1
vote
2answers
26 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
61 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?
3
votes
5answers
343 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
59 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
78 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
48 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
34 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
46 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
43 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
119 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
181 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
772 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
29 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 $1/(z-ξ)$ ...
0
votes
0answers
25 views

If $Z$ is an admissible function, does $f(z) = f(|z|)$?

If $Z$ is an admissible function, does $f(z) = f(|z|)$? For example, if $f(z) = x^3 + 1$ and I am given $2$ points $z_1 = -1+i\sqrt3$ and $z_2 = -1-i\sqrt3$, can I just find the moduli and use that ...
1
vote
1answer
66 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
125 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
66 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. ...
3
votes
0answers
58 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 ...
6
votes
5answers
375 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
2answers
41 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
64 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
86 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 ...
6
votes
1answer
359 views

Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
3
votes
1answer
78 views

Important results of calculus before Newton and Leibniz?

We have all come to know that calculus was invented by Newton and Leibniz, right? But many calculus results were already proven by the time. I have read that Fermat already found how to calculate ...
8
votes
3answers
128 views

Examples of non-Riemann integrable functions that appear “in nature”?

I am teaching an honours calculus class, and am looking for examples on non-integrable functions that occur somewhere real in mathematics. (The standard example of 1 on $\mathbb{Q}$ and 0 elsewhere ...
2
votes
1answer
231 views

How to stop making stupid mistakes

I know that this question has been asked before by others however I just can't get around this problem. I am in my final year of high school and I need to find a solution before my final exams. For ...
5
votes
2answers
90 views

Intuition for differentiating beneath the integral

I apologize in advance for a vague question. There is a theorem: If both $f(x,s)$ and $\partial _sf(x,s)$ are continuous in $x$ and $s$, then $$\partial_s\int_a^bf(x,s)\,dx=\int_a^b ...
17
votes
5answers
473 views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
1
vote
0answers
194 views

Difficulty in learning calculus 3 years after one has learned calculus 2?

Academically speaking, I'm a late bloomer. Long story short, I was a bad kid, teachers told me I wasn't good at math and I believed them. Years later, as an almost 30 year old man, I started going ...
4
votes
4answers
170 views

What does “area” really mean?

My professor had an interesting statement at the beginning of first year integral calculus. What does area really mean? How do we know that the area of a circle is $\pi r^2$? Archimedes used ...
3
votes
2answers
143 views

Knowledge of proofs and proof-writing before studying spivak's or apostol's Calculus

I want to start reading spivak or apostol calculus. But I believe a certain level of proof knowledge is essential to solve the problems. Do these books teach you how to prove results or do we learn ...
2
votes
1answer
88 views

Derivative existence theorem

Has anyone here heard of the Derivative existence theorem? Derivative existence theorem: For $f$ defined on some interval including $a$, $f$ is differentiable at $a$ if and only if there ...
2
votes
3answers
143 views

Decision Calculus text [duplicate]

I was wondering out of these three which would you take calculus by Spivak calculus by Hughes-Hallet calculus by Morris Kline I'm taking calculus III next semester but I want a better book that's ...
20
votes
3answers
721 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
10
votes
5answers
1k views

Calculus self taught? Books?

I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take ...
3
votes
3answers
179 views

Examples where derivatives are used (outside of math classes)

I want to know what is the use of derivatives in our daily life. I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in ...
2
votes
0answers
57 views

(Actual) applications of basic differential and integral methods

If this isn't the place, I apologize: At the end of my calculus class, we asked the students (among other things) what some applications of calculus methods are. Disappointingly, many focused on the ...
5
votes
5answers
202 views

Why isn't there a fixed procedure to find the integral of a function? [duplicate]

Since the integration of a function is the opposite of a the derivative of a function, and there are clear steps to follow when we want to find the derivative of a function, I thought there would be ...
2
votes
1answer
122 views

Understand a weird method of calculus

I see this method of calculus on youtube and my question: is this method valid? How we can understand it? Thanks.
6
votes
6answers
200 views

What is the purpose of the limit?

I haven't taken Calculas yet, but I see the use of limit (approaching zero or infinity) in other classes such as physics. I just wanted some intuitive explanation of the limit. I was thinking that ...
1
vote
1answer
44 views

Why can the limit of a sequence approach a number and converge, but the limit of the series must approach $0$ to converge?

My question may not make much sense because I'm still trying to wrap my mind around infinite sequences and series. I seem to have good working knowledge of when and why to apply a certain tests for a ...
1
vote
2answers
51 views

Improper integral: $ \int_{-1}^{\infty} \frac{dx}{x^2 + \sqrt[3]{x^4 + 1}}$

Decide if the integral $$ \int_{-1}^{\infty} \frac{dx}{x^2 + \sqrt[3]{x^4 + 1}}$$ converges. I decided to write $ \int_{-1}^{\infty} \frac{dx}{x^2 + \sqrt[3]{x^4 + 1}}$ = $ \int_{-1}^{1} ...
1
vote
3answers
215 views

Trouble understanding math proofs

*edit Even though there are already answers to my question, I appreciate anyone that offers their advice! I am not sure if this is the right place to ask this but I usually ask for help here. I am a ...
1
vote
0answers
47 views

A Question Related to the Divergence Test for Series

Suppose $\lbrace x_i\rbrace$ is a sequence of real numbers. If $\lim x_i=0$ then for every $\varepsilon>0$ there exists an $N$ such that $$n\ge N\Rightarrow \vert ...
0
votes
2answers
93 views

How was Integral Calculus discovered/derived? [closed]

Can you please explain in layman terms. I don't know if this is a duplicate, but if I find one I will delete this question. The thing I don't get the most is differentials.
1
vote
0answers
92 views

Question about limits

I am quite new on SE. I see a lot of question about integrals, series, limits. I am wondering if there is a limit to teachers (or textbooks) imagination in these areas.
0
votes
1answer
98 views

Strategy to recognize and solve sequence and series problems?

I've been reading my Stewart Calculus book and I honestly find most of the coverage of sequences and series easy to grasp (excluding power series, Taylor and Maclauren since we haven't covered those ...
2
votes
1answer
67 views

Mathematics sign for ceil and floor functions

What is mathematics sign for ceil and floor ?
2
votes
3answers
426 views

How to master integration and derivation?

We have learnt in school about derivation and integration, however I find my knowledge fairly poor. I mean I have problems with taking the derivative/integrating even simple functions. So I would like ...
9
votes
7answers
192 views

Intuition for a real line vs. a “hyperreal line”

I am a student of pure mathematics but I have no formal background in nonstandard analysis. I came across the concept of a hyperreal field recently, read just a little about them, and followed the ...