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
Homework Equations
[/B]
----------------------
function [C]=mymatmult(A,B)
[L1 C1]=size(A);
[L2 C2]=size(B);
if C1 ~= L2
error('dimension mismatch');
end %if ERROR
C=zeros(L1,C2);
for i=1:L1
for j=1:C2
C(i,j)=A(i,:)*B(:,j);
end %in for
end %out for
end %function...
I've run across a system of recursive functions (call them f and g). The system looks like this:
f(x) = a f(x-1) + b g(x-1)
g(x) = a g(x-1) + c f(x-1)
I also know that f(0) = 0 and g(0)>0. Finally, I know for other reasons that are too complicated to go into here that the system is somehow...
New to composite functions here. Lesson has been vague and unhelpful.. again. Here is what I've worked on so far but not sure on the last equation in particular, or that I have done my multiplication properly when working with a squared set of brackets, multiplied by an number.. (b and c)
Any...
Homework Statement
A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the...
what is the relationship between special functions and integration ?
why integral of some function like (sqrt(ln(x)) and (cos(1/x) and more) are entering us to special functions??
PLEASE HELP ME TO UNDERSTAND.
Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions.
here is my question:
Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists.
The part that has me confused is the y to the power of 3, where...
Hi. Here is one example from my book.
Calculate Fourier transform of signal:
Here is solution:
We can write x(n) as:
,
where x1(n) is u(n+N)-u(n-N-1). We can write:
(we used that cos(n)=(1/2)*(exp(j*n)+exp(-j*n)).
Using properties of Fourier transform of discrete signal:
,
Fourier...
Hi PF!
Can any of you help me determine a good measure for how "different" two functions are from each other?
I've thought of using something like ##\int_\Omega (f-g)^2 \, dx##. Can anyone recommend a good technique and direct me to the theory so I can understand it well?
Thanks so much!
Josh
Homework Statement
Determine whether or not the following functions are harmonic:
u = z + \bar{z}
u = 2z\bar{z}
Homework Equations
z = u(x,y) + v(x,y)i
\bar{z} = u(x,y) - v(x,y)i
A function is harmonic if Δu = 0.
The Attempt at a Solution
Δu = Δz +Δ \bar{z} = u_{xx} + v_{xx} +...
Hello everybody,
For my thermodynamics test I have to tell whether or not a quantity is a state function, which is obviously not all too difficult when regarding entropy, enthalpy etc. on their own. However there are a lot of questions where it is about "H-S" or "G-H". Are these not always...
Write the piecewise function
\[ f(t) = \begin{cases}
2t, & 0\leq t < 3 \\
6, & 3 \le t < 5 \\
2t, & t \ge 5 \\
\end{cases}
\]
in terms of unit step functions.
So here is what i;ve got just guessing , I don't think I'm correct. I really need some help. But I got...
Homework Statement
Homework EquationsThe Attempt at a Solution
For x>b, Ψ(x) = Ae-ikx + Beikx , where k = (√2mE)/hbar
a<x<b Ψ(x) = Ce-ik'x + Deik'x , where k = (√2m(U2 - E)/hbar
This is the problem part
0<x<a Ψ(x) = Fsink''x...
People say that if you could break a function down into these three functions (constant, successor, projection or sometimes called initial/basic functions) using some operators, then it is primitive recursive.
What makes these three functions so special?
y=CektA) First find k. [Hint:Use the given information of y=100 when t=2, and y=300 when t=4 to compute k.]
B) Finally, find the value for C. [Hint use ine of the two pieces of information given in the problem to solve for C. in other words, use either y=100 when t=2 or use y=300 when 4=4 to...
Homework Statement
∫δ(x3 - 4x2- 7x +10)dx. Between ±∞.
Homework EquationsThe Attempt at a Solution
Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would...
Homework Statement
If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x).
Homework Equations
(None I believe.)
The Attempt at a Solution
I know that for a function to be invertible, it must be both one-one and onto...
Homework Statement
1. Range of the function ## \sqrt {x^2+x+1} ## is equal to?
2.ƒ:R---->R is defined as ƒ(x) = x2 -3x +4, then f -1 (2) is equal to?Homework Equations
NA
The Attempt at a Solution
For the first one tried squaring on both the sides but that does not give linear x in terms of...
Homework Statement
Recall that we have defined the Gaussian ##f_s## by ##f_s (t) = \sqrt{s}e^{-st^2}## and shown that ##\hat{f_s}(\lambda) = \frac{1}{\sqrt{2}}e^{\frac{-\lambda^2}{4s}}##.
Show that ##f_3 \ast f_6 (t) = \sqrt{\pi}f_{1/2}(t) = \sqrt{\pi/2}e^{-t^{2}/2}##
The Attempt at a...
1. Homework Statement
A graph of y=f(x) is shown. Find the following function values and justify your answers.
f(30)=
f(-14)=
Homework EquationsThe Attempt at a Solution
I know the graph is periodic, I know it's max and min, and I know it's amplitude because of that. But I don't know what...
Hi All,
Having a tough time with this one and I'm not sure why.
Need to state amplitude, period and phase shift of f(x)=3cos2[x-(π/4)]+1.
Amplitude being 3, period being 2π/2=π and phase shifted (π/4) to the right.
Midline would also be at y=1
Good so far?
Right, so I know that 1/4 phase...
Homework Statement
1 kg air at the pressure ##10^6##Pa and the temperature ##125^\circ C = 398K## expand until the volume is 5 times larger. The expansion is done with change in heat at every moment being ##1/4## of the work done by the gas. Calculate the end pressure.Homework Equations
##dU...
Hello,
I wonder if anyone could settle a disagreement I'm having with one of my peers. The question is 'How many surjective functions are there from a set of size n+3 to a set of size n?'. Now, I've already proven that there are (n+1 choose 2)n! surjective functions from a set of size n+1 to a...
Homework Statement
Specify the domain and range of f(x, y) = arccos(y − x2). Indicate whether the domain is (i)
open or closed, and (ii) bounded or unbounded. Give a clear reason in each case.Homework EquationsThe Attempt at a Solution
y-x2 >= -1
y >= x2 -1
y-x2 <= 1
y <= x2 +1
I sketched it...
I'm given the following Piecewise function when $f:[0,1]\to[0,1]$:
$f(x) = x$ when $x\in\Bbb{Q}$
$f(x) = 1-x$ when $x\notin\Bbb{Q}$
I need to prove that $f$ is continuous only at the point $x=\frac{1}{2}$.
For this problem, I know I need to use the fact that a function $f$ is continuous at a...
Homework Statement
So the test is to take the determinant (D) of the Hessian matrix of your multivar function.
Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point.
For D<0 it's a saddle point, and D=0 gives no information.
My question is, what happens if fxx=0? Is that...
Homework Statement
Prove that sinx+cosx is not one-one in [0,π/2]
Homework Equations
None
The Attempt at a Solution
Let f(α)=f(β)
Then sinα+cosα=sinβ+cosβ
=> √2sin(α+π/4)=√2sin(β+π/4)
=> α=β
so it has to be one-one
[/B]
Hello,
I've been reading about injectivity from Z to N and surjectivity from N to Z and was wondering whether there was some kind of algorithm that could generate these specific types of functions?
Homework Statement
Homework EquationsThe Attempt at a Solution
I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?
On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series.
I get the part about a Taylor series, that's...
I would like to know some general properties of the modulo (remainder) function that I can use to rewrite expressions. For example, say we wanted to prove the following by rewriting the right-hand-side:
$$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} = \frac{n - n \pmod d}{d} $$
I have no idea how...
Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows:
Question:
Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula:
V = 999.87 −...
I have seen written out in various places (including this forum) the effective potential function that comes from the solutions to the Schwarszschild Geodesic. But I haven't been able to find the effective potential functions for other solutions to Einstein's field equations. Are there...
I was thinking about the connection between fields and particles. For instance the scalar field Φ(x) and the field Φ(x)+a both represent the same scalar particle. Because the action ∫∂Φ∂Φdx^4 is unaltered and the propagator <0|[Φ(x)+a,Φ(y)+a]|0> is presumably the same. What about if we replace...
Homework Statement
For positive integers m, k, and n , let mkn be defined as mkn = kmn , where k\frac {m}{n} is a mixed fraction. What is the value of 643 + 364 ?
Homework Equations
I attempt the other few similar questions where the solution are as follow
832 + 382 = \frac {169}{24}
641 +...
Hello! (Smile)I want to determine if $\sqrt{n}$ is $\Theta $ / $O$ / $\Omega$, $o$, $\omega$ of $n^{\sin n}$.
To do so we could calculate the limit:
$$\lim_{n \to +\infty} \frac{\sqrt{n}}{n^{\sin n}}$$
right? But how can we find the limit, although $\lim_{n \to +\infty} \sin n$ does not...
Suppose two people, X and Y, have two different stopwatches. X starts his/her stopwatch as some particle passes an origin. We can model the velocity of the particle by ##\vec{v}(T)##, where ##T## is the reading on the first stopwatch. After an amount of time ##\Delta t##, Y starts his/her...
Homework Statement
Show that the equations
xy^2+zu+v^2=3
x^3z+2y-uv=2
xu+yu-xyz=1
can be solved for (x,y,z) as functions of (u,v) near the point (x,y,z,u,v)=(1,1,1,1,1) and find dy/du at (u,v)=(1,1)
Homework Equations
Multivariable calculus differentiation
3. The Attempt at a Solution
I...
What are the rules if you have a sequence f_n of real-valued functions on \mathbb R and consider the sequence f_n(x_n), where x_n is some sequence of real numbers that converges: x_n \to x. All I have found is an exercise in Baby Rudin that says that if f_n \to f uniformly on E, then f_n(x_n)...
Homework Statement
f(x) = sin5x ; [π/5,2π/5] finding the point c which f'(x) =0. I understand the theorem and how to complete it, my issue is using the triq functions
Homework Equations
f'(x) = 5cos5x
The Attempt at a Solution
5cos5x=0
cos5x=0
5x=π/3
x=π/15
my answer is not correct, I am...
Here are conditions I use to define my problem:
1) I use cumulative distribution of 2 logistic functions g1(x) and g2(x) with g2 is translated to the right of the g1(x) on x-axis.
2) I make a transformation to eliminate both tails of the function which will not have a significant contribution to...
Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...
For example, say we have ##\frac{x^4(x - 1)}{x^2}##. The function is undefined at 0, but if we cancel the x's, we get a new function that is defined at 0. So, in this case, we have ##x^2(x - 1)##, then ##x^2(x - 1)(1)##, and since ##\frac{x^2}{x^2} = 1##, we then have ##\frac{x^4(x - 1)}{x^2}##...
Hello,
I am trying to solve this. This material is not covered in my class, but I still want to know how to do it.
If cos(t)=$\frac{-9}{10}$ where $\pi$ <t<$\frac{3\pi}{2}$ find the values of
cos(2t)=
sin(2t)=
cos($\frac{t}{2}$)=
sin($\frac{t}{2}$)=
Give exact answers, do not use decimal...
Homework Statement
[/B]
By using the Ascoli-Arzela theorem, show that the functions fn(z) = zn in Δ(1)n = 1, 2,..., are not equicontinuous.
Homework Equations
[/B]
A family F of complex-valued functions on A is called equicontinuous if ∀ε > 0, ∃δ > 0 such that |f(z) - f(w)| < ε, ∀z, w ∈ A...
Homework Statement
Let's study harmonic sound waves with frequency ##\omega ##, that is emitted by a long wire. Let's approximate the earth, above which the wire is, with an infinite rigid plate. If the space wasn't limited by the earth, than the velocity potential of the source would be ##\Phi...
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...