What is Functions: Definition and 1000 Discussions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.
Homework Statement
1. Find a formula for
(f g)(x) = ?
2. Find a formula for
(f f )(x) = ?
3. Find a formula for the composition below.
g(h(x)) = 4. Find a formula for the composition below.
(h g)(x) =The Attempt at a Solution
1. f(g(x))
2. f(f(x))
3. (g º h)(x)
4. h(g(x))
Why are these...
I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ...
Can someone please suggest
(i) an online...
Homework Statement
[23/4, 2] 4/(x√(x4-4))
Homework Equations
∫ du/(u√(u2 - a2)) = 1/a(sec-1(u/a) + c
The Attempt at a Solution
I first multiplied the whole thing by x/x. This made the problem:
4x/(x2√(x4 - 4))
Then I did a u substitution making u = x2. Therefore, du = 2xdx. I multiplied by...
Suppose you have a λφ4 theory. Books only seem to calculate counter-terms for 2-pt and 4-pt functions.
But what about 3 particles scattering into 3 particles? Do the counter-terms determined by renormalizing the 2-pt and 4-pt functions cancel divergences in 3x3 scattering?
For example, take...
I'm trying to solve this problem from a high school math competition:
Find all functions f : R → R such that, f(f(x+y)-f(x-y))=xy, for all real x,y.
Any ideas of how to approach it.
I have found that f(0)=0, if x=y f(f(2x))=x^2
As I read in the James Stewart's Calculus 7th edition, he said:
My question is: Is f(x)\rightarrow 0 the same as f(x) = L?
For example,
f(x) = x^2
\displaystyle\lim_{x\rightarrow 5}f(x) = 25
I can say that f(x) = x^2 approaches 25 as x approaches 5.
Therefore, can I say that the...
I have just posted an edit to my (very) recent post:
http://mathhelpboards.com/analysis-50/apostol-continuity-amp-differentiabilty-14190.htmlin the Analysis Forum.
I am having trouble with the following Latex expression:\text{lim}_{x \rightarrow c} f^* (x) = \text{lim}_{x \rightarrow c}...
Homework Statement
Verify by brute force that the three functions cos(θ), sin(θ)eiφ and sin(θ)e−iφ are all eigenfunctions of L2 and Lz.
Homework Equations
I know that Lz = -iћ(∂/∂φ)
I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged...
I'm a bit confused wether or not there is a link between harmonic functions (solutions of the Laplace pde) and harmonic oscillating systems? What is the meaning of "harmonic" in these cases? Thanks!
Hello! (Wave)
The following two functions are given and I want to find their time complexity.
function BinarySearchTreeLookUp(key K,pointer R): Type
if (R==NULL) return nill;
else if (K==R->key) return R->data;
else if (K<R->key)
return(BinarySearchTreeLookUp(K,R->LC))...
Homework Statement
inputs x1(t) = cos(ω1t), x2(t) = cos(ω2t).
Show that output g(t) (sum of x1 + x2) = 0.5cos[(ω2-ω1)t] + 0.5cos[(ω2+ω1)t]
Homework Equations
included in upload of attempted solution. Trig identities.
The Attempt at a Solution
Uploaded in pdf. A lot more has been done on the...
Hi! (Wave)
Find the cardinal number of $C(\mathbb{R}, \mathbb{R})$ of the continuous real functions of a real variable and show that $C(\mathbb{R}, \mathbb{R})$ is not equinumerous with the set $\mathbb{R}^{\mathbb{R}}$ of all the real functions of a real variable. That's what I have tried: We...
Homework Statement
Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B]
Homework Equations
1. z=a+bi
2. re^itheta
The Attempt at a Solution
I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...
Homework Statement
In 2000 the population of a country was estimated to be 8.23 million. In 2010 the population was 9.77 million.
Assume that the number of people P(t) in millions at time t (in years since 2000) is modeled by the exponential growth function.
P(t) = Aekt
Find P(t) giving the...
Homework Statement
Hi!
Does anyone know how to solve the inverse of these functions?
y=(4x^2+2x-2)/(8x^2-4x+6)
y=(x+1)/(x^2)
I would appreciate your help with these exercises.
The Attempt at a Solution
For the first one: 8yx^2-4xy+6y=4x^2+2x-2
For the second exercise:
yx^2=x+1
yx^2-x=1
I was wondering if it's possible to use the TI89 Titanium's built-in solver with programs. More specifically, for compressible flow problems, I'd like to calculate mach number based on area ratio, specify whether the flow is subsonic or supersonic, then do something with the corresponding...
Homework Statement
The function f(x) = xe-3x2 is expressed as a linear combination of the basis functions un(x), which are orthogonal and normalised from minus infinity to infinity.
It is expressed by xe-3x2 = ∑anun(x)
the un(x)'s are even functions of x for n = 0,2,4 and are odd functions of...
I have a question concerning how how we define the differentiation and integration operators. Firstly, I know that functions are typically defined as an ordered triple triple ##(X, Y, f)## such that ##f⊆X×Y##, where ##x \in X## and ##f(x) \in Y##. This all seems nice and fine, but we also define...
Homework Statement
If we shift a curve to the left, what happens to its reflection in the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c) where f is a one-to-one function.
Homework EquationsThe Attempt at a Solution
Initially I did this...
So I was watching this video on Khan Academy, and it talks about graphing functions that have values in multiple dimensions. It shows how to represent a linear function in 3 dimensions with a set of vectors, L ={p1 + t(p1-p2)|t∈R} where p1 and p2 are vectors that lie on the line you want to...
$$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$
the ans the TI gave me was $\frac{\sqrt{6}}{4}$
the derivative can by found by the product rule. but really expands the problem
so not sure how the $\frac{d}{dx}$ played in this.
Homework Statement
For f(x) = abs(x^3 - 9x), does f'(0) exist?
The Attempt at a Solution
[/B]
The way I tried to solve this question was to find the right hand and left hand derivative at x = 0.
Right hand derivative
= (lim h--> 0+) f(h) - f(0) / h
= (lim h--> 0+) abs(h^3 - 9h) / h...
I have been looking at my old calculus textbook because to my dismay I seem to have forgotten most of the calculus I learned. I am given 3 cases of ##(f+g)(x) ##.
Case 1 both f and g are even:
I know ##f(x) = f(-x) ## and ##g(x)=g(-x) ## for the domain of the function. I can reason by...
Homework Statement
Let F be the set of one-to-one functions from the set ##{1,2,..,n}## to the set ##{1,2,...,m}## where ##m \geq n \geq 1##. Then how many functions f in F satisfy the property ##f(i)<f(j)## for some ##1 \leq i \leq j \leq n##
Homework EquationsThe Attempt at a Solution...
Hello,
I am just doing my homework and I believe that there is a fault in the problem set.
Consider the set of functions defined by
V= f : R → R such that f(x) = a + bx for some a, b ∈ R
It is given that V is a vector space under the standard operations of pointwise
addition and scalar...
Hi everyone,
I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation.
I have tried searching on the web but have had difficulty in finding something which...
Is it okay to define a local operator as an operator whose matrix elements in position space is a finite sum of delta functions and derivatives of delta functions with constant coefficients?
Suppose your operator is M, and the matrix element between two position states is <x|M|y>=M(x,y).
It...
I have learned that they are called so because they cannot be expressed with the help of elemental methods of mathematics such as addition, subtraction, multiplication and division. But then isn't the whole of mathematics itself based on the elemental methods?
Homework Statement
A player hits a volleyball when it is 4 ft above the ground with an initial vertical velocity of 20 ft/s (equation would be h = -16t2 + 20t + 4). What is the maximum height of the ball?
Homework Equations
quadratic formula
The Attempt at a Solution
t = -20 ±√202 - 4(-16)(4)...
hi! i don't quite know how to start solving for this. i understand the problem and what it's asking for but i have no idea how to start solving for it.
In a volleyball game, a player from one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball...
Homework Statement
This is from Apostol's Calculus Vol. 1. Exercise 1.15, problem 6.(c)
Find all x>0 for which the integral of [t]2 dt from 0 to x = 2(x-1)
Homework Equations
[t] represents the greatest integer function of t.
The Attempt at a Solution
[/B]
Integral of [t]2 dt from 0 to x...
Generating functions defined in terms of algebraic operations on real valued variables are used to enumerate answers to certain combinatorial problems. ( This morning, the exposition http://www.cs.cornell.edu/courses/cs485/2006sp/lecture%20notes/lecture11.pdf is the first of many hits on the...
Homework Statement
I am working through some maths to deepen my understanding of a topic we have learned about. However I am not sure what the author has done and I have copied below the chunk I am stuck on. I would be extremely grateful if someone could just briefly explain what is going on...
Hello PF, I've got a curiosity question someone may be able to indulge me on:
The set of implicit functions covers a certain function-space - the set of all functions that can be represented by an implicit relation. Parametric functions also covers a function-space, that at least overlaps...
Assign the first instance of The in movieTitle to movieResult.
Sample program:
#include <iostream>
#include <cstring>
using namespace std;
int main() {
char movieTitle[100] = "The Lion King";
char* movieResult = 0;
<STUDENT CODE>
cout << "Movie title contains The? ";
if...
Find a formula for g(x)= sin(arccos(4x-1)) without using any trigonometric functions.
I have the answer key right in front of me, but i still get how to start it off or the steps in solving these kind of questions or how to do it at all :/
Thanks!
Hello,
I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf
In page 6, immediately after equation (3.15), its written that "functions of the form v(t) are dense in L^2". I have been looking for proofs online which verifies the above statement but...
Hello everyone,
I know that the integral of an odd function over a symmetric interval is 0, but there's something that's bothering my mind about it.
Consider, for example, the following isosceles trapezoidal wave in the interval [0,L]:
When expressed in Fourier series, the coefficient...
Hi
I am struggling immensely with understand some aspects of chemical thermodynamics:
1) Let's say I have a solid with N atoms and am examining the ionization of individual atoms, and I am supposed to think of the electrons as ideal gasses.
Or,
2) a solid or liquid is in thermal equilibrium...
Homework Statement
Suppose we have the function ##f(z) = x + iy^2## and a contour given by ##z(t) = e^t + it## on ##a \le t \le b##.
Find ##x(t)##, ##y(t)##, and ##f(z(t))##.
Homework EquationsThe Attempt at a Solution
Well, ##x(t)## and ##y(t)## are rather simple to identity. However, I am...
When you have a rational function, such as:
3x-5/x-1
After attaining things like the x and y intercepts and asymptotes, how do you know how many "pieces" of the graph there are? With linear functions/equations, you know it's a single line. Even quadratic graphs are a single piece - albeit...
1.
I have to find the area between
$x = 2y^2$
and
$x = 1 - y$
I find the intersection points
$ 1 -y = 2y^2$
$2y^2 + y - 1= 0 $
$(2y - 1)(y + 1)= 0$
so y = 1 and -1
However, x = y - 1 is not a vertical line so I am not sure how 1 and -1 can be intersections. Also, when I plug these...
I need to find the area bounded by:
$y = \sqrt{x}$, $y = x/2$, and $x = 9$.
I found that the intersecting point is 4 and $y = \sqrt{x}$ is the smaller function between 4 and 9 so:
$$\int_{4}^{9}\frac{x}{2} - \sqrt{x} \,dx$$
and I get
$$ \left[ \frac{x^2}{4} - \frac{2x^{3/2}}{3}\right]_4^9...
Hi,
I have this problem to find the area between 2 curves:
$y = x^2$
and
$y = \frac{2}{x^2 +1}$
I found that the points of intersection are -1 and 1 and it is symmetrical.
I get
$2\int_{0}^{1} \ \frac{1}{x^2 + 1} - x^2 dx$which I am unable to solve. I have tried u-substitution but I end up...
Hi,
I need to find the area between these 2 functions:
$$y = |2x|$$
and
$$y = x^2 - 3$$
So I need to find the points of intersection:
$$|2x| - x^2 + 3 = 0$$
for which I get
x = 3, -1
However, since there are no negative x values in y = |2x| I get
$x = 3, 1$
I find that $y = |2x| $is...
I need to find the are between $$y1 = 12 - x^2$$ and $$ y2 = x^2 - 6$$.
Since y1 is greater, I subtract y2 from y1 getting:
$$ \int 18 - 2x^2$$ which is $$18x - 2x^3 / 3$$,
The intersecting points are $$x = -3 and x= 3$$.
So I find $$18x - 2x^3 / 3 from x = 3 to x = -3$$(I'm trying to...
OK, so I posted this a few days ago:
https://www.physicsforums.com/threads/subtracting-the-overlap-of-functions.784184/#post-4925108
What I've come to discover is that I want to understand how I can subtract f(x) on domain [b,c] from g(x) on domain [a,d].
I want to be able to disregard both...
If $$f(z)=u(x,y)+iv(x,y)$$ is analytic in a domain D, then both u and v satisfy Laplace's equations
$$\nabla^2 u=u_{xx} + u_{yy}=0$$
$$\nabla^2 v=v_{xx} + v_{yy}=0$$
and u and v are called harmonic functions.
My question is whether or not this goes both ways. If you have two functions u...