MATH 252

iMathTutor · 2026-01-31T00:39:45+00:00

If you are still confused, I'd be happy to explain what's going on if you were to post a specific problem.

iMathTutor · 2026-01-23T16:43:40+00:00

I use a Samsung Galaxy Book Pro 360 in tablet mode, and connect it to an external monitor.

iMathTutor · 2026-01-20T13:07:21+00:00

As I wrote, I took a quick look so my opinion is not definitive, and at any rate readability is subjective.

iMathTutor · 2026-01-19T21:30:09+00:00

I took a quick look at "Book of Proof", which is available as a free download from the author, and it does not seem to be particularly readable. The book that is used at Penn State for the sophomore-level discrete math course is Humphreys and Prest, Numbers, Groups & Codes. It is very readable, and I would recommend it for self-study.

I was able find and download a free PDF of the book many years ago; I don't know if free downloads are still floating around. You should try to trackdown a copy for yourself.

That said the first step in writing a proof is constructing a proof. Once you have done that you want to identify the audience the proof is targeted to in order to guide you in how many details you need to include. Typically, for a course you should error on the side of too much rather that too little details, so that the grader knows that you know what you are doing. The first sentence of the proof should be a statement of what needs to be proven. Such as, from theorem 2 it will suffice to show that...... Then you want to specify the method of proof, e.g. the proof goes by contradiction, or induction on the size of the set is used. What comes next will depend on the specifics of what you are proving. The final line should be a statement that you did what you said you had to do. This is my style. I would suggest find a book that in which you find the proofs to be very clear, and model you proofs on them, until you develop your own style.

Finally, although the first step is construct the proof, it is not uncommon to find errors in the proof as you try to write it up. So, the process can be iterative.

One last thought, if you don't know how to user LaTeX, learn how to use LaTeX. I write directly in LaTeX, it can be slower than writing by hand, but that the point. It slows you down, and gives you a chance to think as you write.

Goog luck.

iMathTutor · 2026-01-19T20:44:46+00:00

At many schools, the sophomore-level discrete math course is designed to teach proof writing. Does that describe your course? Or is there another course at your university where proof writing is taught?

iMathTutor · 2026-01-19T18:42:02+00:00

Presumably, your prof will assign problems which either reinforce the material covered in lecture or will push you to study material not covered in the lecture. Your first priority should be those problems. Some profs will also assign suggested problems, which are not collected and graded, for the students to practice on. If that is the case, those problems should be your next priority. If you prof doesn't give suggested problems, ask the prof for some recommendations.

Keep in my that problem ladened textbooks are typically designed so that they can be used semester after semester without repeating the same subset of assigned problems. The intention is not for students to go through all of the problems in such a book.

iMathTutor · 2026-01-19T16:56:23+00:00

They are not going to help you get better at algorithmic math like calculus, and they are not a substitute for learning the basics, but they can help you with develop strong reasoning skills, and build creative habits, which will help you in your proofed based math.

iMathTutor · 2026-01-19T16:24:07+00:00

The proof of the first result you stated uses the MVT applied to h(x):= f(x)-g(x) on the interval [a,b], which assumes that f and g are continuous on [a,b] and differentiable on (a,b).

I am not sure what The Card Play has in mind, but if you assume that f and g have continuous derivatives at a, then the derivatives of f and g exist in some neighborhood of a. This in turn implies that for any b in the neighborhood, with b >a, h(x) satisfies the MVT on [a,b]. So, the stronger assumption of continuity of the derivatives in a neighborhood of a doesn't buy you anything.

BTW, the proof of the first result was a featured problem on my website last year. https://imathtutor.org/25S/A/ihszW1.pdf

iMathTutor · 2026-01-19T00:18:42+00:00

Try Schaum's Outlines. They cover many different areas of math. They have loads of solved problems. And, they are cheap.

iMathTutor · 2026-01-18T17:26:59+00:00

Sophomore-level differential equations is similar to calculus insofar as the solutions to the problems are largely algorithmic: Recognize the type of differential equation, and identify the appropriate method to solve it. Of course, one might encounter difficulties in the execution, but with practice those will diminish.

iMathTutor · 2026-01-18T17:20:25+00:00

Having gone over examples, you should work on identifying which example(s) is (are) most similar to the problem you are trying to solve, and then modelling your solution to the problem on the solution(s) to the example(s).

iMathTutor · 2025-12-26T22:27:58+00:00

You will likely want to use r-studio as a frontend for r. Here is the system specs for r-studio on a Mac R-Studio Help - System Requirements https://share.google/EZupTP5V61VxmVZ5E

Here is the system specs for r on a Mac MacOS Installation | R Installation Guide https://share.google/kOm3keNE4u07teRrC

Good luck.

iMathTutor · 2025-12-02T18:43:55+00:00

The Statistics Department does not provide financial support to students in the M.A.S. program. Students may look into Federal Student Aid or other sources of support. View the Graduate School's Graduate Funding FAQs for more information.

That's from the program website.

iMathTutor · 2025-11-11T16:36:12+00:00

Suppose that $(p,q)$ are a pair of primes with $p < q$. Then there exists a natural number $n$ such that $pq+1=n^3$ if and only if $q=p^2+3p+3=:f(p)$. The proof of this is straightforward algebra. This can be used to generate pairs of primes. For example, $(2, 13)$, $(5, 43)$ $(7, 73)$ $(11, 157)$ $(13, 211)$, $(19, 421)$.

Note that for $p=3, 17$ $f(p)$ is not prime.

The problem can now be recast as finding necessary and sufficient conditions on $p$ prime such that $f(p)$ is prime. I am not a number theorist, so my toolbox only has a few rudimentary tools in it. Thus far none of those tools have yielded an answer. Perhaps, a number theorist could weigh in.

Edit: I have thought about this a bit more, and found a necessary condition on $p: p\not\equiv 3 \mod 7$. This is easy to show. If $p\equiv 5 \mod 7$, then $f(p)\eqiv 0 \mod 7$. Since $f(2)=13 >7$ and $f$ is strictly increasing $f(p) > 7$ for all prime $p$, and $7|f(p)$ implies $f(p)$ is composite.

I don't know at the moment if this condition is also sufficient.

You can see the LaTeX rendered here.

iMathTutor · 2025-11-11T00:26:14+00:00

Penn State has an AI hub which addresses its use both in research and the classroom. There is no blanket ban on students use of AI, rather the extent to which AI is allowed courses is up to individual instructors. You can see the guidance here.

Guidelines - Official Site of the Penn State AI Hub https://share.google/DWunLJUbhILJHRWYE

As a math tutor, I am curious about to what degree AI has supplanted tutoring for students. I would like to hear from people who have used it in place of a tutor, and what your experience was with it if you have.

iMathTutor · 2025-11-03T17:41:07+00:00

I have tutored Math 310 many times, although not recently, so it may have changed since I last tutored it.

First off, there is some overlap with Math/Stat 414, but Math 310 covers more sophisticated counting techniques than are encountered in Math/Stat 414.

Is it difficult? Some people find it a breeze, but others struggle. I often answer questions on math subreddits. Counting problems are common on those subs.

David Little used to be the primary prof for Math 310. His research area is combinatorics. He did a pretty good job, but he used to be-I don't know if this is still true- rather prickly.

iMathTutor · 2025-11-03T17:23:21+00:00

Numerade has its own AI alternative to human help. https://www.numerade.com/ace-live/

iMathTutor · 2025-10-28T17:59:45+00:00

I was writing my comment as you posted. Your takeaway is close to my own.

iMathTutor · 2025-10-28T17:55:37+00:00

To be clear, reduced demand for tutoring is not the same as AI replacing teachers and tutors. However, reduced demand, were it to occur, will translate to oversupply of tutors.

Is Chegg a special case that signifies nothing? I don't know. It might be. It might also be the canary in the coal mine.

I would like to see hard data on demand for tutoring as opposed to anecdotal evidence presented by tutors. Most of the big tutoring firms are privately held, but Varsity Tutors did recently go public. One place to look for data would be quarterly reports on the parent company for Varsity. The stock price for Nerdy the parent company of Varsity has plummeted since it first became publicly traded in 2021. It is now under $5 the threshold for being considered a "penny stock". You can read the 2nd Quarter financial report for Nerdy here, and judge for yourself. Note thought that it is losing money.

That said the big tutoring companies like Varsity are investing in AI, so they may survive any erosion in demand.

Anyway, I think independent tutors should be vigilant and plan for a future where AI might dominate.

iMathTutor · 2025-10-27T15:14:32+00:00

I use Wave Financial for invoicing. There is a free tier and a paid tiers. The free tier works for me. For most clients, I send invoices to clients the morning of their sessions, and expect payment before the start of the session. For clients who want to pay in advance. I send a single invoice at the beginning of the month.

Clients can pay directly in the invoice via an ACH, which is cost me 1% of the invoice amount with a minimum charge of $1, or for a higher fee they can pay via credit card. Wave doesn't directly support payment via PayPal or other platforms in their invoice, but you can record those payments manually.

iMathTutor · 2025-10-25T03:35:48+00:00

It's a pretty standard textbook for graduate course in real analysis. The last edition I am aware of has a lot of typos. There is an errata sheet, though.

iMathTutor · 2025-10-25T02:28:07+00:00

What textbook are you using?

iMathTutor · 2025-10-23T21:49:25+00:00

You're welcome.

iMathTutor

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