In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
this had ahttps://mathhelpboards.com/threads/2-2-21-ivp.27772/ but wanted to add tikz graph
orifinally authored by Klazs van Aarsen
\begin{tikzpicture}%[scale=.6]
[declare function = {radius(\phi)=sqrt((3*sin(\phi)+cos(\phi)) / (sin(\phi)^3 -cos(\phi)^3)); },]
% \draw[help lines] (-3,-2) grid...
Hello, I have a few questions and I'd appreciate if you can please help me.
1. If I want to say "for every ## i \in \Bbb N ## and ## 0 \leq j \leq i ## define ## A_{i,j} := i ## and ## B_{i,j} := i \cdot j ## ",
then is the logical formula used for this is as such?:
## \forall i \in \Bbb N...
##A=(-1,1)##
##B=[0,1)##
Define ##f:A\longrightarrow B## by ##f(x)=x^2##
Set ##X=A##.
##f(X)=\{f(x):x\in X\}=\{x^2:x\in(-1,1)\}=B##
##f^{-1}(f(X))=\{x\in A:f(x)\in f(X)\}=\{x\in (-1,1):x^2\in B\}=X##
Now choose a non-zero point ##y\in f(X)##. There are two pre-images of this point: ##x,-x\in...
ok I don't think MHB will process a newcommand but I don't know how to put this in the after \begin{tikzpicture} line
the problem with posting pic here is eventually they get remove and OP is useless...
this tikz code renders in overleaf but I also have many newcommands in preamble
Summary:: According to Yale’s University PHYS: 200:
v*(dv/dt) = d(v^2/2)/dt
Could someone explain how has he reached that conclusion? He claims to be some standard derivation rules, but I can’t find anything about it.
As much as I can tell: (dv/dt)* v = v’ * v = a* v
thanks!
[Moderator's...
Last week I encountered a problem in my graduate project. The project was about designing an autonomous and mobile robot that picks up 9 glass tiles from a stack point and place them into a 3x3 matrix with minimum tolerance.
I am using a DC motor with an infinite turn potentiometer for closed...
What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept...
I'm trying to see if I can calculate the peak draw weight of my bow based on the draw length and the velocity of the arrow and a known shape of a curve, but I'm not quite sure how to make such a function, if there even is such a way.
This is the shape of the draw weight plotted against...
Hi All
I am currently doing Master in data science. I came across the function PDF probability density function which is used to find cumulative probability(range) of a continuous random variable.
The PDF probability density function is plotted against probability density in y-axis and...
My trial :
I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##.
So any help about the right answer is appreciated!
Hello!
Let's say we have a wave function. Maybe it's in a potential well, maybe not, I think it's arbitrary here. This wave function is one-dimensional for now to keep things simple. Then, we use a device, maybe a photon emitter and detector system where the photon crosses paths with the wave...
The ac signal is converted to DC signal which is connected to a capacitor to filter the DC signal. The filter DC signal is step down for 12volt to 5 volt using a voltage regulator. The regulated DC signal is connected to a crystal oscillator that converts the DC signal to a square wave signal...
{\catcode`\^^M=12 \endlinechar=`\^^J \catcode`\^^J=5 This is an M:
This is a middle line.
This is a "J":
}
As I see it, the TeX processor would first need to feed this to the input processor to be transformed into a token list.
The input processor would see the code I posted above with their...
I'm used to calculating Jacobians with several functions, so my only question would be how do I approach solving this one with only one function but three variables?
I think our function becomes (s^2+sin(rt)-3)/since we are looking for J(f/s). So then would our Jacobian simply be J=[∂f/∂s...
Since z=0, the only variable that counts is x.
So the solution would be:
$$\frac {f \left(a + \Delta\ x, b \right) - f(a,b)} {\left( \Delta\ x\right)}$$
Hello, I don't know to solve this exercise:
Let $\mathcal{B}_\mathbb{R}$ the $\sigma-algebra$ Borel in $\mathbb{R}$ and let $\mu : \mathcal{B}_\mathbb{R} \rightarrow{} \mathbb{R}_{+}$ a finite measure. For each $x \in \mathbb{R}$ define
$$f_{\mu} := \mu((- \infty,x]) $$
Prove that:
a)...
I followed a demonstration in one of my electromagnetism books, but it is not clear to me.
My problem is at the starting point.
The book begins by considering the office defined in the following way:
$$Q=\int d^4xJ^\alpha(x)\partial_\alpha\theta(\eta_\beta x^\beta)$$
where...
hi guys
I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by
$$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$
i was thinking about taking substitutions in order to...
I have no problem in following the literature on this, i find it pretty easy. My concern is on the derived function, i think the textbook is wrong, it ought to be,
##S^{'}(t)##=##\frac {4t} {\sqrt{1+4t^2}}=0## is this correct? if so then i guess i have to look for a different textbook to use...
Hi everyone! I have a 8th order transfer function, you can see it in the first image:
% Transfer function
num = [2.091,0,203.3,0,-2151,0,-1.072e05];
den = [1,0,-830.4,0,-1.036e05,0,-5.767e05,0,2.412e07];
tf = tf(num, den)
I need to use a PID, so I'm trying to use a compensator, adding poles...
Let ##\quad z=h(x, y)##
and
##x=f(t) ; y=g(t)##
Let the change in the function z be given by ##\Delta z=h(x+\Delta x, y+\Delta y)-h(x,y)##
We can also write the change as
##\begin{aligned} \Delta z=h &(x+\Delta x, y)-\\ & h(x, y)-h(x+\Delta x, y) \\ &+h(x+\Delta x, y+\Delta y)...
Hi everybody
We can't differentiate ##x^x## neither like a power function nor an exponential function. But ##x^x=e^{x\mbox{ln}x}##. So
##\dfrac{d}{dx}x^x=\dfrac{d}{dx}e^{x\mbox{ln}x}=x^x(\mbox{ln}x+1)##
And here comes the doubt: prove the domain of ##x^x## is ##(0, +\infty)##
Why is only...
$\tiny{6.5.95 Kamehameha HS}$
Express y as a function of x. $\quad C>0$
$3\ln{y}=\dfrac{1}{2}\ln{(2x+1)}-\dfrac{1}{3}\ln{(x+4)}+\ln{C}$
rewirte as
$\ln{y^3}=\ln{(2x+1)^{(1/2)}}-\ln{(x+4)^{(1/3)}}+\ln{C}$
then e thru and isolate y
i think
looks like it will be ugly
##x## is a function ##f(\alpha)## of ##\alpha##:
$$\displaystyle x\, = \,\ln \left( {{\rm e}^{ 0.6931471806\,{\alpha}^{-1}}}-{{\rm e}^{ 0.2876820724\,{\alpha}^{-1}}} \right)$$
and ##y## is a function ##g(\alpha)## of ##\alpha##:
$$\displaystyle y\, = \,\ln \left( {{\rm e}^{...
Hi,
I have the following function, which is computed by: (x+n)/(x+y+n+m),
where x, y are real numbers
n, m are natural numbers
What techniques I can use to smooth the function preventing it to jump up or down at an early stage.
I would appreciate your suggestion.
Thanks
I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time.
The time-domain form is,
s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}}
Here, \mu , \Omega and \lambda are constants.
A quick...
Let f be a 2 variables function.
1) ##f(x,y)=g(x)+h(y)\Rightarrow df=g'(x)dx+h'(y)dy\Rightarrow\int df=g(x)+k(y)+h(y)+l(x)=f(x,y),\textrm{ if } k=l=0##
2) ##f(x,y)=xy\Rightarrow df=ydx+xdy\Rightarrow\int df=2xy+k(y)+l(x)\neq f(x,y)##
Why in the second case the function cannot be recovered ?
I am not that super expert of statistics, so feel free to shift my formulation of the problem into the right one.First, for a physicist, the basic formulation of the problem. Let us say that you have a gravitational field and you have a fully symmetric problem on a flat world without other...
Ron Larson stated:
"The domain of a function can be described explicitly or it can be implied by the
expression used to define the function. The implied domain is the set of all real
numbers for which the expression is defined."
1. How is a function defined explicitly?
2. How is a function...
Here is the fuzzy definition of a function as presented by Ron Larson.
Definition of Function
A function f from a set A to a set B is a relation that assigns to each element x
in the set A exactly one element y in the set B. The set A is the domain (or set
of inputs) of the function f, and...
Hi,
I am looking for changing the logit f(z) = 1/(1+exp(-z)), where z range is [-inf,+inf]. I want to adapt it as follows:
if z > 0.5 then f > 0.5
z < 0.5 then f < 0.5
Thanks
So I've been programming the BDF methods and for some reason I have an issue with the Backward Euler technique.
Given the differential equation y" + y = 0 (with y(0) = 2, y'(0) = 0), my backward Euler solution goes like this:
Obviously this is not possible as the function should be a...
Use the graph to investigate
(a) lim of f(x) as x→2 from the left side.
(b) lim of f(x) as x→2 from the right side.
(c) lim of f(x) as x→2.
Question 20
For part (a), as I travel along on the x-axis coming from the left, the graph reaches a height of 4. The limit is 4. It does not matter...
My apologies. I posted the correct problem with the wrong set of instructions. It it a typo at my end. Here is the correct set of instructions for 28:
Use the graph to investigate limit of f(x) as x→c. If the limit does not exist, explain why.
For (a), the limit is 1.
For (b), the limit DOES...
$\tiny{ACT.trig.01}$
What is the period of the function $f(x)=\csc{4x}$
$a. \pi \quad b, 2\pi \quad c. 4\pi \quad d. \dfrac{\pi}{4} \quad e. \dfrac{\pi}{2}$
well we should know the answer by observation
but I had to graph it
looks like $\dfrac{\pi}{2}$
Hello! So I need to find the potential function of this Vector field
$$
\begin{matrix}
2xy -yz\\
x^2-xz\\
2z-xy
\end{matrix}
$$
Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not...
Let ##f,g:\mathbb{R}^2\longrightarrow\mathbb{R}## be defined, and denote ##D=f(\mathbb{R}^2)##. Assume without loss of generality that ##g(\mathbb{R}^2)\equiv f(\mathbb{R}^2)##.
Define a function ##\varphi_f:D\longrightarrow \mathbb{R}^2## as follows: ##\varphi_f(z)=\{(x,y):f(x,y)=z\}##, and...
Summary:: Stoke stream function
[Mentor Note -- Thread moved from the technical forums, so no Homework Template is shown]
Why the quantity of fluid that crosses the surface of revolution formed by the vector OP is ?
If you are told something holds if the limit exists, and given a function f (specifically not piecewise defined), is it enough to show that the limit as x approaches c = the function evaluated at c?
With a piecewise defined function, it is easy to check both sides of a potential discontinuity...
Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##.
Attempt : I argue for or against each statement by putting it down first in blue and my answer in red.
##g(x)## is always positive : The exact...
Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##.
Proof: First we'll regard the case ## l = \infty ## ( the case...