Tagged Questions
6
votes
2answers
115 views
If $\left| f'(x) \right| \leq A |f(x)|^\beta $ then f is a constant function
Problem Let $f(x)$ be a differentiable function on $[a,b]$ satisfying $f(a)=0$. If there exist $A \ge 0$ and $\beta \ge 1$ such that the inequality
$$\left| f'(x) \right| \leq A \left| f(x) ...
2
votes
0answers
49 views
Symmetry between differentiation and integration [duplicate]
I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!!
as experience told me, ...
4
votes
3answers
80 views
What is the domain of $x^x$ when $ x<0$
I know that $x^x$ for all $x>0$
but what is negative values for that function which give a real number
for example $$f(-1)=(-1)^{-1}=-1\in R$$
I try to put sequence for that but i faild
is ...
1
vote
2answers
242 views
A less known definition of the definite integral of a continuous function
The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110.
(link to full book) (screenshots: page ...
2
votes
1answer
67 views
Prove that $(1+1/x)^x$ is concave for $x>0$
From the graph it looks like $(1+1/x)^x$ is concave for $x>0$. But in this post, I can only prove that it is concave for $x\ge 1$. It is of interest to see a proof for $x>0$.
2
votes
2answers
49 views
alternating series test for $\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}$
I must determine if this series converges (using specifically the alternating series test) $$\sum_{n=4}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}.$$
I know the necessary and sufficient conditions are:
The ...
2
votes
2answers
85 views
Lipschitz continuous
Let $\delta$ be an interval in $\mathbb{R}$. Recall that a function $f$ is called Lipschitz continuous on $\delta$ with Lipschitz constant $L$ if there holds
$|f(x) - f(y)| \leq L|x-y|$ for all $x,y$ ...
4
votes
3answers
106 views
Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$
Is it OK how I want to do?
...
1
vote
0answers
56 views
Continuity of integer part function of $1/x^3$
Check the continuity of $$f(x)=\left[\dfrac{1}{x^3}\right]\mathrm{Sgn}\big(\sin(\pi x/2)\big)$$ where $[\cdots]$ is the integer part of $\dfrac{1}{x^3}$
Is there someone who knows the answer?
1
vote
1answer
43 views
Find local maxima of this quadratic function
How can I find local maxima of this quadratic function?
$$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$
which ...
29
votes
4answers
817 views
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The question is:
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?
Edit
I should ...
5
votes
1answer
91 views
A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$
Is there a special trick to calculate this integral?
$$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$
for $\lambda>0$.
6
votes
1answer
46 views
Exercise on convergent series
I am stumped by the following exercise (3.24 in Biler--Witkowski's book "Problems in mathematical analysis"):
Let $f$ be a continuous, increasing function from $[0,+\infty]$ to itself. Show that ...
4
votes
0answers
95 views
Open Problem in Fixed Point Theory [Prize]
This open problem appeared on the bulletins of Evans Hall at Berkeley this week.
I hope this doesn't violate StackExchange policy (the solution carries a $500 prize), but I thought why not re-post ...
4
votes
0answers
35 views
Differential calculus on Banach space
I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly.
Problem
Given the ...
6
votes
0answers
184 views
A curious theorem by Peano
Let $f$ be defined on $[a,b]$ and there differentiable.
Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
1
vote
3answers
110 views
For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function?
Alternatively, how does the definition of a limit guarantee that if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
1
vote
1answer
18 views
compare of domains $D_a$ and $D_c$
Let $f(x)$ be a smooth function on $[-1,1]$, such that $f(x)>0$ for all $x\in(-1,1)$,$f(-1)=f(1)=0$. consider $\gamma\subset\Bbb{R}^2$ the graph of the $f(x)$. Let $T_a$ the symmetry with respect ...
0
votes
0answers
62 views
Geometrical Inequality
Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals
$AC$ and $BD$ intersects at $E$. If the shortest height of the
triangle $ACD$ equals the radius of the incircle of the triangle ...
3
votes
3answers
104 views
Prove that there is at least one real solution to the equation…
$x^{17}+\frac{243}{1+x^4}=120$
Can anyone show me how to approach this problem..? Any help would be great, thanks.
1
vote
1answer
77 views
Convexity of $x^2f(x)$
Given a function $f$ which is decreasing and convex on $(0,\infty)$, is it possible to find a simple condition on $f$ such that
\begin{equation}
2f(x) + 4xf^\prime(x) + x^2f^{\prime\prime}(x) \geq 0.
...
1
vote
1answer
52 views
Show that this is a diffeomorphism
I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
2
votes
1answer
35 views
proving differentiability for a function in a point
Given a differentiable function $f:D\backslash\{a\}\rightarrow\mathbb R$ and $\lim_{x\rightarrow a}f'(x)=c$ and $f$ is continous in $a$, I want to prove that $f$ is differentiable in $a$ and ...
2
votes
1answer
105 views
The range of the derivative of a differentiable function
I read somewhere that, given a function $f$ differentiable on $[a,b]$, the range of $f'$ can be
(1) a closed interval or
(2) an open interval or
(3) a half-open interval or
(4) an unbounded interval
...
1
vote
2answers
97 views
Integral with hyperbolic functions
I need to compute:
$$ \int_{x^2+y^2=1} \frac{\sinh(x)dy- \sin(y)dx}{\cosh(x)-\cos(y)}$$ where the circle $x^2 + y^2 = 1$ is oriented anticlockwise. So, can somebody show me how? I found the ...
1
vote
1answer
45 views
Prove for continuous f and g, f(x)<g(x) there exists k such that f(x)+k<g(x)
Suppose that $f$ and $g$ are continuous on $[a,b]$ and for each $x$, it holds that $f(x)<g(x)$. Prove that there exists $\alpha>0$ such that for each $x$, it holds that $f(x) + \alpha <g(x)$ ...
2
votes
1answer
37 views
Unique solution differential equation proof
Prove that there is a $\delta>0$ such that there is a unique solution of the differential equation $y'(t)=\sin(y(t))$ with $y(0)=1$ on the interval $[-\delta, \delta]$. How large can you choose ...
3
votes
1answer
48 views
$f(x) = e^{-x^2}$ series representation
Let $f(x) = e^{-x^2}$, defined for $x \in \mathbb{R}$. Find a series representation of a function $F:\mathbb{R} \to \mathbb{R}$ such that $F(0)=0$ and $F'(x)=f(x)$ for each $x$.
I know that the ...
2
votes
2answers
63 views
questions about integration by substitution
I've calculated the following integral by substitution $z=x^2$
$$\int x\cdot\cos(x^2)\,\mathrm d x=\frac12\int\cos(z)\,\mathrm dz=\frac12\sin(z)+c=\frac12\cos(x^2)+c$$
with $c\in\mathbb R$.
My ...
3
votes
3answers
106 views
Is there a way to compute $\lim\limits_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$ without using L'hopital?
I can compute
$$\lim_{x\to\pi/3}\frac{\sqrt{3+2\cos x}-2}{\ln(1+\sin3x)}$$ using L'hopital and the limit equals $\frac{\sqrt{3}}{12}$, but is there another way to compute this limit without using ...
1
vote
1answer
63 views
Question on the perimeter of any quadrilateral
Is it true that the perimeter of any convex quadrilateral inside a unit circle
is no more than $4\sqrt{2}$?
1
vote
1answer
81 views
Use the Mean Value Theorem to prove the Binomial Inequality?
Binomial Inequality: $\forall x \in \mathbb{R}, x\geq-1, \forall n \in \mathbb{N}: (1+x)^{n} \geq 1+nx$
I need to prove this using the Mean Value Theorem: If $f$ is continuous on $[a,b]$ and ...
1
vote
1answer
67 views
Evaluation of a limit with integral
Is this limit
$$\lim_{\varepsilon\to 0}\,\,\varepsilon\int_{\mathbb{R}^3}\frac{e^{-\varepsilon|x|}}{|x|^2(1+|x|^2)^s}$$
with $s>\frac{1}{2}$, zero?.
The limit of a product is the product of limit, ...
1
vote
1answer
43 views
Definite integral with no closed form antiderivative [duplicate]
Now from wikipedia I know that $$\int_{-\infty}^\infty e^{-x^2}dx=\sqrt\pi.$$ Also on wikipedia they have the following claim $$\int_{-\infty}^\infty x^{2n}e^{-x^2}dx=\frac{(2n-1)!!}{2^n}\sqrt\pi.$$ ...
0
votes
0answers
85 views
Holes in $\mathbb{R}^2$ [closed]
I heard one time that the $\mathbb{R}^2$ plane has holes. What does this mean?
0
votes
1answer
73 views
Evaluating the limit: $\lim_{x\to\infty}\sqrt[x]{1+\sin(x)}$
How to evaluate the following limit?
$$\lim_{x\to\infty}\sqrt[x]{1+\sin(x)}$$
1
vote
1answer
84 views
Partials and maximization
If we have that the contours of a response surface are elliptical and the response is given by the following function:
$$\large \exp\left(-\left(w^2 + \frac{1}{4}l^2 -\frac{1}{4} \cdot w \cdot ...
6
votes
1answer
76 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
1
vote
1answer
83 views
Does there exist such a pentagon that can be covered by a circle?
Does there exist a pentagon in which every two nonadjacent vertices
is connected by a diagonal and the minimal height of every triangle formed by the
sides and diagonals of the pentagon whose two ...
1
vote
1answer
98 views
$f$ not differentiable at $(0,0)$ but all directional derivatives exist
Consider the function :
$$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto
\begin{cases}
0 & \text{for } (x,y)=(0,0) \\
\frac{x^3}{x^2+y^2} & \text{for } (x,y) \neq ...
2
votes
2answers
67 views
Quadratic surface maximization and Hessians
If we have that the contours of a response surface are elliptical and the response is given by the following function:
$$\large \exp\left(-\left(w^2 + \frac{1}{4}l^2 -\frac{1}{4} \cdot w \cdot ...
2
votes
1answer
54 views
An example of differentiability in $\mathbb{R}^n$ everywhere but not at origin.
I came across such problem:
$g:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^1$-function with $g(\theta+\pi)=-g(\theta)$ for all $x$. Define a function $f: \mathbb{R}^2\rightarrow \mathbb{R}$ as ...
1
vote
1answer
72 views
Boundedness of a function
I consider this function of $z=x+iy$ with $y>0$:
$$f(z)=\bigg|\frac{1}{\alpha-i{z}}\bigg|$$
with $\alpha> 0$ ($\alpha\in\mathbb{R}$).
Is it bounded?
making calculation we have
...
2
votes
0answers
200 views
These unknown uniformly differentiable functions
Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point).
Given $\epsilon>0$, choose a partition $P \, : \, ...
2
votes
0answers
45 views
Discontinuous for rationals
Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals.
I guess it would be nice ...
1
vote
2answers
52 views
product rule for matrix functions?
Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
3
votes
2answers
48 views
Showing a function is not one-to-one near the origin
Let $$f(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right) \text{ if } x \neq 0 \\ 0 \text{ if } x=0 \end{cases}$$
I'm trying to show this is not one-to-one near $0$. I was given a hint to consider ...
1
vote
1answer
52 views
infinite sum limit how to find the following
Hi what is the limit of the following sum:
$$\lim \limits_{n\rightarrow\infty}\frac{2}{n^2}\sum\limits_{j=0}^{n-1}\sum\limits_{k=j+1}^{n-1}\frac{k}{n}$$
Thanks a lot!
4
votes
1answer
182 views
Can one define the derivative of a function using tangent cones? Does such a notion already exist?
I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
5
votes
4answers
263 views
What is a simple example of a limit in the real world?
This morning, I read Wikipedia's informal definition of a limit:
Informally, a function f assigns an output f(x) to every input x. The
function has a limit L at an input p if f(x) is "close" to ...
