For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.
175
votes
8answers
15k views
Is $dy/dx$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula ...
63
votes
6answers
4k views
How can you prove that a function has no closed form integral?
I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/subtraction
...
62
votes
6answers
2k views
Find a real function, $f$, s.t. $f(f(x)) = -x$.
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:
Find a function $f: ...
56
votes
8answers
2k views
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
To prove the convergence of
$$\sum_{n=1}^{\infty} \frac1{n^p}$$
for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.
I am wondering if there is a ...
52
votes
3answers
2k views
Is there an integral that proves $\pi > 333/106$?
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves ...
46
votes
17answers
4k views
Striking applications of integration by parts
What are your favorite applications of integration by parts?
(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)
Thanks for your ...
44
votes
3answers
11k views
What is the practical difference between a differential and a derivative?
I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative:
So, a derivative is the rate of change of a function ...
43
votes
7answers
1k views
What's so “natural” about the base of natural logarithms?
There are so many available bases. Why is the strange number $e$ preferred over all else?
Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
42
votes
7answers
3k views
Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$
I was wondering the following. And I probably know the answer already: NO.
Is there another number with similar properties as $e$. So that the derivative of $\exp(x)$ is the same as the function ...
41
votes
4answers
2k views
Is there a function with a removable discontinuity at every point?
If memory serves, ten years ago to the week (or so), I taught first semester freshman calculus for the first time. As many calculus instructors do, I decided I should ask some extra credit questions ...
40
votes
6answers
3k views
Ways to evaluate $\int \sec \theta \, d \theta$
The standard approach for showing $\int \sec \theta \, d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a ...
39
votes
4answers
1k views
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
I have a question:
Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that
$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$
How can I ...
39
votes
1answer
1k views
How to determine with certainty that a function has no elementary antiderivative?
Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental ...
38
votes
3answers
3k views
What is the importance of Calculus in today's Mathematics?
For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
37
votes
3answers
2k views
Why is $1^{\infty}$ considered to be an indeterminate form
From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
37
votes
2answers
884 views
Is there a bounded function $f$ with $f'$ unbounded and $f''$ bounded?
Is there a $C^{2}$-function $f:\mathbb{R}\to\mathbb{R}$ that is bounded and such that $f'(x)$ is unbounded, but $f''(x)$ is bounded again?
For example, $f(x)=\sin(x^2)$ is bounded and has ...
36
votes
1answer
1k views
The $100$th derivative of $(x^2 + 1)/(x^3 - x)$
I am reading a collection of problems by the Russian mathematician Vladimir Arnol'd, titled A Mathematical Trivium. I am taking a stab at this one:
Calculate the $100$th derivative of the function ...
36
votes
5answers
595 views
What is so special about $\alpha=-1$ in the integral of $x^\alpha$?
Of course, it is easy to see, that the integral (or the
antiderivative) of $f(x) = 1/x$ is $\log(|x|)$ and of course for
$\alpha\neq 1$ the antiderivative of $f(x) = x^\alpha$ is
...
36
votes
3answers
604 views
Calculus conjecture
When I was a senior in high school in 2001, as I took calculus, I made the following conjecture that proves resistive to attack. It goes like this:
For every positive integer $n$, there are exactly ...
35
votes
2answers
533 views
Symmetry of function defined by integral
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$
You can use, for ...
34
votes
13answers
4k views
Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$
How to prove
$$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
34
votes
6answers
1k views
Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge?
Can someone give a simple explanation for why the harmonic series
$$\frac 1 1 + \frac 12 + \frac 13 + \cdots $$
doesn't converge, but just grows very slowly?
I'd prefer an easily ...
34
votes
5answers
1k views
Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$
$$ \sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$$
Does it converge or diverge? Can we have a rigorous proof that is not probabilistic? For reference, this question is supposedly a mix of real ...
33
votes
4answers
1k views
A circle rolls along a parabola
I'm thinking about a circle rolling along a parabola. Would this be a parametric representation?
$(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$
A gives us the radius of the circle, B changes the frequency ...
32
votes
11answers
3k views
How to convince a layman that the $\pi = 4$ proof is wrong?
The infamous "$\pi = 4$" proof was already discussed here:
Is value of $\pi$ = 4 ?
And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this ...
32
votes
4answers
3k views
Lesser-known integration tricks
I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
32
votes
6answers
1k views
A question on Taylor Series and polynomial
Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$.
For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial.
Is true that $f(x)$ is a ...
31
votes
8answers
5k views
What is the meaning of the third derivative of a function at a point
What is the geometric, physical or other meaning of the third derivative of a function at a point?
(Originally asked on MO by AJAY)
If you have interesting things to say about the meaning of the ...
30
votes
8answers
2k views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim\limits_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my ...
30
votes
3answers
1k views
$e$ to 50 billion decimal places
Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. ...
30
votes
4answers
815 views
Rain droplets falling on a table
Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
30
votes
2answers
713 views
Is there any connection between Green's Theorem and the Cauchy-Riemann equations?
Green's Theorem has the form:
$$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$
The Cauchy-Riemann equations have the ...
30
votes
2answers
600 views
Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$
Prove that
$$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
30
votes
1answer
463 views
Prove that $\sum_{k=1}^{\infty} \large\frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$
Prove that
$$\sum_{k=1}^{\infty} \frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$$
29
votes
5answers
1k views
Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I've got troubles in computing the below integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$
I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
29
votes
2answers
2k views
$ \lim_{n \to \infty }\sin \sin …\sin n$
I need your help with evaluating this limit:
$ \lim_{n \to \infty }\sin \sin ...\sin n$,
we apply the $\sin$ function $n$ times.
Thank you.
29
votes
2answers
765 views
Prove $\int_0^1 \left| \frac{f^{''}(x)}{f(x)} \right| dx \geq4$
$f''(x)$ is continuous in $[0,1]$, $f(0)=f(1)=0$, $f(x)\neq 0$ when $x \in(0,1)$, try to prove:
$$\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4.$$
28
votes
5answers
852 views
$\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate:
$$\lim_{n\rightarrow\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$
By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. ...
28
votes
4answers
2k views
Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
I'm trying to find
$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$
I tried couple of methods: Stolz, Squeeze, D'Alambert
Thanks!
Edit: I can't use Stirling.
28
votes
2answers
897 views
Evaluating $\int P(\sin x, \cos x) \text{d}x$
Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.
For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.
Is there a general method which allows us to evaluate the ...
28
votes
4answers
2k views
$n$th derivative of $e^{1/x}$
I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula
$$\frac{\mathrm d^n}{\mathrm ...
28
votes
4answers
696 views
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The question is:
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?
Edit
I should ...
27
votes
2answers
632 views
Making trigonometric substitutions rigorous
I've been tutoring some basic calculus, and it made me think about something pretty basic.
Let me explain the problem by example:
Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}dx$. It ...
26
votes
9answers
1k views
What is $dx$ in integration?
When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.
$$\displaystyle \int f(x)\, dx$$
When he came to explain the meaning of the ...
26
votes
4answers
1k views
How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$
$$\int{\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}}\mathrm dx$$
This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone ...
26
votes
5answers
862 views
Are there ways of finding the $n$-th derivative of a function without computing the $(n-1)$-th derivative?
Say we have a function $f(x)$ that is infinitely differentiable at some point.
Is it possible to find $f^{(n)}(x)$ without having to find first $f^{(n-1)}(x)$? If so, does it take less effort than ...
26
votes
13answers
1k views
List of interesting integrals for early calculus students
I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
25
votes
4answers
622 views
Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}dx$?
There is a well-known trick for integrating $\int_{-\infty}^\infty e^{-x^2}dx$, which is to write it as $\sqrt{\int_{-\infty}^\infty e^{-x^2}dx\int_{-\infty}^\infty e^{-y^2}dy}$, which can then be ...
25
votes
4answers
702 views
A limit wrong using Wolfram Alpha
I want to calculate the following limit:
$$\displaystyle{\lim_{x \to 0} \cfrac{\displaystyle{\int_1^{x^2+1} \cfrac{e^{-t}}{t} \; dt}}{3x^2}}$$
For that, I use L'Hopital and the Fundamental Theorem ...
24
votes
4answers
2k views
How can a structure have infinite length and infinite surface area, but have finite volume?
Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.
One Dimension - Clearly this structure is infinitely long.
Two Dimensions - Surface Area = ...
