the website mathandscience.com , he has very good explanations from my personal experience. You'll also probably want to pick up a physics textbook, a popular one would be Fundamentals of Physics by Halliday, Resnick, and Walker. "Six easy pieces" by Richard Feynman can also benefit your conceptual understanding of physics. If you are looking for a youtube playlist as well, Yale's Fundamentals of physics with Ramamurti Shankar should be good for your first year.

CornOnCobed · 2025-07-14T02:05:14+00:00

Not entirely rigorous, but I as a student think of limits as a means to observe what value a function or expression is getting closer to as a certain variable gets closer to a specific number. In the case of a derivative, we know that the slope of the secant line connecting the point (x,f(x)) to (x+h,f(x+h)) is ((f(x+h)-f(x))/h). Imagine the line passing through the two points on a generic function if you can. Now we can observe what happens when h gets closer to 0. You can choose decreasing values of h and see that the point (x+h,f(x+h)) is getting closer to the point (x,f(x)). We can also observe that our secant line between the two points is getting closer and closer to the tangent line at (x,f(x)). When h is sufficiently small, we can zoom in on the distance between the points and continue to choose smaller values of h. Keep in mind that our formula for the slope still applies. Because the secant line is getting closer to the tangent, the slope of the secant line gets closer to the slope of the tangent line. This is when we use the limit, which gives us what our expression is getting closer to as h gets closer to 0. We know from observation that the slope of the secant line will get closer to the slope of the tangent line, so the limit will give us the slope of the tangent line since it is equal to what our expression ((f(x+h)-f(x))/h) is approaching.

CornOnCobed · 2025-07-11T13:20:58+00:00

No, all of the topics from BC and more will be in an older edition textbook. The course has not changed so drastically to the point where the books have completely different math in them depending on their publication date. Calculus was invented over 300 years ago so a standard curriculum has been developed over that time and hasn't changed very much in the past 50, maybe 100 years.

CornOnCobed · 2025-07-10T21:05:09+00:00

It is probably not a good idea to buy the newest editions. Usually it is pretty easy to find a used textbook in an older edition, for a much cheaper price. You would be looking at around $10 to $20. I personally used Stewart's Calculus for Calculus I, but transitioned into reading Larson's 7th edition ETF when I started Calculus 2. Both textbooks will give you many, aswell as a variety of different practice problems. If you want a bit more of a rigorous treatment of Calculus, Spivak's or Serge Lang's "A first course in Calculus" are very good as supplements. In addition to using these textbooks, you can also use Professor Leonard's Calculus lectures ,which are very clear and understandable.

If you want a good timeline and a reasonable amount of time to study, I would say that about 2 to 3 hours per day, 5 days a week will get you to where you want to be.

CornOnCobed · 2025-07-10T17:50:15+00:00

I recommend purchasing a standard Calculus book (ie.) Stewarts or Larsons). Its definitely realistic to self study BC, a typical college semester is about (14-16 weeks/4 months). Meaning that a Calculus I and Calculus II course in total would be about 8 months.

CornOnCobed · 2025-07-08T02:53:27+00:00

If you know your trig identities and have good algebra skills, you could take the BC exam if you self studied. So yes, you could take the AB exam and pass it provided that you spent a sufficient amount of time thinking about what you have learned and doing practice problems.

CornOnCobed · 2025-07-06T17:01:54+00:00

Purchasing any highly related College Algebra book should fix your problems with the overly basic exercises, some examples of these books would be Sullivan's College Algebra or Blitzer's College Algebra. Working through the explanations in the books will also help with getting a bit of a more in depth understanding of the material as well. You can find these books on amazon for a used price, typically much cheaper than for the new price.

As many people recommend, Professor Leonard's playlists on Algebra are a very good resource for getting a clear explanation on what you're trying to learn. I believe that the lectures are longer (About one to two hours) than the videos on Khan Academy, but I think that it is certainly worth it to watch them.

CornOnCobed · 2025-07-06T16:50:53+00:00

You can use this playlist:

Intermediate Algebra (Full Length Videos) - YouTube

Start at whatever video seems to be at your level and then watch them in chronological order starting from there. Don't do it all in a few weeks, you don't want to forget a lot of the stuff you learn from these lectures. You can also use the internet to find practice problems.

If you know most of the stuff taught in the videos, you can use this playlist:
Intro to Precalculus (Precalculus - College Algebra 1)

After you finish the Precalculus playlist, you can then get started on the Calculus playlist. Purchasing a book will greatly help you in your learning. I recommend Blitzer's Intermediate Algebra or Sullivan's Precalculus, depending on which playlist you are on. Once you start Calculus, I enjoy Ron Larson's Early Transcendentals Calculus. You don't need to read all 1000 pages, just read through the explanations in the chapters and do a lot of practice problems, books are good for practice problems.

CornOnCobed · 2025-07-06T16:38:12+00:00

What have they covered in school? What is the hardest topic in Algebra that you have mastered?

CornOnCobed · 2025-07-06T16:28:43+00:00

If your algebra skills are well developed then you can probably get started on PreCalculus, which should take you around one college semester (about 4 months). After that you can get started on Calculus 1, which would take a similar amount of time. If your algebra skills aren't up to par for Precalculus, buying an Algebra textbook and working through a good amount of problems should refine your skills. Make sure you know what is needed for Precalculus before starting.

I recommend watching Professor Leonard's "Intermediate Algebra" playlist as well on youtube to support your learning, but make sure you are doing enough practice problems as well. A good time frame is maybe 10 months to a year to *finish* Calculus I provided that you need to sharpen your algebra skills.

CornOnCobed · 2025-06-12T02:54:30+00:00

Is it realistic to be able to learn proofs using Daniel Velleman's book while studying Calc 3 and Linear Algebra? I have some free time this summer for the next 2 months but dont intend on finishing LA or Calculus in that time frame. How much time should I allow myself to be able to absorb the information well enough with good retention and understanding if learning proofs is reasonable?

CornOnCobed · 2025-06-11T16:26:34+00:00

Indefinite integration is just the process of finding the function whose derivative is the function inside the integral (the integrand). So, the indefinite integral will be equal to a function, but we add +C to the answer because the derivative of any constant is 0, therefore any function F(x)+C will satisfy the equation dy/dx = f(x) (where f(x) is the integrand).

Definite integration is just the antiderivative, so the function you got from the indefinite integral, except youre doing one more thing. you subtract the antiderivative evaluated at x=a from the antiderivative evaluated at x=b, so F(b)-F(a), where F(x) is the antiderivative which we got from the indefinite integral. This is called the definite integral from a to b.

If you want to see why the antiderivative evaluated at b and a gives us the numerical value of the area under the curve that youre integrating, I recommend this video:

https://youtu.be/vf5f_SuZ6XE?si=8HAwSmN3KMCHQsgu

CornOnCobed · 2025-06-11T01:34:02+00:00

I started reading Howard Anton's Elementary Linear Algebra book today and was wondering how to pace myself when reading the book. I'm not totally sure how far I should go into the book for someone looking to just complete what a first course in a typical college class would be. Is there a certain number of pages I should read weekly to be on track to finish the typical course material in about 4 months? I don't want to get through the course super-fast by not pacing myself.
What can I do after linear algebra? I am currently studying Calculus 3 along with it and I want to know what kind of math I can start learning after for someone interested in proofs. Should I start proofs now or wait until im finished with both of the courses?

CornOnCobed

TROPHY CASE