3
votes
5answers
590 views

How should I self-study calculus?

So I already took Pre-Calc, and ended up with a B both semesters. I am an incoming senior in high school. My special-ed case manager won't let me take it because she doesn't want to see me panic ...
0
votes
2answers
74 views

What are the differences between mathematics courses taken by engineering majors and by math majors? [closed]

I am curious to know what are the differences between mathematics taken by engineering students and by math majors. Let's say in terms of the approach, depth and in the topics covered. And even within ...
2
votes
0answers
45 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
1
vote
3answers
149 views

Is tutor essential for success in mathematics? [closed]

Everyone in my Pre-Calc - Calc I class is failing, except the kids who go tutor. They get top percentile ranks in the class. Should I drop maths all together so I don't have to invest in a tutor? I ...
1
vote
4answers
88 views

What functions are most useful after the ones learned in high school?

I have learnt how to use trig functions, hyperbolic trig functions, exponentials and logs and simple things like polynomials, ellipses, hyperbolas and rational functions but lately when doing calculus ...
1
vote
1answer
164 views

Gilbert Strang's books on calculus and linear algebra?Are they for math majors?

I would to know what are the best resources to use to teach and learn elementary subjects (calculus,linear algebra),I remember when learning calculus, I used Spivak's book which had wonderful ...
1
vote
1answer
136 views

Comparing $\pi^e$ and $e^\pi$

Comparing $\pi^{e}$ and $e^{\pi}$ I read the answer there but I didn't understand one thing. How I should know to put $\dfrac{π}e-1$ instead of $x$? If I had this question on a test, I had no idea ...
3
votes
0answers
133 views

Any comments on Lax's “Calculus with Applications, 2e”

There's a new calculus book titled Calculus with Applications by Peter Lax (2nd edition of an old one). I really liked his linear algebra and functional analysis books, and I was wondering if this ...
4
votes
0answers
52 views

Where to post a Calculus review guide?

I created a PDF document (using LaTeX) in which I wrote relevant review materials and Calculus problems for Calculus 1, 2, and 3. Is there an appropriate forum where I could try to post this to ...
1
vote
1answer
44 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
6
votes
1answer
504 views

Is calculus not rigorous?

While studying single and multivariable calculus during my first year some people complained that calculus wasn't rigorous enough, when I asked about this no one seemed to be able to really specify ...
4
votes
1answer
107 views

Lecture notes ready for $\LaTeX$

Are there on the internet lecture notes in calculus in .tex or .txt format, that is, ready to be edited/modified/re-used and compiled using $\LaTeX$? EDIT: now I am specifically asking for calculus, ...
3
votes
2answers
148 views

Bridging the gap of understanding function terminology in math for a programmer.

I'm a computer programmer by profession with no formal CS education. When I read in mathematics the terminology used around a function, I get confused. For example, I was reading up on some calc and ...
15
votes
4answers
2k views

Is there a(n elementary) function whose derivative we cannot integrate?

Say, for example, I take a reasonably-complicated function $f(x)=\tanh[\ln(x^x)]$, and differentiate it to get $$f'(x)=\frac{4x^{2x} [1+\ln(x)]}{(x^{2x}+1)^2}.$$ Now, to integrate this, I imagine, ...
6
votes
1answer
57 views

What is the difference between a calculus and an algebra? [duplicate]

You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra. Is there any ...
6
votes
2answers
105 views

Advice for self-studying Inequalities and Calculus

I'm interested in self-studying the following books over the next year or so: Spivak's Calculus (I'm already in Ch. 5 and it is very slow going) The Cauchy-Schwarz Master Class by J. Michael Steele ...
9
votes
2answers
451 views

Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
3
votes
0answers
71 views

Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
3
votes
1answer
57 views

Is there a reason that sine substitution is preferred to cosine substitution?

When integrating by trigonometric substitution that has $\sqrt{a^2-x^2}$ in the integrand, the general recommended approach is to make the substitution $x = a \sin\theta$, and this substitution is the ...
3
votes
2answers
123 views

I want to learn math from ground-up, basic to advanced, beginner to expert

I want to learn math. I've learned math long time ago, but i hardly remember anything. I really want to relearn but have no idea where to begin. I want to learn math by reading through good books, ...
2
votes
1answer
101 views

Structured Self-Learning Program for Calculus I & II

I'm interested in a organised program which comprehensively covers the topics of Calculus I and Calculus II. I've recently finished taking my secondary school's university-level Calculus I course, ...
0
votes
1answer
32 views

How should this definition of a family with corresponding index set be interpreted?

Does the following definition of a family imply that there exist a surjection from $I$ onto the family ? Or should it be interpreted as a injective partial function from $I$ to the family ? ...
9
votes
7answers
431 views

Beautiful Indefinite Integrals. [closed]

These are some of the integrals with beautiful solutions I came across- $$\int \frac{x^2}{(x\sin x+\cos x)^2} dx$$ $$\int\frac {1}{\sin^3x+\cos^3x} dx$$ $$\int \frac{1}{x^4+1}dx$$ I'd love if you ...
3
votes
0answers
61 views

How to practice applied mathematics calculation skill

As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students ...
3
votes
1answer
94 views

Does Discrete Math require Calculus?

So I'm transferring to UC Berkeley and they require Discrete Math or 1 other math class for me to declare my Psychology major. So it is highly recommended to take Calculus for maturity before ...
1
vote
2answers
99 views

How can we relate calculus, trigonometry etc in real life

I have always wondered what does trigonometry, calculus, logarithms solve real world problems? Where do they apply in real life? Is there any simple book where I can understand it?
2
votes
3answers
135 views

Most important things to be proficient in before Calculus 1?

What are the main things one should be proficient in before taking Calculus 1? Please be specific.
13
votes
1answer
284 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
3
votes
1answer
163 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
1
vote
2answers
43 views

Cycloidal coincidence?

For the cycloid $$x=a(t-\sin t)\ ,\quad y=a(1-\cos t)$$ we have, as is easily seen, $$\frac{dx}{dt}=y\ .$$ Does this have any geometrical or physical significance? Or is it just a meaningless ...
2
votes
0answers
149 views

Calculus book advice

I'm reading Thomas Calculus now but I don't think it includes Mellin transform or Riemann-Stieltjes integration... Can you recommend an advanced calculus book which includes all of this stuff?
4
votes
6answers
413 views

Jumping back into Calculus III

At the age of 30 I am going back to school for Electrical Engineering. Because of the way higher education works, all of my previous college coursework is being transferred, which does not allow you ...
3
votes
2answers
63 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
3
votes
1answer
89 views

Best proof of some theorems in calculus

I would like to choose (among the miriads of proofs) a well-structured, elegant, neat, clear proof of the first fundamental theorem of calculus; the second fundamental theorem of calculus; the mean ...
0
votes
1answer
61 views

Where is the Pinching/Squeeze theorem in Spivak Calculus?

So I got Spivak Calculus 3. Edition. I'm starting it now but I want to know if there is the squeeze theorem clearly explained in the book, as I can't find it in the Appendix(pinching theorem, ...
0
votes
1answer
38 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
1
vote
0answers
54 views

Why is there no universally accepted mathematical definition of turbulence?

Why is there no universally accepted mathematical equation for the definition of turbulent flow?
1
vote
1answer
73 views

Calculus book with good explanation on squeeze theorem?

I'm trying to learn squeeze theorem and it's quite difficult to get the concepts. Could anyone suggest some good books which explain this in depth, along with limits, continuity, etc? So far, I have ...
1
vote
4answers
185 views

Which book among these would you recommend for first year calculus?

I'm struggling a bit with functions(limits, squeeze theorem, etc). I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would ...
1
vote
4answers
262 views

Applications of inflection points

Recently, I was teaching maxima, minima and inflection points to first year engineering students. I motived extrema by giving practical examples of optimization problems, but when a colleague asked me ...
16
votes
11answers
902 views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
0
votes
1answer
30 views

Complex polynomial decomposition - Residue Theory

I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2 $ I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ ...
1
vote
1answer
73 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let $2$ points $z_1 = (-1+i\sqrt3)/2$ and $z_2 = (-1-i\sqrt3)/2$ I am trying to show that there is no point $w$ on the line segment from $z_1$ to ...
2
votes
1answer
149 views

A beautiful identity of $\sin(x)$ [duplicate]

When I was in high school, I had proved that $$\sin^2(x)-\sin^2(y)=\sin(x-y)\sin(x+y) $$ I think it is beautiful since it resembles the identity $a^2-b^2=(a+b)(a-b)$. But I can not find it in ...
0
votes
2answers
105 views

Intuitive idea behind the probability density function

as an application of Calculus, I am currently teaching some material about continuous random variables. My main example is the height $X$ of a French male chosen randomly in the French population. ...
4
votes
0answers
74 views

Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
8
votes
5answers
454 views

Why has $\int \sin (\sin x) dx$ not been solved yet?

I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation. What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what ...
0
votes
4answers
212 views

Calculator similar to Desmos but for 3D

Is there a calculator with functionality similar to Desmos but in 3 dimensions? I am looking to learn about families of quadric surfaces so I am looking for a 3D calculator with sliders.
4
votes
0answers
77 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
1
vote
2answers
117 views

Convention verses memory: The quotient rule v product rule for derivatives

I have long wondered why the product rule is taught the way it is. ${ d(UV)=Udv+Vdu}$ Don't get me wrong, I am not a complete NOB when it comes to calc, but the quotient rule states $${d(\frac ...