Fraction Definition and 640 Threads

Homework Statement Proof that: ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2}## using the epsilon-delta definition of a limit. (Problem below) Homework Equations ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2} \iff \forall ε>0, \existsδ>0, \forall x ∈ \mathbb{R}\backslash\{-1\}: 0 < |x-1| < δ...

Hi, $$\frac{3 }{ 2(x+1)} + \frac{1}{x+1} = \frac{5}{6}$$ Can you help me on this problem. I'm not sure on how to solve this and please be kind enough to explain your steps little bitMany Thanks:)

In this I think you should cross multiply ; I got 1/2 by doing so and I am not sure whether it is correct.Can you help me to solve this and be kind enough to explain your steps a little. :)

For example, say I want to convert 1/7 to its representation as a repeating decimal? Is the fastest way just to do long division, or is there a faster way?

Homework Statement After undergoing through 90° compton scattering, the fraction of energy lost by photon is a) 10% b) 20% c) 50% d) zero e) none of these Homework Equations ∆λ= h/moc (1-cosΦ) The Attempt at a Solution What i m doing is that, i get scattered photon energy and subtracting...

I have this fraction $$x^2 / (x^2 + 9)$$ I'm not sure how to approach this problem since the denominator can't be further factored. What is the right approach for this type of problem?

I have this partial fraction: $$ 18 = (x^2 + 9) + (Bx + C)(x + 3)$$ which the textbook says is equal to: $$(B + 1)x^2 + (C + 3B)x + (9 + 3C)$$ But I don't follow this step. How do I derive this?

$$ \int_{}^{} \frac{1}{ e^x + {e}^{-x}}\,dx $$ I have this integral, and I'm not sure how to approach it. I tried u-substitution with $u = e^x + {e} ^{-x}$, but that seemed to go to a dead end. I'm not sure how to apply partial fractions to this problem. Is there a better way?

Homework Statement Hello! Here is my second post on the subject partial fraction decomposition. The subject looks pretty easy to learn, but when I try exercises, I do not get to the correct answer. Please, take a look at the exercise below and help me to see my mistakes. Homework Equations...

Homework Statement Hello! I am doing a chapter on partial fraction decomposition, and it seems I do not understand it correctly. Here is the exercise doing which I get wrong answers. Please, take a look at the way I proceed and, please, let me know what is wrong in my understanding. Homework...

I have this expression: $\frac {x^2 + 1 - 1}{x^2 + 1}$ Is there a way to simplify this expression and get: $1 - \frac {1}{x^2 + 1}$ My professor wrote it on the board and I didn't follow the reasoning.

Homework Statement c & d are positive integers & 1/c = 1 + 1/d Now tell which is greater c or d? Homework Equations 1/c = 1 + 1/d The Attempt at a Solution 1/c = (d+1)/d c = d/ (d+1) c = 1 + d Thus if c =10 therefore d = 9 But this is not a correct answer. Some body please guide me. Zulfi.

Homework Statement I am trying to factor a denominator so I can do a partial fraction decomposition to solve using a Laplace transform. denominator= 2s^3+3s^2-3s-2 Homework EquationsThe Attempt at a Solution

How powerful are continued fraction representations? From what I understand, they could be used to exactly represent some irrational numbers So, could they represent any root of an nth degree polynomial equation? Specially where n>4, since 5th degree roots are not guaranteed to have an...

I ran across this years ago. It’s defined by the addition rule: a/b + c/d = (a+c)/(b+d) (Sounds crazy – until you think of x/y as meaning There’s a pile of x things on one side, and one of y things on the other.) Applying it in almost a Pascal's Triangle sort of way generates ... er, a...

Hi, I need to know the official name of these type of fraction problems? This is what I wrote on a flash card I couldn't upload the flashcard because of file size. Example problem - 4/3 times x = 6/7 These are the instructions I wrote on the flash card. 1. Divide sides by left (least)...

Problem: Prove that any positive fraction plus its inverse is greater than or equal to two. Proof: $$\frac{a}{b}+\frac{b}{a}\ge2$$ $$\frac{a^2+b^2}{ab}\ge2$$ $${a^2+b^2}\ge2ab$$ $$a^2+b^2 - 2ab\ge0$$ $$a^2 - 2ab + b^2\ge0$$ $$(a-b)^2\ge0$$ This is true for all a and b: Case 1...

Hi, please could someone give me some advice on how I would convert the mol fraction to percentage? i.e. I want a gas mixture of Argon and Carbon dioxide with only 0.02 mol fraction of CO2, what would this be as a percent? I tried using this...

Hello, can anyone help please? Many thanks! 'Jing set a goal of hitting 9 home runs.When the season was over,she had hit 12 home runs.Express the portion of her goal Jing reached as a fraction,a decimal,and a percent.Round to the nearest whole percent.'

Hello everybody! I have to decompose to simple fractions the following function: $$V(z)=\frac{z^2-4z+4}{(z-3)(z-1)^2}$$. I know I can see the function as: $$V(z)=\frac{A}{z-3}+\frac{B}{(z-1)^2}+\frac{C}{z-1}$$, and that the terms A, B, C can be calculated respectively as the residues in 3...

In the attached picture, how is it justified to factor out an 'i' in the exponent of the 2 and the 5?

Homework Statement Decompose \frac{2(1-2x^2)}{x(1-x^2)} I get A = 2, B =-1, C = 1, but this doesn't recompose into the correct equation, and the calculators for partial fraction decomposition online all agree that it should be A = 2, B = 1, C = 1. Here is one of the online calculator results...

Homework Statement 2*3*4*5/4+3*4*4*6 Homework Equations 2*3*4*5/4+3*4*4*6 The Attempt at a Solution 2*3*5/1+3*4*6 Knowledge gap 1 - I do not understand why if top and bottom are divided through by 4 there should be a 4 left in the denominator and why the rest of the numerators and...

Homework Statement Simplify (6x+12y)/(10a +5) times (100a2 - 25)/(9x2 - 81y2) Homework Equations (6x+12y)/(10a +5) times (100a2 - 25)/(9x2 - 81y2) The Attempt at a Solution (6x+12y)/(10a +5) times (100a2 - 25)/(9x2 - 81y2) = 3(2x+4y)/(10a+5) times (4a - 1)/(3x-27y) =...

Homework Statement Simplify: (a2 + 2ab + b2)/(a2 - b2) times (a2 - 2ab + b2)/(a2 - b2) Homework Equations (a2 + 2ab + b2)/(a2 - b2) times (a2 - 2ab + b2)/(a2 - b2) The Attempt at a Solution Did I simplify this correctly?? (a+b)(a+b)/((a-b)(a+b)) times (a-b)(a-b)/((a-b)(a+b)) = ((a+b)/(a+b))...

Homework Statement I would just like to know if I got this right. ((x2 + 2x + 1)/(18cw3) times ((12c3w)/(x2 - 1)) = (12c3wx2 + 2x12c3w + 12c3w)/(18cw3)(x2 - 18cw3 Homework Equations ((x2 + 2x + 1)/(18cw3) times ((12c3w)(x2 - 1)) The Attempt at a Solution (12c3wx2 + 2x12c3w +...

Homework Statement Homework Equations not sure The Attempt at a Solution First, I'm a bit confused as to what the mole fraction of the products mean. Can someone explain? Is it something to do about the total number of carbon and hydrogen atoms in methane, ethene and propene? Just can't...

Hi, I have a paper that for a gas mixture writes Ar/25%O2 implying that 25% of the atmosphere of the gas is oxygen. I have another paper that writes 0.02 mol frac O2 for a Ar/CO2 atmosphere. Please could someone tell me how I would write the 0.02 mol frac O2 in the form Ar/X%CO2 and what X...

Homework Statement Find the Inverse laplace transform of: http://www4c.wolframalpha.com/Calculate/MSP/MSP14541hg721e74730d4fb00004644i96f59549h1d?MSPStoreType=image/gif&s=30&w=201.&h=40. Result...

I had a student calculate ##\tan\left(\frac{5\pi}{6}\right)## on a TI-84 calculator and he had a rounding error in the 6th or 7th decimal place. This isn't really a big deal, but he asked and we quickly found out it was because he was in fraction display mode. I don't know too much about...

Homework Statement divide in partial fractions: x^3 -3x^2+x-12 / x^4+5x^2+4 Homework EquationsThe Attempt at a Solution I factored x^4+5x^2+4 to (x^2 +1) (x^2+4) x^3 -3x^2+x-12/(x^2 +1) (x^2+4) = A (x^2+1)/(x^2 +1) (x^2+4) + B (x^2+4) /(x^2 +1) (x^2+4) they all have the same...

Ok so I took partial fraction decomposition in Calc II, and now I'm taking it again in Differential Equations course. The problem is that I don't really understand what I'm doing. I understand the procedure when having simple real roots, for example 2x+1/(x+1)(x+2), it becomes A/(x+1) + B/(x+2)...

...to a percentage? I know that 0.5 = 5/10 = 1/2 = 50%. But I'm not sure why multiplying 0.5 and 1/2 by 100 gives us them as a percentage. Take 1/2. Multiplying it by 100 means multiplying the numerator by 100. So 1/2 becomes 100/2 which gives 50 (50%). I can see that it works. But I don't...

In a certain problem, but almost no others I've encountered, the problem instructed to convert 14/5 to a mixed number, 2 and 4/5. I've noticed that mixed numbers almost never appear in Algebra (let alone Trig). Is it better to leave it as an improper fraction or to mix it up? (Get the pun, mix it?)

I encountered this question recently, and didn't really understand what they did in the solution. This time I posted the entire question to assist with knowing what is going on. Basically, I multiplied both sides by 2pi/3, but the question wanted me to multiply both sides by 3/2pi, the...

I'm interested in the following inequality (which may or may not be true) Theorem 1: ##( \sum_{i=1}^n \frac{a_i} {n}\ )( \sum_{i=1}^n \frac{1} {b_i}\ ) > \sum_{i=1}^n \frac{a_i} {b_i}\ ## Where ##n ≥ 2, a_1 < a_2 < ... < a_n## and ##b_1 < b_2 < ... < b_n##. My attempt at a proof: 1) When n =...

Homework Statement I want to express the following expression in its Taylor expansion about x = 0: $$ F(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)} $$ The Attempt at a Solution First I tried to rewrite the function in partial fractions (its been quite a while since I've last...

Because technically, the numerator is smaller than the denominator as −2<−1 I know it's an extremely stupid question. I mean I know that I can just multiply −1 to the numerator and the denominator and I'll get 2/1 which is greater than one. But what exactly is happening here? A number that...

So PWM can simulate a decrease in voltage when useing a stable amplitude. 10v50% duty cycle is 1/2 the amplitude meaning 5v. And if this is through a standard resistor then you will see 1/2 the current. But what about a sine wave? What fraction or percentage does this come out to be? What...

Homework Statement An unstable ion of mass ##M##, energy ##E## emits a massless particle of energy ##E_\nu## at angle ##\theta##. In the rest frame of the ion, find ##E_\nu^*## and ##cos \theta^*##. Ions are now accelerated to ##\gamma=100## and a detector with radius ##r=20m## is placed...

Homework Statement What is the partial fraction decomposition in ##\mathbb{R}[X]## of ##F = \frac{1}{X^{2n} - 1 } ##, ##n\ge 1##. Homework EquationsThe Attempt at a Solution Is this correct ? ## F = \frac{1}{2n}(\frac{1}{X-1} - \frac{1}{X+1} + 2 \sum_{k = 1}^{n-1} \frac{ \cos...

Why are atoms taken to be spheres, and not of some other shape, in the calculation of the packing fraction of different crystal lattices? In other words, what experimental evidence and theoretical reasoning motivates this form of the atomic shape?

(x-3)/(x^2+4x+3) After i factor the denominator what do i do next to find A and B? =(x-3)/(x+3)(x+1) =A/(x+3)+B/(x+1)

Homework Statement There is an electronic transition between the first energy state and the ground state of a neon atom, emitting a wavelength of 746nm. The question asks for the energy of the transition and an estimate of the fraction of atoms in a sample that is thermally excited at 300K...

Homework Statement There is a statement in a book : " Graph of P vs ##\chi## is a straight line which ##cannot## pass through origin" Homework EquationsThe Attempt at a Solution But if mole fraction of a component is zero then it can't form vapours because of which its partial pressure will be...

So let me start off by saying I am horrible at math and always have been. Despite considering myself highly intelligent, I suck at math. It's what led me into computer programming instead of physics like I wanted. Simply, I want to ask, say I have a fraction or a ratio that can't be simplified...

Homework Statement I was working on a physics problem that involves integrals and I stumbled upon this: m*g*sin θ − k*x = m*dv/dt =m*v*dv/dx → (m*g*sin θ − k*x)/m =v*dv/dx →∫[(m*g*sin θ − k*x)/m]dx =∫v*dv Notice that we multiplied by dx on both sides and dx has been canceled from the right...

Homework Statement Consider a cryogenic chemical propulsion system with Isp (specific impulse) = 450 s Enter the required propellant fraction as a proportion with at least two decimal places. The ΔVelocity is provided. ΔV = 9.6 km/s Homework Equations Rocket equation: ΔV = Vex x Ln (initial...

Homework Statement At 1 atm and 0° C, a 5.04 L mixture of methane (CH4) and propane(C3H8) was burned producing 20.9 g of CO2. Assume complete combustion. 1. How many moles total of methane and propane were present before combustion? 2. How many moles of carbon dioxide were present after the...

Homework Statement Find lim_{x->- \infty} \; \frac{(x^6+8)^{1/3}}{4x^2+(3x^4+1)^{1/2}} Homework Equations N/A The Attempt at a Solution Factoring out \frac {(-x^6)^{1/3}}{-x^2} leaves me with \frac{(-1-8x^{-6})}{-4+(3+x^{-4})^{1/2 }} Taking the limit at infinity gives me...