55

Ferdinands, in his short note "Finding Curves with Computable Arc Length", also comments on the difficulty of coming up with suitable examples of curves with easily-computable arclengths. In particular, he gives a simple recipe for coming up with examples: let $$f(x)=\frac12\int \left(g(x)-\frac1{g(x)}\right)\,\mathrm dx$$ for some suitably differentiable $...


38

A software engineer probably does not need to study calculus, and it is less likely to be useful than graph theory, elementary logic, study of algorithms, etc. Of course, if you are implementing algorithms for use in science and engineering, calculus and numerical methods for approximating calculus operations will show up all of the time. AI, on the other ...


32

But it must start to repeat after some digit, no matter how large, she retorted. This sounds like a misunderstanding that you could do something about by showing her a concretely defined different irrational where it is clear to see that the digits cannot repeat, such as $$ \sum_{n=1}^\infty 10^{-n^2} $$ If you manage to convince her that is is possible for ...


28

I would try to convince her that most numbers are irrational and not even address $\pi$ - showing her that the question of "Do the digits of $\pi$ repeat?" is nontrivial (and, indeed, deep) is probably more valuable than convincing her that they don't, especially since convincing her of $\pi$'s irrationality could make it seem like $\pi$ is special when this ...


28

Another example: you can get $$ \sqrt{1 + [f'(x)]^2} = ax + \frac 1{2ax} $$ by taking $f(x) = \frac 12 a x^2 - \frac 1{4a} \ln(x)$ for any constant $a$. A possibly helpful way of reframing the question: we would like to know for which "nicely integrable" functions $g(x)$ is there a "reasonable" $f(x)$ satisfying $\sqrt{1 + [f'(x)]^2} = g(x)$. In other ...


26

I fell into the trap of memorisation. Work through the proof, understand what is being done. You will not need to memorise. You will remember key plot points simply because you've spent time on the proof and relevant definitions (learn math by doing it). You can carry on from there independently. Also, use the whole semester to study. Don't jumpstart your ...


26

You do not study proofs because you need to memorize them, you study them because you want to understand them, and when you understand them you should be able to more or less easily reproduce them to someone, if needed. When you understand proofs of some statement and when you also understand that statement then you can try to find some other proofs of that ...


23

Calculus is a fundamental mathematical science - Learn it to broaden your mind and not necessarily to be graded at.it. It is fundamental for scientific computing. Programming in scientific filed specially engineering require background. I am surprised that you are studying engineering without calculus!!!


23

This example $$ y = a\cosh \frac{x}{a} $$ is quite simple for computations.


22

One of the things that makes a cone simpler than a cube is that it is an “algebraic object” that can be defined by a simple polynomial identity ($x^2 + y^2 - z^2 = 0$). Taking cross-sections preserves this algebraic nature (since an infinite plane is also algebraic) so we end up with a quadratic curve in two variables, which is a fairly nice object. In some ...


22

I will dissent here (as often) and say: DON'T. The problem here is looking at everything as something that needs to be computed, "solved", or otherwise manipulated into some set, pat form. As it is well-known, few of these integrals are amenable to exact representation in terms of anything encountered at this point (if anything at all). Any exercise you ...


21

Elementary algebra begins with the study of linear equations. Quadratic equations naturally come next. Their graphs are ellipses, parabolas and hyperbolas. The Greek geometers knew them as sections of a cone. That geometry is often left out of beginning math courses today. The quadratic functions are particularly useful in physics. I think they are not ...


19

You can try $f(x)=\dfrac{\sqrt{a^2e^{2ax}-1}-\tan^{-1}\sqrt{a^2e^{2ax}-1}}{a}$, which has arc-length $e^{ax}-1$ and isn't too hard to work with as long as you remember $\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}x$. But other than that, you could always define your function as an unsolved integral, $f(x)=\int\sqrt{L'(x)^2-1}\ \mathrm{d}x$. Then even when the ...


18

Sometimes, such question have much more meaningful interest than just exercises. The process of learning mathematics, studying in the right way and getting to properly understand the true meaning of them is something that not only bothers lower-level or standard level students (of any age) but also anyone who is involved with mathematics. The process of ...


15

This is just an opinion, based on my own experience as a working mathematician. Let me say honestly that a professor seldom studies a subject as a student would do. In our times, publishing has become an urgence, so that we must publish as soon as possible. Therefore, we do not usually move very far away from our expertise, unless we are the few outstanding ...


15

My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some other 3D (or any dimensional) object? Because there's nothing else. Look at the common 3D solids. The cross sections of a cuboid (and in fact any other polytope) are just a bunch of straight lines connected together, so this is ...


12

A rigorous proof may not be possible at this level of mathematics. However, you may be able to get away with a more informal approach. First, convince her that pi equals some generalized continued fraction, such as: $$ \pi = 3 + \frac{1^2}{6 + \frac{3^2}{6 + \frac{5^2}{6 + \frac{7^2}{\ddots}}}} $$ (or pick another example which converges faster?) You can ...


12

Hint: A great introduction into the concepts of mathematics is What Is Mathematics? An Elementary Approach to Ideas and Methods by R. Courant and H. Robbins. This classic is mathematically profound and good to grasp. It is from my point of view a wonderful example how mathematical thoughts can be provided in a pedagogically valuable manner to ...


11

The nursery rhyme may be referring to the fact that power series are "very, very good" within their radius of convergence (absolute convergence, term-by-term differentiation and integration, uniform convergence, etc.) while they diverge outside the radius of convergence (which is "horrid" as far as series are concerned). The curl in the middle of her ...


10

Understanding a proof means, you need to understand the full idea as a whole, getting every line of a proof but not getting the whole picture is not actual understanding. So, if you understand the proof, no need to memorize it. It will not harm to understand proofs outside your course.


9

There are several types of objects that can be analyzed as a curve defined by the zeroes of a second-degree polynomial in two variables: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. This includes circles, ellipses, parabolas, hyperbolas; all with the same basic equation form, but differing in their relationship between A, B, and C. It's handy to have one term ...


9

TL;DR: ultimately you do study cross-sections of other cone-like shapes, you just don't know about it unless you've studied some algebraic geometry. The cross-sections of a cone lie in perspective. Imagine you are sat at the origin in $\mathbb{R}^{3}$ and you look out along a ray (=half-line) which lies inside the cone. Fix a plane $\Pi$ which does not go ...


7

In group theory, "visual" explanation of the group $D_4$ (or sometimes called $D_8$, which is dihedral group of order $8$ or degree $4$) was really exciting to me. Even with some elementary knowledge about groups, if someone defines $D_4$ as $$\langle x, a\ |\ a^4 = x^2 = e, axa = x \rangle$$ it might seem meaningless or too abstract to understand. When I ...


7

For instance, $$\frac13+\frac13=\frac12+\frac16$$


6

One way in which it can be helpful to memorize a proof is to learn the techniques. Diagonalization comes up in both Cantor’s proof of the countability of the rationals and Turing’s proof of the undecidability of the Halting Problem, among others. I’ve had the chance to apply it to several other problems since I saw it. The original proof of the Fourier ...


6

The way I interpret this problem, the salaries go as follows: Person 1: He gets paid $2000$ for the first year, $2300$ for the second year, $2600$ for the third year, etc. So for the $n$th year he gets paid $2000+300(n-1)$. Person 2: He gets paid $1000+1100$ for the first year, $1200+1300$ for the second year, $1400+1500$ for the third year, etc. So he ...


6

Suppose if they get $100$ each half year. Then the following are the possible outcomes $$1^{st}\mbox{ year }\ \ \ \ \ \ 1000+1100=2100$$ $$2^{nd}\mbox{ year }\ \ \ \ \ \ 1200+1300=2500$$ $$3^{rd}\mbox{ year }\ \ \ \ \ \ 1400+1500=2900$$ $$4^{th}\mbox{ year }\ \ \ \ \ \ 1600+1700=3300$$ Suppose if they get $300$ each per year. Then the following are the ...


6

I believe this is a poorly worded math problem, sixth grade or otherwise. The statement $$m \neq n \neq 0$$ is possibly meant to be something like $$m, n \neq 0$$ instead. This would be to ensure that $m = n$ and $m = -n$ can't both be true simultaneously as it only occurs when $m = n = 0$. This is only a guess, but it would mean the question would then ...


6

Challenge: divide the surface of the mug into seven connected regions such that each one is adjacent to the other six. In other words, embed $K_7$ on a torus.


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