Inverse Definition and 1000 Threads

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...

Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$$(A)\, \dfrac{2}{27} \quad (B)\, \dfrac{1}{54} \quad (C)\, \dfrac{1}{27} \quad (D)\, \dfrac{1}{6} \quad (E)\, 6$ok not sure what the best steps on this would be but assume we first find...

Can anyone explain the logic behind the answer? Taken from HiSet free practice test

Hi, Although I'm using trigonometric form of Fourier transform, first I'd discuss both, exponential and trigonometric forms, for the sake of context. Now proceeding toward the main question and we would only be using trigonometric form. % file name...

The test data and notes are attached - showing that the inverse square calculations fail to reasonably predict the actual dimming of light over a test distance of 168 mm. Did I err in my test design or my calculations?

First, the inverse quantum Fourier transform is ##\left | k \right > =\frac 1 {\sqrt {2^n}} \sum_{j=0}^{2^n-1} e^{-2 \pi ijk / 2^n} \left | j \right >##, and it is equal to ##\left | k_1 , k_2 , \dots , k_n \right > \rightarrow \frac { \left ( \left | 0 \right > + e^{-2 \pi i 0.k_n} \left...

The mass of an object moving at speed v increases such that $$\frac{m'}{m}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$ and its apparent frequency decreases such that $$\frac{\nu'}{\nu}=\sqrt{1-\frac{v^{2}}{c^{2}}}$$ so $$\frac{\nu'}{\nu}=\frac{m}{m'}$$ but equating the energies $$ h\nu= mc^{2}$$...

##132,289≡1973* 67 + 98## ##1973≡98*20+13## ##98≡13*7+7## ##13≡7*1+6## ##7≡6*1+1## now in reverse my attempt is as follows, ##1≡7-6## ## 1≡7-(13-7)## ##1≡2*7-(1973-20*132,289+1340*1973)## ##1≡2*7-(1341*1973-20*132,289## which is correct but my interest is in finding the inverse of 1973 help?

In "An Introduction to Modern Cosmology" by Andrew Liddle, page 130, paragraph A2.3 Luminosity distance, explains why the inverse square law does not hold at very far distances. One reason given is the expanding universe. (Another was the geometry of the Universe.) Could there be also...

It is obvious that there is a one-to-one relationship between real numbers (defined to include infinity) and their multiplicative inverses (assuming we map the inverse of zero to infinity and vice versa). Thus, one should be able to replace the distance between two points in space with it's...

Hello, I am using a code on EUCLID future mission. The original author of this code has set a value for the density of galaxy equal to : ng = 354543085.80106884 I think this is expressed in inverse steradian. I think that EUCLID mission has a 30 arcmin^-2 value for density of galaxies...

Hi, I think I am stuck in my understanding of "inverse" probability distributions. This is a question I would like to have help understanding. I want to figure out the distribution of number of trials for a given fixed number of successes and given probability for success for Bernoulli trials...

If I'm given a function ##f(x)##, say it has continuos first derivative, then I expand it as ##f(x + \Delta x) = f(x) + (df / dx) \Delta x##. If instead, I'm given ##f^{-1}(x)## how do I go about expanding it? Will this be just ##f^{-1}(x + \Delta x) = f^{-1}(x) + (df^{-1} / dx) \Delta x##?

Homework Statement Hello. My colleague and I have been struggling with this assignment where we have to: - Given the position and orientation of the end-effector of a robotic arm with 6 degrees of freedom get the joint angle values (inverse kinematics). The robotic arm in question is the...

Homework Statement I plotted a graph where Capacitance is at y-axis and Inverse Distance is at x-axis. It looks like a positive, increasing function. I am asked of a best fit, also an equation. But i am not sure whether to use quadratic or linear fit. I am also asked of what the slope...

Square law? i raise this question because of recently reading some QM, and realizing that for significantly short periods of time, it becomes hard to detect the mathematical patterns. E.g. in the double slit experiment, the standard pattern doesn’t appear after just a few photons. It takes...

Hello, guys. I'm currently working on a physics problem that requires me to evaluate the inverse Laplace of the function in the attached file. When b = 0, "y" vanishes, and all one has to do is to look up the Laplace table for the inverse. However, non-zero b has been giving me a headache. I...

Homework Statement Find the inverse function of ##f(x) =x^4+2x^2, x>0## Homework Equations ##f(f^{-1}(x)) = x## The Attempt at a Solution My only progress so far is ##x^4+2x^2 = x^2(x^2+2)## Then I am stuck. Since my progress is close to nothing so I don’t expect a complete...

Suppose we are given $A^{-1} = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 3 & 0 \\ 4 & 2 & 2 \end{bmatrix} = \left[\begin{array}{rrr|rrr} 1&4&0&1&0&0\\ 2&3&0&0&1&0\\ 4&2&2&0&0&1...

using this equation ##1-\sqrt{1-x^2/c^2}## where c = 1 and x = 0.0 - 1.0 the speed of c for example ##1-\sqrt{1-.886^2/1^2}## = y = 0.5363147619 gives me the y values. How do I find the inverse? How do find for x inputting the values of y? Thank you.

I'm forking this off another thread where I brought it up but it was getting OT. It is good enough for a first approximation but it is certainly not exact. Consider a test mass one radius from a spherical body. Work out the contributions form two points diametrically opposed on the surface...

Do exist examples of attraction forces with such a type potential ##V(\boldsymbol r)\sim-\frac{1}{|\boldsymbol r|^2}, \quad |\boldsymbol r|\to 0## in physics ?

Hi, I've a question that asks me to determine matrix A , where A= ${S}^{-1}$* B* S They have given matrices S and B in the question. I think the answer is A=B, since S * ${S}^{-1}$ would give me the identity matrix and anything multiplied by the identity matrix is itself. Is this correct?

I am interested in evaluating light intensity variation in a digital image. A colleague wants to apply an inverse square law correction to account for distance variation. I am trying to justify that in this case, the inverse square law does not apply. Treating each pixel as a detector, it has...

I am working on a proof problem on ordered field from a textbook, which lists additive and multiplicative properties similar to the ones here: The followings are what I was able to come out -- I just wanted to make sure that they are acceptable: (a) By the multiplicative inverse property...

Homework Statement find the inverse of the function ##y=2^{x}## ok i know the steps but why is this question awarded 1 mark...Homework Equations 3. The Attempt at a Solution [/B] ## y= 2^{x}## →## ln y= xln 2## →## x= ln y/ln 2##

Homework Statement In calculus, I learn that the derivative of the inverse function is g'(x) = 1/ f'(g(x)) Homework Equations So.. The Attempt at a Solution Can someone give me an example of where I need to know this, or is this just a math exercise. Is there a relatively simple physics...

Homework Statement Not for homework, but just for understanding. So we know that if a matrix (M) is orthogonal, then its transpose is its inverse. Using that knowledge for a diagonalised matrix (with eigenvalues), its column vectors are all mutually orthogonal and thus you would assume that...

Homework Statement For ##y=f(x)##, find the slope of the tangent line to its inverse function ##f^{-1}## at the indicated point P. ##f(x) = -x^3-x+2## , ##P(-8,2)## Homework Equations The Inverse Function Theorem: ##(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}## The Attempt at a Solution So...

Hello, I am trying to find the inverse of f(x) = x \div (x+4) IIRC i need to replace f(x) with y and solve for x. I've tried to do y = x \div (x+4) becomes y(x+4) = x then xy + 4y = x but I can't reduce the amount of x to one. What am I doing wrong in this problem?

I'm currently carrying out an analysis on waveforms produced by a particular particle detector. The Nyquist-Shannon sampling theorem has been very useful for making an interpolation over the original sample points obtained from the oscilloscope. The theorem (for a finite set of samples) is given...

One of our homework problem asks: If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1. Here's how I attempted to solve the problem: -3=3x-1 3x=-2 x=-2/3 Is this the correct way to solve the problem?

Hi all, just had a question about tensor/matrix notation with the inverse Lorentz transform. The topic was covered well here, but I’m still having trouble relating this with an equation in Schutz Intro to GR... So I can use the following to get an equation for the inverse...

Homework Statement Find the area bounded by arcsinx, arccosx and the x axis. Hint-you don't need to integrate arcsinx and arccosx Homework Equations All pertaining to calculus The Attempt at a Solution I drew the correct graph and marked their intersection at (1/√2, pi/4) and painstakingly...

Homework Statement Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1 Homework Equations All trigonometric and inverse trigonometric identities, special usage of double angle identities here The Attempt at a Solution I can get the answer by puting x=cosy, the term inside...

i have heard how our broadcasts will be seen by aliens far away or whatever. but i realize those signals are going to "attenuate" by d^-2 anyway... how come in astronomy we can see light sources millions of light years away? shouldn't those signals be far too weak to detect after such a long...

Let ##G## and ##H## be groups, and let ##\phi : G \to H## be an isomorphism. I want to show that ##\phi^{-1} : H \to G## is also an isomorphism. First, note that ##\phi^{-1}## is clearly a bijection as ##\phi## is its inverse. Second, let ##a,b \in H##. Since ##\phi## is surjective, there exist...

If we have a relation, ##R##, and it's inverse, ##R^{-1}## they behave such that a point on ##R##, say (a,b), corresponds to the point (b,a) on ##R^{-1}## This is a reflections across the line y=x. This relation does not mean that ##R^{-1}## is a function. For example, Let ##R## be...

Hello! (Wave) I want to find $f(t)$ if its Laplace transform is $F(s)=\frac{1}{s(s^2+1)}$. We use the following formula, right? $$f(t)=\frac{1}{2 \pi i} \lim_{T \to +\infty} \int_{a-iT}^{a+iT} e^{st} F(s) ds$$ But how can we calculate the integral $\int_{a-iT}^{a+iT} e^{st}...

Hi PF! I'm reading an article and there is a differential equation cast as an operator equation: $$f_n-d_x^2 f_n = \lambda f$$ where ##f_n = \partial_n f##, which is derivative of ##f## normal to a given parameterized curve. The author casts the ODE as $$B[f_n] = \lambda A[f_n]:\\ B[f_n] \equiv...

Homework Statement Evaluate and express your answer in radians: $$cot^{-1}\left(1\right)$$ Homework EquationsThe Attempt at a Solution I start by identifying that the domain of Arccotangent is all real numbers. So 1 is in the domain. From here, I looked at the unit circle and saw that...

Let ##f:U\subset R^n\rightarrow V\subset R^n## be a biyective function of class ##C^m(m\geq 1)##, ##U## and ##V## are open sets in ##R^n##. I know from the inverse funtion theorem that when ##J(f)\neq 0## in a point of the the domain a local inverse exists, however, given the above conditions...

Hi PF! One way to solve a simple eigenvalue problem like $$y''(x)+\lambda y(x) = 0,\\ y(0)=y(1)=0$$ (I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...

Homework Statement Homework Equations log(y) = mlog(x)+log(k) y=kxm The Attempt at a Solution Determine the exponent m and coefficient k of the power law that best fits your data. Is the acceleration directly or inversely proportional? Taking some points on the graph to get the slope (0.78 -...

Homework Statement Y=(8s-4)/(s²-4) Homework EquationsThe Attempt at a Solution I rearranged the right side as: 8*(s/(s²-2²))-2*(2/(s²-2²)) Using the Laplace transform chart given in the class I was able to identify these as the transforms of hyperbolic sine and hyperbolic cosine making the...

Homework Statement ##f:= tanh = \frac{e^x-e^{-x}}{e^x+e^{-x}}## Prove that ##f^{-1}(x)= \sum\limits_{k=0}^{\infty} \frac{x^{2k+1}}{2k+1}## for all x in (-1,1) The Attempt at a Solution I also found the inverse function to be: ##f^{-1}(x)= \frac{1}{2}ln(\frac{1+x}{1-x})## I tried working...

Homework Statement I'm not asking how to do this question This is a work done by one of my students And the highlighted part it seems to be the correct answer that the teacher gave. I cannot make any sense out of these two questions Perhaps one of you might shed some light on to...

Suppose I have a matrix M = A + εB, where ε << 1. If A is invertible, under some assumptions I can write e Neumann series M-1 = (I - εA-1B)A-1 But if A is not invertible, how can I expand M-1 in powers of ε? Thanks in advance

In QM, the inverse distance operator ##\hat{r}^{-1}## appears often because of the association to Coulomb potential. The operator of inverse momentum, ##\frac{1}{\hat{p}}## is a lot more rare. In the book "Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers...

Hi i’m having problems with the following equations: X(w)=2/(-1+iw)(-2+iw)(-3+iw) This then becomes the following equation according the the tutorial, although there is no explanation as to how: X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw The commas indicated the end of each fraction to make it easier...