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# Blog

### application of derivatives in business and economics examples

Then, right click to view or copy to desktop. Examples include profit & loss, average cost, and interest accumulation. Webcomic #1 - "Volume: A Math Guy's Business Model" (10-21-11) Real world applications of derivatives and limits. For example: f(x) = sin x. The second derivative of MC is positive for all values of Q, therefore the MC function is convex, and is at a relative minimum when q is equal to 8. Application of the derivative (2) Business and Economics Page 4 of 15 Example 5: A rectangular tennis court of 1800 square meters is to be fenced with 2 types of materials. The shorter sides are made with fence material costing $100 per meter and the other sides with fence material costing$50 per meter. Search for: Application of Derivatives. Link to worksheets used in this section. cost, strength, amount of material used in a building, profit, loss, etc.). Derivatives have been traded for centuries, with early examples including tulip bulb options in Holland and rice futures in Japan during the 17th century. The application of derivatives exists in Mathematics, … If they sell x widgets during the year then their profit, in dollars, is given by, $P\left( x \right) = 30,000,000 - 360,000x + 750{x^2} - \frac{1}{3}{x^3}$ How many widgets should they try to sell in order to maximize their profit? One of the most important application is when the data has been charted on graph or data table such as excel. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. There is buyer and seller for each contract. Applications of the Derivative Optimization Problems in Economics. A real estate broker charges a commission of 6% on Sales valued up to $300,000. Example 3: Find the optimum points of the profit function and determine what level of production Q will maximize profit. In this chapter we seek to elucidate a number of general ideas which cut across many disciplines. c02ApplicationsoftheDerivative 180 AW00102/Goldstein-Calculus December 24, 2012 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics In recent years, economic decision making has … In words: To perform marginal analysis on either profit, revenue or cost, find the derivative function for the one quantity out of these three that you are estimating for. APPLICATIONS OF DERIVATIVES IN BUSINESS. Business • In the business world there are many applications for derivatives. We have looked at the definite integral as the signed area under a curve. We are thankful to be welcome on these lands in friendship. Solution: f (2 =2 2 2 3 2 100 106) 32−× +×+ = Examples 2 . Also learn how to apply derivatives to approximate function values and find limits using L’Hôpital’s rule. Applications of the Derivative In this section, we focus on the applications of the derivative. Thus, C(x) is described as the cost function. Add 200P to both sides of the demand equation. We use the derivative to determine the maximum and minimum values of particular functions (e.g. Derivatives describe the rate of change of quantities. Despite the fact that the definition of the derivative is rather abstract (using the limit of the ratio of the increments of the function and the independent variable), the fields of its applications are extremely diverse. Derivatives are met in many engineering and science problems, especially when modelling the behaviour of moving objects. III. Example: The total cost in thousands of Pesos to manufacture x electric Below examples of a derivative illustrate of the most common derivatives. This lets us compute total profit, or revenue, or cost, from the related marginal functions. Application of Derivatives. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Applications. Derivatives are financial instruments like equity and bonds, in the form of a contract that derives its value from the performance and price movement of the underlying entity. To check the temperature variation. The examples are used in mathematical Once it has been input, the data can be graphed and with the applications of derivatives you can estimate the profit and loss point for certain ventures. Section 4-14 : Business Applications. Using the demand equation to derive total revenue as a function of q requires the following steps:. A derivative is any instrument whose value depends upon the value of another instrument or index known as the “underlying.” The value of the derivative is derived from the value of the underlying. Section 4.7 Applications to Business and Economics Math 1a Introduction to Calculus April 2, 2008 Announcements Problem Sessions Sunday, Thursday, 7pm, SC 310 Ofﬁce hours Tues, Weds, 2–4pm SC 323 Midterm II: … Furthermore, economics has differentiation tools like marginal cost and marginal revenue as its basic necessities. A common question in Economics is how many units to produce to create the maximum profit. It can be used to measure: How cost and revenue are changing based on how many units are built and sold; How profit can be maximized for a specific quantity of sales and/or units produced; How a population is changing over time Start by taking first and second derivatives: First, we need to know that profit maximization occurs when marginal cost … A company can produce a maximum of 1500 widgets in a year. a. These four categories are what we call the 4 basic types of derivative contracts. In business calculus (and also in economics and social sciences), derivatives have many applications. This becomes very useful when solving various problems that are related to rates of change in applied, real-world, situations. To calculate the profit and loss in business using graphs. APPLICATIONS OF DERIVATIVES Derivatives are everywhere in engineering, physics, biology, economics, and much more. ( The actual equation used was y = -x3 + 9x2 - … It is impossible to provide all types of derivative examples, since there thousands of such derivatives, and which vary in every situation. APPLICATION OF DERIVATIVES 195 Thus, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t. Let us consider some examples. To determine the speed or distance covered such as miles per hour, kilometre per hour etc. Also, they are the oldest form of derivatives. y = f(x), then the proportional ∆ x = y. dx dy 1 = dx d (ln y ) Take logs and differentiate to find proportional changes in variables Solution: f'(x) = cos x. f”(x) = sinx = 0 x = nπ, n ∈ z. f (2). For instance, derivatives exist with payments based on the level of the S&P 500, the temperature at Kennedy Airport, or the number of bankruptcies among a group of selected companies. The market demand curve for the good your monopoly produces is. In this article, we will list down and explain those 4 types: Type 1: Forward Contracts. Examples of Derivative Assets. Applications of Derivatives to Business and Economics. Linearization of a function is the process of approximating a function by a line near some point. Application III: Differentiation of Natural Logs to find Proportional Changes The derivative of log(f(x)) ≡ f’(x)/ f(x), or the proportional change in the variable x i.e. Let y = C(x), where y represents the cost to manufacture x items. Definition: A derivative is a contract between two parties which derives its value/price from an underlying asset.The most common types of derivatives are futures, options, forwards and swaps. Following are the main examples of derivative assets. The methods of differentiation find great application in estimating various quantities of interest. In business and economics there are many applied problems that require optimization. Section 6 Use of Partial Derivatives in Economics; Some Examples Marginal functions. Economic derivatives can be traded on an exchange. In Figure 3 above, we saw that f(4) = 21. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Example 1 Find the rate of change of the area of a circle per second with respect to its radius r when r = 5 cm. In Mathematics, the derivative is an expression that gives the rate of change of a function with respect to an independent variable. For sales valued at more than$ 300,000, the commission is $6,000 plus 4% of the sales price. Differentiation has applications to nearly all quantitative disciplines. Application of Derivatives in Real Life. The derivatives of these quantities are called marginal profit function, marginal revenue function and marginal cost function, respectively. The derivative is defined as something which is based on some other thing. Forward contracts are the simplest form of derivatives that are available today. where q is the market and firm’s quantity demanded, and P is the market price in dollars.. Futures; Options; Futures: Future contract is an agreement between two parties that specifies the provision of certain product (financial or tangible) at a certain future date and at a specified price. THE DERIVATIVE AND MARGINAL COST, REVENUE, AND PROFIT Jaromír Zahrádka Institute of Mathematics, Faculty of Economics and Administration, University of Pardubice Abstract: The article is focused to describe several examples of economical and business problems which are solved by using derivations. Click to enlarge image. The derivative; maxima, minima, and points of inflection One very important application of the quotient property above is the special limit known as the derivative function. Applications of Derivatives to Business and Economics. Derivative Examples. This chapter covers concepts relating to the application of derivatives to find the maxima or minima of functions used in business, economics, and the social sciences, especially cost, revenue, and profit. Solution 2The area A of a circle with radius r is given by A = πr. If we have, or can create, formulas for cost and revenue then we can use derivatives to find this optimal quantity. Section 7.8 Economics Applications of the Integral. The exchange provides the product specifications; for example, the non-farm payrolls economic derivative may be a monthly auction. 1. Examples 1: Let f. 3 2 ( ) 2 3. x x x x = − + + 100 Find. Lesson 22: Applications to Business and Economics 1. Examples of Derivatives (With Excel Template) Let’s try to understand derivatives through the below examples. _____ Provide all types of derivative examples, since there thousands of such derivatives, much. Optimum points of the demand equation lets us compute total profit, or cost, and much.! 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First, we need to know that profit maximization occurs when marginal cost and then! = sin x L ’ Hôpital ’ s try to understand derivatives the., derivatives have many applications for derivatives we need to know that profit maximization when...


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Volume 345 - International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions (HardProbes2018) - Heavy Flavours & Quarkonia
Quarkonium production in heavy ion collisions: coupled Boltzmann transport equations
X. Yao,* W. Ke, Y. Xu, S. Bass, B. Muller
*corresponding author
Full text: pdf
Pre-published on: 2019 January 11
Published on:
Abstract
By coupling the Boltzmann transport equations of both quarkonium and open heavy quarks, we investigate their dynamical evolution inside the quark-gluon plasma and study quarkonium production in heavy ion collisions. The Boltzmann transport equation of quarkonium is derived from the open quantum system formalism and effective field theory of QCD by assuming quarkonium interacts weakly with the plasma. The dissociation and recombination terms in the Boltzmann equation are calculated in potential nonrelativistic QCD. It is shown that the combination of quarkonium dissociation, recombination, open heavy quark diffusion and energy loss can drive the system of quarkonium and open heavy quarks to detailed balance and kinetic thermalization. By solving the transport equations with initial momenta of quarkonia and heavy quarks sampled from PYTHIA and a hydrodynamic medium, we can calculate the nuclear modification factors of bottomonium and describe the data at both RHIC and LHC energies. The azimuthal angular anisotropy coefficient $v_2$ of $\Upsilon$(1S) in 5.02 TeV peripheral Pb-Pb collisions is also predicted.
Open Access

format: plain


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Let be $\phi:X\rightarrow \mathbb{R}$ a lower semi-continuos function then $X = \cup_{n=1}^{\infty}\phi^{-1}(-n,\infty)$.

Why this?

-

This is true for any function from $X$ to $\Bbb R$; it has nothing to do with semi-continuity.
$$X=\varphi^{-1}[\Bbb R]=\varphi^{-1}\left[\bigcup_{n\in\Bbb Z^+}(-n,\to)\right]=\bigcup_{n\in\Bbb Z^+}\varphi^{-1}\big[(-n,\to)\big]$$
simply because $\Bbb R=\bigcup\limits_{n\in\Bbb Z^+}(-n,\to)$.
What mean $\to$?, infinite positive or negative? –  user46060 Oct 26 '12 at 15:45
@user46060: I (and others) write $(a,\to)$ instead of $(a,\infty)$, and $(\leftarrow,a)$ instead of $(-\infty,a)$. –  Brian M. Scott Oct 26 '12 at 15:50
Understand, in my case the union is $\cup_{n=1}^{\infty}$ and not $\bigcup_{n\in\Bbb R}$ or these are equivalents? –  user46060 Oct 26 '12 at 16:18
@user46060: Yes, of course. If you read the whole line, the $\Bbb R$ in the last union was an obvious typo. I’ve fixed it now. –  Brian M. Scott Oct 26 '12 at 16:20


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# Can we compare correlations between groups by comparing regression slopes?

In this question they ask how to compare Pearson r for two independent groups (such as males vs females). Reply and comments suggested two ways:

1. Use Fisher's well-known formula using "z-tranformation" of r;
2. Use comparison of slopes (regression coefficients).

The latter could be easily performed just via a saturated linear model: $Y = a + bX + cG + dXG$, where $X$ and $Y$ are the correlated variables and $G$ is a dummy (0 vs 1) variable indicating the two groups. The magnitude of $d$ (the interaction term coefficient) is exactly the difference in coefficient $b$ after model $Y = a + bX$ conducted in two groups individually, and its ($d$'s) significance is thus the test of difference in slope between the groups.

Now, slope or regression coef. isn't yet a correlation coef. But if we standardize $X$ and $Y$ - separately in two groups - then $d$ will be equal to the difference r in group 1 minus r in group 0 and therefore its significance will be testing the difference between the two correlations: we're testing slopes but it appeares [as if - ?] we're testing correlations.

Is that I've written correct?

If yes, there's left the question which is a better test of correlations - this one described or Fisher's one? For they will yield not identical results. What do you think?

Later Edit: Thanking @Wolfgang for his reply I nevertheless feel I miss understanding why Fisher's test is more correct a test for r than comparison-of-slope-under-standardization approach described above. So, more answers are welcome. Thank you.

Everything that you have written is correct. You can always test out things like that with a toy example. Here is an example with R:

library(MASS)

rho <- .5  ### the true correlation in both groups

S1 <- matrix(c( 1,   rho,   rho, 1), nrow=2)
S2 <- matrix(c(16, 4*rho, 4*rho, 1), nrow=2)

cov2cor(S1)
cov2cor(S2)

xy1 <- mvrnorm(1000, mu=c(0,0), Sigma=S1)
xy2 <- mvrnorm(1000, mu=c(0,0), Sigma=S2)

x <- c(xy1[,1], xy2[,1])
y <- c(xy1[,2], xy2[,2])
group <- c(rep(0, 1000), rep(1, 1000))

summary(lm(y ~ x + group + x:group))

What you will find that the interaction is highly significant, even though the true correlation is the same in both groups. Why does that happen? Because the raw regression coefficients in the two groups reflect not only the strength of the correlation, but also the scaling of X (and Y) in the two groups. Since those scalings differ, the interaction is significant. This is an important point, since it is often believed that to test the difference in the correlation, you just need to test the interaction in the model above. Let's continue:

summary(lm(xy2[,2] ~ xy2[,1]))$coef[2] - summary(lm(xy1[,2] ~ xy1[,1]))$coef[2]

This will show you that the difference in the regression coefficients for the model fitted separately in the two groups will give you exactly the same value as the interaction term.

What we are really interested in though is the difference in the correlations:

cor(xy1)[1,2]
cor(xy2)[1,2]
cor(xy2)[1,2] - cor(xy1)[1,2]

You will find that this difference is essentially zero. Let's standardize X and Y within the two groups and refit the full model:

x <- c(scale(xy1[,1]), scale(xy2[,1]))
y <- c(scale(xy1[,2]), scale(xy2[,2]))
summary(lm(y ~ x + x:group - 1))

Note that I am not including the intercept or the group main effect here, because they are zero by definition. You will find that the coefficient for x is equal to the correlation for group 1 and the coefficient for the interaction is equal to the difference in the correlations for the two groups.

Now, for your question whether it would be better to use this approach versus using the test that makes use of Fisher's r-to-z transformation.

EDIT

The standard errors of the regression coefficients that are calculated when you standardize the X and Y values within the groups do not take this standardization into consideration. Therefore, they are not correct. Accordingly, the t-test for the interaction does not control the Type I error rate adequately. I conducted a simulation study to examine this. When $\rho_1 = \rho_2 = 0$, then the Type I error is controlled. However, when $\rho_1 = \rho_2 \ne 0$, then the Type I error of the t-test tends to be overly conservative (i.e., it does not reject often enough for a given $\alpha$ value). On the other hand, the test that makes use of Fisher's r-to-z transformation does perform adequately, regardless of the size of the true correlations in both groups (except when the group sizes get very small and the true correlations in the two groups get very close to $\pm1$.

Conclusion: If you want to test for a difference in correlations, use Fisher's r-to-z transformation and test the difference between those values.

• Does the Fisher transformation have any advantages over the other test? – mark999 Aug 7 '11 at 9:37
• Turns out I was a bit too quick. See my edits. That should hopefully answer your question. – Wolfgang Aug 7 '11 at 10:04
• So, @Wolfgang, you hold that comparison-of-slope-under-standardization approach is a valid comparison of r. Fisher's famous alternative is actually an approximation of that. Did I understand you right? – ttnphns Aug 7 '11 at 10:06
• See my edits. I was too quick with my initial conclusions. – Wolfgang Aug 7 '11 at 10:27
• @Wolfgang, later added EDIT of your reply states that Fisher is better. Comparison-of-slope-under-standardization approach is inadequate because "standard errors ... when you standardize ... do not take this standardization into consideration". Please explain me how they ought to take standardization into account so that comparison-of-slope-under-standardization approach become as valid as Fisher's test. – ttnphns Aug 7 '11 at 10:33


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# Resistance

### After reading this section you will be able to do the following:

• Define resistance and how we measure it.
• Discuss the similarities between resistance in a wire and the resistance in a water hose.

Resistance is another important property that can be measured in electrical systems. Resistance is measured in units called ohms. Resistance is a term that describes the forces that oppose the flow of electrical current in a conductor. All materials naturally contain some resistance to the flow of electron current. We have not found a way to make conductors that do not have some resistance.

If we use our water analogy from the voltage lesson to help picture resistance, think of a hose that is partially plugged with sand. The sand will slow the flow of water in the hose. We can say that the plugged hose has more resistance to water flow than does an unplugged hose. If we want to get more water out of the hose, we would need to turn up the water pressure at the faucet. The same is true with electricity. Materials with low resistance let electricity flow easily. Materials with higher resistance require more voltage (EMF) to make the electricity flow.

The scientific definition of one ohm is the amount of electrical resistance that exists in an electrical circuit when one amp of current is flowing with one volt being applied to the circuit.

Resistivity (ρ) is an intrinsic property of a material relating to the materials resistance to the flow of electrical current. A high resistivity indicates that a material is not a good conductor of electricity. Resistivity is determined by measuring the electrical resistance over some length of material with a constant cross-section. The materials resistivity is the product of the resistance value and the cross-sectional area divided by the length as given in the following equation.

$\rho=\frac{RA}{l}$

where
ρ = resistivity (ohm meters)
R = electrical resistance of a uniform specimen of the material (ohms)
l = length of the specimen (measured in meters)
A = cross-sectional area of the specimen (meters 2)

Conductivity is calculated by simply taking the inverse of resistivity. A material with a resistivity of 1/58 ohm-meter will have a conductivity of 58 MegaSiemens per meter.

## Is resistance good or bad?

Resistance can be both good and bad. If we are trying to transmit electricity efficiently from one place to another through a conductor, resistance is undesirable in the conductor. Resistance causes some of the electrical energy to turn into heat so some electrical energy is lost along the way. However, it is resistance that allows us to use electricity for heat and light. The heat that is generated from electric heaters or the light that we get from light bulbs is due to resistance. In a light bulb, the electricity flowing through the filament, or the tiny wires inside the bulb, cause them to glow white hot. If all the oxygen were not removed from inside the bulb, the wires would burn up.

An important point to mention here is that the resistance is higher in smaller wires. Therefore, if the voltage or EMF is high, too much current will follow through small wires and make them hot. In some cases hot enough to cause a fire or even explode. Therefore, it is sometimes useful to add components called resistors (see picture below) into an electrical circuit to restrict the flow of electricity and protect the components in the circuit.

### Review

1. Resistance is the opposition to electrical current.
2. Resistance is measured in units called ohms.
3. Resistance is sometimes desirable and sometimes undesirable.


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# Find each of the following quotients to detect a pattern. then explain how you can use

###### Question:

Find each of the following quotients to detect a pattern. then explain how you can use this pattern to mentally divide 0.0096 by 3

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### What is the value of x?

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### Which three of the following actions were taken by the Council of Trent?denounced the supremacy of the pope in the Catholic Churchcondemned

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### A bag contains 2 black balls, 4 yellow balls and 4 white balls. Event A is defined as drawing a white ball on the first draw and event

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### Which number in the placement set {1,2,3,4} makes 6z + 4 = 22 a true statement?

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### Under a dilation of scale factor 3 with the center at the origin, if c (6,9), what will be the coordinates of the point c'?

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### Using the discriminant , how many real solutions does the following quadratic equation have? 2x^2+7x-15=0a. no real solutions

Using the discriminant , how many real solutions does the following quadratic equation have? 2x^2+7x-15=0a. no real solutions b. two real solutions c. three real solutions d. one real solutions...

### Refer to group members' personal motives, goals, attitudes, and values. a. social forces b. psychological

Refer to group members' personal motives, goals, attitudes, and values. a. social forces b. psychological forces c. cognitive forces d. physiological forces...

### What are Constant/control variables

What are Constant/control variables...

### .Put the verbs in brackets in the correct past form. 1 While ... (wait ) for the bus i(meet) two of my friends.

.Put the verbs in brackets in the correct past form. 1 While ... (wait ) for the bus i(meet) two of my friends....

### 1. in which sentence does the verb agree with the subject? a. mosquitos and rain makes an outdoor picnic

1. in which sentence does the verb agree with the subject? a. mosquitos and rain makes an outdoor picnic pretty miserable. b. books and television entertain us on a rainy day. c. the ducklings and their mother waddles toward the pond. 2. in which sentence does the verb agree with the subject? a....

### What is the graph of the function, ƒ, is shown. what is ƒ(0)?

What is the graph of the function, ƒ, is shown. what is ƒ(0)?...

### ¿Cual era la actividad económica que impulsaloa la economia novohispana durante el siglo XV111?

¿Cual era la actividad económi ca que impulsaloa la economia novo hispana durante el siglo XV111?...
Find the length of side x in simplest radical form with a rational denominator. $Find the length of side x in simplest radical form with a rational denominator.$...


text:
## geerky42 2 years ago The radius, r, of the base of a circular cylinder increases by 2 feet per second while the height, h, decreases by 1 foot per second. How fast is the surface area of the cylinder changing when the height of the cylinder is 50 feet and the radius of the base is 40 feet? I got $$440\pi$$ square feet per second, am I right?

1. geerky42

My work: __________________________________________________________________ Since it is known that surface area of this cylinder is $$SA = 2\pi r^2 + 2\pi rh$$$\dfrac{dSA}{dt} = 4\pi r\dfrac{dr}{dt} + 2\pi \dfrac{dr}{dt} +2\pi r \dfrac{dh}{dt}$We are given that r = 40, h = 50, $$\dfrac{dr}{dt} = 2$$, $$\dfrac{dh}{dt} = -1$$ I plugged in these values and I got $$\boxed{440\pi}$$ __________________________________________________________________ But there is no option for $$440\pi$$, I'm not sure if there is typo in options or I made mistake somewhere in my work...

2. klimenkov

Your solution seems to be okay, (but you forgot $$h$$ in the second addend). Maybe there is a typo or the surface area is computed without bottom and top of the cylinder.. Better to ask the person that has gived you this task.


text:
# Thread: Solve System of Equations

1. ## Solve System of Equations

Solve this system of equations.

y = x^(3/2)
y = e^(-x)

My Answer is x = e^(-2x/3).
Is this right?

2. ## Re: Solve System of Equations

Originally Posted by nycmath
Solve this system of equations.

y = x^(3/2)
y = e^(-x)

My Answer is x = e^(-2x/3).
Is this right?
Nah. The solution to this system is in terms of the Lambert W function which I'd be surprised if you've been taught about.

Lambert W function - Wikipedia, the free encyclopedia

$x=\dfrac 3 2 W(\frac 2 3)$

You sure you have the problem correctly written?

3. ## Re: Solve System of Equations

Originally Posted by nycmath
Solve this system of equations.

y = x^(3/2)
y = e^(-x)

My Answer is x = e^(-2x/3).
Is this right?
Far from being the "right answer", it is not even an "answer"! You must have x= a single number. No matter how complicated the right side there cannot be an "x" on the right side.

4. ## Re: Solve System of Equations

Originally Posted by nycmath
Solve this system of equations.

y = x^(3/2)
y = e^(-x)

My Answer is x = e^(-2x/3).
Is this right?
Part of your answer will be the values of x which satisfy this equation. The only way to do that (unless you know the Lambert W Function) is using numerical methods or getting a decimal approximation with a calculator (which also uses numerical methods).

You then need to get the corresponding y values.


text:
# How do you find the slope of a line parallel to the graph of each equation y=1/2x+2.3?

Jun 29, 2018

See a solution process below:

#### Explanation:

This equation is in slope intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y = \textcolor{red}{\frac{1}{2}} x + \textcolor{b l u e}{2.3}$

Therefore, the slope of the line in the problem is: $\textcolor{red}{m = \frac{1}{2}}$

Two parallel line, by definition, will have the same slope

Therefore, the slope of any line parallel to the line in the problem is also:

$\textcolor{red}{m = \frac{1}{2}}$

Jun 29, 2018

$\text{slope } = \frac{1}{2}$

#### Explanation:

• " Parallel lines have equal slopes"

$\text{the equation of a line in "color(blue)"slope-intercept form}$ is.

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$y = \frac{1}{2} x + 2.3 \text{ is in this form}$

$\text{with slope } = \frac{1}{2}$

$\text{slope of parallel line } = \frac{1}{2}$


text:
# Problem 5: Standing 30 meters away from jet e engine vou measure sound level of 140 decibels. Assuming the waves

###### Question:

Problem 5: Standing 30 meters away from jet e engine vou measure sound level of 140 decibels. Assuming the waves from the engine are expanding in a uniform sphere; how far would You need to be from the jet engine for the sound levelto drop to 65 decibels?

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text:
# Are my serial downvotes inappropriate?

I recently read an answer regarding L'Hospital rule and I don't even understand a line what this answer is talking about. There are several others of such answers (which have been deleted now):

All of them are related to a magic "Taylor's formula" from non-standard analysis, which can be seen from the first linked answer. I know nothing but the name of non-standard analysis and I have never seen people do calculus this way before.

I was not intending to target one particular user. But all these low-quality answers are from this same person, it seems that I am now.

Are my serial downvotes to such answers inappropriate? If yes, would anyone suggest what one should do?

[Added:] Here is the list of undeleted(deleted now) answers I have downvoted:

(I really want to know if an expert on nonstandard analysis would stand up and shout at me: "go home and learn more math! These answers are all valid!" )

• Series are related, but I think you meant "serial", as in many votes to one user. – Simply Beautiful Art Aug 24 '17 at 21:38
• The second deleted answer is outright nonsense, in general, there should be another correction term. – Simply Beautiful Art Aug 24 '17 at 21:40
• @SimplyBeautifulArt: Fixed now. Thanks. – Jack Aug 24 '17 at 21:40
• Yeah, this user is definitely misinterpreting Taylor's theorem. I don't believe the deletion was bad, despite them all coming from the same user. – Simply Beautiful Art Aug 24 '17 at 21:42
• It depends on how one defines "targeting" a user. Focusing in on one user to the exclusion of other moderating actions, may not fare well upon the crunching of numbers, by the daily scripts detecting "serial voting". I am not suggesting you were serially downvoting or deleting for any reason other than enhancement of site quality. You made a lot of good calls; but maybe review such a user's posts less rapidly; as is, many of the posts in question are from a year ago. Bottom line: Your honesty here is applaud-able. And it's a good question, in general!! – Namaste Aug 24 '17 at 22:10
• From the viewpoint of non-standard analysis, some of the added answers (the one you just added) do make sense. It is a little bit handwaving but the spirit is mostly correct. – achille hui Aug 25 '17 at 3:38
• @achillehui: No they do not. In a lot of those cases the user claims that L'Hopital's rule is false, just because his weird manipulations arrive at a contradiction. Also, he is willy-nilly using the idea of nilsquare elements all over. – user21820 Aug 25 '17 at 6:44
• @user21820 I didn't bother to look at all the answers. – achille hui Aug 25 '17 at 6:46
• @achillehui: That's okay. I'm just stating it in case people get the wrong impression. I look sufficiently at every single answer that I delete. =) – user21820 Aug 25 '17 at 6:53
• @Jack The second answer from the second list has no downvotes at the moment. Perhaps you can check the other answers, but I guess that your votes have been reversed by the automated script which removes serial votes. (Also on the user's profile I see today +8 reversal - Voting corrected. Of course, I cannot see whether the reversed votes were yours - voting is anonymous.) – Martin Sleziak Aug 25 '17 at 8:51
• Is it not ironic that the mentioned user's username is "selfawareuser"? – Frpzzd Aug 26 '17 at 15:17
• Let me add something to the discussion: I had a brief virtual encounter with that user last year, when I detected him using 3 separate accounts; of them, one was for posting, and another one for upvoting his own posts. When confonted with this fact, he quickly deleted 2 out of those 3 accounts. Regarding his posts - some of them are ok, but there is a lot of non-standard analysis in answers to simple high-school questions where the OP is obviously looking for "standard" analysis. I myself have downvoted and voted to delete many of this user's posts. – Alex M. Aug 26 '17 at 15:45
• @AlexM. I no longer see it there. I'm pretty sure I saw it when I posted my comment - maybe I should have made a screenshot. (Maybe all the post related to those reputation changes were deleted and that's why we no longer see the reputation changes.) But I guess it is not very difficult to find some other user profiles with voting reversals. Since this seems only tangential to the issue at hand, we can continue this discussion in chat, if needed. – Martin Sleziak Aug 26 '17 at 15:55
• @rocksNwaves ?? – Simply Beautiful Art Aug 27 '17 at 23:17
• Maybe a recent discussion on meta might be of interest to you in relation to this: Someone is doing “targeted serial downvoting” against me? (Just a guess, but I thought it might be useful to let you know about that discussion.) – Martin Sleziak Dec 1 '17 at 5:55

In my mind there are at least three sides to this, in general not only for this particular case.

1. At one level the votes are appropriate. They are based on your best judgment of the post. To communicate your evaluation of a post is exactly what votes are for.

2. At another level you focused your votes on one user. This could raise some red flags and indeed it seems from Martin's comment that a script picked up on the pattern and annulled your votes in part. This is unfortunate for various reasons. This could be seen as a lack of the script, but let me get to the third side.

3. I do know the situation of stumbling over a user and finding many of their contributions lacking and feeling the urge to take some action. But, ultimately, I mostly try to resist the urge to take actions in bulk. On the one hand, to avoid the second point. On the other hand, and more importantly, I feel it can be better to give the user direct feedback via a comment and to give them some chance (that is in particular time) to fix their stuff.

Sure, sometimes there will be no change in behavior and then the situation changes a bit. Yet to vote massively on one user's contribution is not necessarily the best option.

If there is a massive and persistent problem signal it to the moderators (for example, via a custom flag on one of the concerned posts).

If the problem is not massive the best course of action might actually be not to focus on it. Deal with the posts occasionally if you come across them, otherwise trust that somebody else will take care of them. (Yes, maybe, nobody will do that, but then realistically there are any number of other users' posts that would also require your attention.)

• (+1) I think such an extreme case should be dealt with by moderators. – Simply Beautiful Art Aug 27 '17 at 23:23

I think the community's overreacting on this one. The user under question seems to be basing their viewpoint on calulcus on this book. I haven't looked at the arguments too carefully, but it's not clear to me that the these arguments fall into the category of "plain wrong." In particular, aren't they just working in the dual numbers and then using the phrase "let $\varepsilon \rightarrow 0$" as sloppy terminology for "apply the $\mathbb{R}$-algebra homomorphism $\mathbb{R}[\varepsilon] \rightarrow \mathbb{R}$ defined by $\varepsilon \mapsto 0$?" Really they should find better notation for this, and they should probably disclose that they're not working in the real line, but apart from that, most of the answers I glanced at seemed okay. Obviously if the results they're deducing directly contradict ordinary real analysis that's a problem, and the idea that L'Hopital's rule is "just wrong" is itself just wrong, but nonetheless seems a bit rash to delete all their answers like that. We should restrict ourselves to deleting the blatantly incorrect ones. And, I should add that the whole thing seems to be a device for throwing away "terms that won't end up mattering anyway," so mathematically, the interesting question is: under what circumstances can this device be used safely, without deducing statements about the real line that are already known to be false? I don't know the answer to that question, but surely the answer isn't "never."


text:
## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Numerical Godeaux surfaces with an involution in positive characteristic

Soonyoung Kim

#### Abstract

A numerical Godeaux surface $X$ is a minimal surface of general type with $\chi(\mathcal{O}_{X})=K_{X}^{2}=1$. Over $\mathbf{C}$ such surfaces have $p_{g}(X)=h^{1}(\mathcal{O}_{X})=0$, but $p_{g}=h^{1}(\mathcal{O}_{X})=1$ also occurs in characteristic $p>0$. Keum and Lee~[9] studied Godeaux surfaces over $\mathbf{C}$ with an involution, and these were classified by Calabri, Ciliberto, and Mendes Lopes~[4]. In characteristic $p\ge 5$, we obtain the same bound $|\mathrm{Tors}\,X|\le 5$ as in characteristic 0, and we show that the quotient $X/\sigma$ of $X$ by its involution is rational, or is birational to an Enriques surface. Moreover, we give explicit examples in characteristic 5 of quintic hypersurfaces $Y$ with an action of each of the group schemes $G$ of order 5, and having extra symmetry by $\mathrm{Aut}\,G\cong\mathbf{Z}/4\mathbf{Z}$, hence by the \textit{holomorph} $H_{20}=\mathrm{Hol}\,G=G\rtimes\mathbf{Z}/4\mathbf{Z}$ of $G$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 8 (2014), 113-118.

Dates
First available in Project Euclid: 3 October 2014

https://projecteuclid.org/euclid.pja/1412341994

Digital Object Identifier
doi:10.3792/pjaa.90.113

Mathematical Reviews number (MathSciNet)
MR3266744

Zentralblatt MATH identifier
1338.14042

Subjects
Primary: 14J29: Surfaces of general type

#### Citation

Kim, Soonyoung. Numerical Godeaux surfaces with an involution in positive characteristic. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 113--118. doi:10.3792/pjaa.90.113. https://projecteuclid.org/euclid.pja/1412341994

#### References

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• C. Liedtke, Algebraic surfaces of general type with small $c_{1}^{2}$ in positive characteristic, Nagoya Math. J. 191 (2008), 111–134.
• C. Liedtke, Non-classical Godeaux surfaces, Math. Ann. 343 (2009), no. 3, 623–637.
• R. Miranda, Nonclassical Godeaux surfaces in characteristic five, Proc. Amer. Math. Soc. 91 (1984), no. 1, 9–11.
• Y. Miyaoka, Tricanonical maps of numerical Godeaux surfaces, Invent. Math. 34 (1976), no. 2, 99–111.
• M. Reid, Godeaux and Campedelli surfaces. (http://homepages.warwick.ac.uk/ masda/surf/Godeaux.pdf).
• M. Reid, Surfaces with $p_{g}=0$, $K^{2}=1$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1978), no. 1, 75–92.
• Q. Xie, Kawamata-Viehweg vanishing on rational surfaces in positive characteristic, Math. Z. 266 (2010), no. 3, 561–570.


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# jim morrison tribute- Jabès

Photo by: Bertil Nilsson

WIDE, the margin between carte blanche and the white page. Nevertheless it is not in the margin that you can find me, but in the yet whiter one that separates the word-strewn sheet from the transparent, the written page from the one to be written in the infinite space where the eye turns back to the eye, and the hand to the pen, where all we write is erased, even as you write it. For the book imperceptibly takes shape within the book we will never finish… There is my desert.
~
Edmond Jabès


text:
Abhisar one year ago A wheel having a radius of 10 cm is coupled by a belt to another wheel of radius 30 cm. 1st wheel increases its angular speed from rest at a uniform rate of 1.57 rad/s2. The time for the 2nd wheel to reach a rotational speed of 100 rev/min is (assume that the belt does not slip)

1. Compassionate

There's actually a really simple equation for this. It's kind of like a system of equations. Since you have two variables, you'll need two equations. Did your book or course give you any equations to use? Do you have an idea of what you need to do?

2. Abhisar

Well, I know that linear velocity for both the wheels will be equal and I need to equate them to find the angular velocity of second.

3. Compassionate

Hmm. Marvelous. Give me a minute to write this down.

4. Abhisar

Sure..

5. Compassionate

I'm at McDonald's so my wifi is really slow. Bare with me.

6. Abhisar

ok..

7. Compassionate

I almost got it but my computer is freezing. I gotta use the restroom. Give me another minute. It's really freezing up and the wifi keeps going out. But I think I have it.

8. Abhisar

Sure, No problem

9. Abhisar

Well, I think I got it doe.....

10. Compassionate

Show me what you got.

11. Abhisar

$$\sf α_11=1.57rad/s^2$$ $$ω_2=100rad/m$$ =100×2π/60 =10π/3rad/s linear velocity of belt is equal ∴ω1r1=ω2r1 and α1r1=α2r2 ∴α2=(1.57)(1030) =1.573rad/s^2 $$\sf ω=ω_0+αt$$ 10π/3=0+1.573×t t=20sec.

12. Compassionate

That's what I got.

13. Compassionate

I actually kept getting 21.666 repeating, but 20 looks good.

14. Compassionate

If I redo it, I got 20. My first two times I was going say 21.7, but decimals are... uncommon. I'll show my work

15. Abhisar

Sure..

16. Compassionate

$\omega _{average} = \frac{ \Delta \theta }{ \Delta \tau }$ On the final step, I put in the known factors to isolate Wheel B, and simply solved for t, which gave me 21.67

17. Compassionate

I suspect I may be wrong.

18. Abhisar

I am not sure if that would be correct. I am myself confused >_<

19. Compassionate

You know what, I just realized w(a) is for average, but I am assumming you're looking for the Vf or Vi, even then, I think I royally messed up my average. I'm more of a newtonian kinda guy ;) 20 sounds good.

20. Abhisar

Thanks c:

21. Abhisar

I'll drop an explanation, may be some one else need it too. Speed of second disk will be $$\sf \Large {\frac{100 \times 2\pi}{60}=\frac{10 \pi}{3}}rad/s$$ Since both the wheels or disks are connected, linear velocity and linear acceleration of each of them will be same

22. Abhisar

So, we can write that, $$\sf a_1=a_2\\\Rightarrow\alpha_1\times r_1=\alpha_2\times r_2\\ \therefore \alpha_2=0.52$$ Now, using equations of motion, $$\sf \omega=\omega_0+\alpha t \\ 10\pi/3=0+0.52\times t\\ \therefore t=20.1 \approx 20sec$$

23. IrishBoy123

for connected wheels you can say $$r_1 \theta_1 = r_2 \theta _2$$ provided the belt does not slip from there, it follows that $$r_1 \dot \theta_1 = r_2 \dot \theta _2$$ and $$r_1 \ddot \theta_1 = r_2 \ddot \theta _2$$ so the acceleration of the larger wheel is $$\ddot \theta = \frac{1.57}{3} rad/s^2$$ using the acircular analogue for $$v = u + at$$ ie $$\dot \theta_f = \dot \theta_i + \ddot \theta t$$ with $$\dot \theta_i = 0$$ gives $$t = \dfrac{\dot \theta_f}{\ddot \theta} = \dfrac{2\pi \frac{100}{60}}{\frac{1.57}{3}} \approx 20s$$ so i guess i agree with you guys:p

24. Astrophysics

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text:
# Estimating products

#### All in One Place

Everything you need for better grades in university, high school and elementary.

#### Learn with Ease

Made in Canada with help for all provincial curriculums, so you can study in confidence.

#### Instant and Unlimited Help

0/6
##### Intros
###### Lessons
1. Introduction to Estimating Products:
2. Example of estimating the product of 49 × 58
3. Key terms for estimating products
4. Estimating products by rounding
5. Estimating products by clustering
6. Estimating products using compatible numbers
7. Underestimating and overestimating products
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##### Examples
###### Lessons
1. Estimating Products by Rounding
Estimate the product using estimation by rounding. Then, find the exact product to check your answer.
1. 695 × 487 =
2. 89.32 × 514 =
3. 0.71 × 2.64 =
4. 6$\frac{2}{5}$ × 8$\frac{6}{8}$ =
5. 11$\frac{1}{9}$ × 3$\frac{6}{10}$ =
2. Estimating Products by Clustering
Estimate the product using estimation by clustering. Then, find the exact answer to check your estimate.
1. 71 + 68 + 73 + 65 =
2. 0.318 + 0.299 + 0.256 + 0.341 =
3. 11$\frac{7}{10}$ + 12$\frac{2}{5}$ + 12$\frac{1}{4}$ + 11$\frac{1}{2}$ + 11$\frac{4}{5}$ =
3. Estimating Products with Compatible Numbers
Estimate the product using compatible numbers. Is the estimated product less than or greater than the exact product?
1. $\frac{3}{4}$ × 7$\frac{5}{6}$ =
2. 25$\frac{3}{5}$ × $\frac{5}{8}$ =
4. Comparing estimated products
Compare the products by first estimating each product using compatible numbers. Then, use symbols (for less than <, greater than >, or about equal to $\approx$) to show which product will be greater.

8$\frac{1}{2}$ × $\frac{4}{7}$ vs. $\frac{4}{9}$ × 10$\frac{1}{3}$
1. Estimating Products Word Problem - 1
If a box of twelve mangoes costs \$47.88 and weighs 5.34 pounds:
1. About how much will 350 boxes of mangoes cost? Use the estimation by rounding method.
2. About how much will 350 boxes of mangoes cost? Use the estimation with compatible numbers method.
3. Find the exact cost of 350 boxes of mangoes? Are your estimates from part a) and b) more or less than the exact cost?
4. About how many pounds will 350 boxes of mangoes weigh? Use the estimation by rounding method.
5. About how many pounds will 350 boxes of mangoes weigh? Use the estimation with compatible numbers method.
6. Will the boxes of mangoes weigh more than or less than 1,500 pounds? Are your estimates from part c) and d) more or less than the exact cost?
2. Estimating Products Word Problem - 2
Elsa used 4$\frac{3}{4}$ flasks full of blue liquid in her science experiment. Charlie wants to use 2$\frac{1}{5}$ times as much as Elsa. About how many flasks of blue liquid would Charlie need for his experiment?
1. Use the estimation by rounding method to estimate the number of flasks.
2. Use the estimation with compatible numbers method to estimate the number of flasks.
3. Calculate the exact product (exact answer) of how many flasks Charlie needs.
###### Topic Notes

In this lesson, we will learn:

• How to estimate the answer to multiplication statements
• The three methods for estimating products: estimation by rounding, estimation by clustering, and estimation with compatible numbers
• How to check and compare your estimated products with the exact answer

Notes:

• An estimation is a rough calculation of what the exact answer could be around. It is less exact but easier (faster) to calculate!

• When estimating, it is helpful to remember the rules for rounding numbers:
• If the number to the right of the digit you are rounding to is $\geq$ 5, round UP; if the number is < 5, round DOWN
• For mixed fractions, round to the nearest whole number: if the fraction part is $\geq$ $\frac{1}{2}$, round UP. If the fraction part is < $\frac{1}{2}$, round DOWN.

• Three methods to estimate products are: estimation by rounding, estimation by clustering, and estimation with compatible numbers.

• Estimation by Rounding:
• 1. Round
• Round each factor to its greatest place value
• For mixed fractions, round to the nearest whole number
2. Multiply the rounded factors
• Multiplying mixed fractions requires converting back to improper fractions first

• Estimation by Clustering:
• 1. Round all the addends to the same place value
2. Do all the estimates cluster around the same number?
3. Multiply: [cluster number] × [number of addends]

• Estimation with Compatible Numbers:
• 1. Look at the denominator of the proper fraction (fraction that is <1)
2. Look at the whole number in the mixed fraction
3. Change the whole number to a “compatible” number
• A compatible number is something that is close to your original number (i.e. 1-2 more or less), but it’s a multiple of the denominator
4. Multiply, using cross cancellation!

• You can compare the exact product and the estimated product to see how close they are
• An underestimate happens when you round DOWN a factor; the estimated product is LESS than the exact product
• An overestimate happens when you round UP a factor; the estimated product is MORE than the exact product


text:
Disabled workers living facilities in the organizational environment, relationships and relevance of the exhaustion of the working environment and clients

Title & Authors
Disabled workers living facilities in the organizational environment, relationships and relevance of the exhaustion of the working environment and clients
Kim, Min-Cheol;

Abstract
This study aims to explore ways to provide a high quality work environment services to social workers to analyze the relevance of relationships and exhaustion of the organizational environment, and the working environment and clients with disabilities living facility workers Gwangju and Jeonnam 252 people were targeting professionals working with people with disabilities living facility located on the main results are as follows. Organizational environment, work environment, organizational environment results by setting the relationship with the client as independent variables conducted a regression analysis to investigate the factors that influence independent of exhaustion ($\small{{\beta}}$=-.200, p=.001) and Job environment ($\small{{\beta}}$=-.460, p<.001) was a significant independent explanatory factor was 31.0%. However, relationships with clients ($\small{{\beta}}$=-.059, p=0.258) were not significant factors in exhaustion. Consequently, practitioners must always have the confidence to actively work with pride and facilities are needed to improve the organizational environment and the working environment.
Keywords
Burnout;Disabled Living Facilities;Exhaustion;
Language
Korean
Cited by
References
1.
Korea Institute for Health and Social Affairs Survey on disability. (2014)

2.
Soo-Chan Choi, Sang-A Kim, Ji-Young Jeon, Woong-Sub Park "Job-Concerned Factor Giving Effect to Burnout of Welfare Facility Worker -Focusing on the Mediation Effect of Job Stress", Mental Health and Social Work, Vol. 29 No 5-31. (2008)

3.
Su-Wha Min "A case study on the working conditions and quality of service of caregivers at facilities for the people with disabilities", Graduate School Soongsil University. (2001)

4.

5.
Hyun-Suk Shin "The Impacts on the Service Quality of a Living Facility for the Disabled of the Empowerment of Living Rehabilitation Teachers", Department of Social Welfare Graduate School, Daegu University Kyungbuk, Korea. (2006)

6.
Maslach, C. "Burn-out", Human behavior, Vol.5, No9, 161. (1976)

7.
Yu-Jung Lee "study on the influence of care workers' emotional labor on their burn-out in the skilled nursing home for the old", school of social yonsei university. (2004)

8.
Sun-Duk Kwon "A Study on the Status and exhausted exhaustion factor of social workers: workers mainly in Seoul Social Welfare", Yonsei University Graduate School of Public Administration, Master's thesis. (2001)

9.
Mi-Sun Kim "A study on the relationship between burnout and social support and mental health social worker", Ewha Womans University, master's thesis. (2002)

10.
Gae-Soon Kong "A Study on the exhaustion of child abuse prevention center agents", challenges of localized social welfare field, in 2004 South Korea Social Welfare Society Fall Joint Conference Kit. (2004)

11.
Dae-Hee Lee "A Study on the Effect of the Work Environment on Social Workers' Burnout in Elderly Long-term Care Facilities", Elderly WelfareGrauate School of Social Welfare Soongsil University. (2008)

12.
Young-Chun Kim "study of relationship between strategic human resource management practice and organizational effectiveness : focused of mediating role of human capital and moderating role of employee alignment and of management control system", Dept, of Business Administration Graduate School of Hongik Univ. (2001)

13.
Ji-Young Lim "Study on the Antecedents Affecting Exhaustion of Emotion-laborWorkers in Short-term Daycare Center for the Elderly", Ewha Womans University, Graduate School of Social Welfare, Department of Social Welfar. (2008)

14.
Young-Mi lee "Study on "Burnout" of social workers", school of social yonsei university. (1990)

15.
Hae-Young Choi "Factors Affecting Burnout of social workers - Focusing on the welfare and hospital", school of social yonsei university. (1994)

16.
Kyu-Won Shin "A Study on Burn-out of the Employees in In-home Care Center", Department of Social Welfare Graduate School of Social Welfare, Daegu University Gyungbuk, Korea. (2007)

17.
Yo-Han Yang "Factors affecting the exhaustion of social workers with disabilities living facilities", Chonbuk National University, Graduate School of Public Administration, Master's thesis. (2011)


text:
# Beats: frequency of resulting wave vs. beats frequency

• I
The beats frequency heard from the interference of two sound waves with frequencies ##f_1## and ##f_2## is $$\nu=|f_1-f_2|$$

Nevertheless the frequency of the resulting wave is not ##\nu## but the mean value of the two frequencies
$$f_{resulting}=\frac{f_1+f_2}{2}$$

As far as I understood ##\nu## is the frequency at which the maxima of intensity are heard, while ##f_{resulting}## is indeed the frequency of the resulting wave.

In particular I can notice that ##f_{resulting} > \nu## always.

The distincion between the two is clear in theory, but in practice I still have doubts.

Suppose to have an instrument that can measure sound waves iff the frequency is, say, bigger than ##f_{min}## and lower than ##f_{max}##.

Now suppose that two waves interefere in a way such that ##f_{resulting} > f_{min}## but ##\nu <f_{min}##, or, in a way such that ##\nu < f_{max}## but ##f_{resulting} > f_{max}##.

What does the instrument measure in these cases?

Which of the two frequencies "determine" the upper of lower limits for the frequency that can be measured by such instrument?

sophiecentaur
Gold Member
2020 Award
The beats frequency heard from the interference of two sound waves with frequencies ##f_1## and ##f_2## is $$\nu=|f_1-f_2|$$

Nevertheless the frequency of the resulting wave is not ##\nu## but the mean value of the two frequencies
$$f_{resulting}=\frac{f_1+f_2}{2}$$

As far as I understood ##\nu## is the frequency at which the maxima of intensity are heard, while ##f_{resulting}## is indeed the frequency of the resulting wave.

In particular I can notice that ##f_{resulting} > \nu## always.

The distincion between the two is clear in theory, but in practice I still have doubts.
This link says it all, I think.
The reason for hearing a beat of twice the frequency that you would expect is that you hear the two peaks in loudness and your ear ignores the phase inversion of the sum at frequency (f1+f2)/2
The ear is an imperfect instrument for measuring what you present it with and that accounts for the apparent paradox, I think.
The instrument that you describe would be like an AM receiver, I think and, to measure the beat, it would need a wide enough bandwidth to include the two input tones. But, as with an AM receiver, it only needs to admit those tones and needs no response at the difference frequency - to take an extreme case, a 1MHz AM receiver doesn't need any response to audio signals in its antenna input in order to receive audio signals carried on the carrier.

Svein
$\sin(x)+\sin(y)=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$
Let $x=\omega_{1}t$ and $y=\omega_{2}t$.

sophiecentaur
$\sin(x)+\sin(y)=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$
Let $x=\omega_{1}t$ and $y=\omega_{2}t$.


text:
# If a monotone sequence is convergent, does this imply boundedness?

I was proving a problem which states that: let $\{x_n\}_n$ be a monotone sequence of real numbers. Show that $\{x_n\}_n$ is convergent if and only if it is bounded.

I have proven that if the sequence is monotone and bounded, it is convergent. But must I prove the reverse case where I assume that the sequence is monotone and convergent and prove boundedness? I assume this is trivial because if the sequence is increasing or decreasing and is bounded, then the limit must be the supremum or the infimum.

• perhaps what you mean by "trivial" here is that any convergent sequence in $\mathbb{R}$, monotone or not, must be bounded. so certainly that direction is a more relaxed exercise. maybe try showing that an unbounded sequence cannot be a Cauchy sequence, and thus, a fortiori, cannot be convergent. – David Holden Mar 12 '15 at 1:36

It's trivial, but for a different reason: a convergent sequence in any metric space is bounded. Suppose $\{x_n\}$ isn't bounded. Then no open ball centered at $x$ contains all points of the sequence. So for each $k$, we can find a point $x_{n_k}$ such that $d(x,x_{n_k})>k$. Take $\varepsilon=1$. Then for any $N$, we have a positive integer $n_k$ such that $d(x,x_{n_k})>k>1$. Hence, $x_n$ does not converge to $x$, a contradiction.

In fact, any Cauchy sequence is bounded. It's a good exercise to prove that as well.

Needless to use a proof by contradiction: if $(x_n)$ converges to $\ell$ in a metric space, then for every $\varepsilon>0$ there exists $N$ such that $d(x_n,\ell)<\varepsilon$ for all $n>N$.

Here every $x_n$ belongs to the ball $$B(\ell, \max(\varepsilon, d(x_0,\ell),d(x_1,\ell),\dots, d(x_N,\ell)).$$

• There's various ways to prove this. I gave a few here: math.stackexchange.com/questions/1180332/… – Math1000 Mar 12 '15 at 1:48
• You beat me to it. I was going to say very nearly the same thing. Convergence puts the tail in any given ball, so there are at most finitely many other points at the head of the sequence to consider. +1 for you. – MPW Mar 12 '15 at 1:49

First, assume the sequence is monotone increasing i.e. $x_n \le x_{n+1}$ for all $n$.

What does it mean that the sequence converges? It means: there is some $x \in \mathbb{R}$ such that given $\varepsilon > 0$ there is an $N \in \mathbb{N}$ such that if $n \ge N$ then $$|x_n - x| < \varepsilon$$ Hence, we have $x_n - x < \varepsilon$ so that $x_n < x + \varepsilon$ for all $n \ge N$. Now, $x_n$ is monotone, so what can you conclude about $x_n$ for $n < N$?

To wrap everything up, just pick $\varepsilon = 1$. Then there is an $N$ as above, and so for all $n \ge N$, $x_n < x + 1$. Hence, the sequence is bounded by $x + 1$.

• When you say, what can you conclude, do you mean that we can conclude since it is convergent and increasing, that this sequence is increasing to $x.$ If we just pick an epsilon, we have that it is bounded? – H5159 Mar 12 '15 at 2:01
• @Frumpy No I mean that since $x_n$ is monotone, what can we conclude about the relation between $x_n$ for $n < N$ and $x_N$? – Ryan Sullivant Mar 12 '15 at 3:08
• @Frumpy Well actually, I guess what you are saying is morally correct, just jumping ahead a few steps. – Ryan Sullivant Mar 12 '15 at 3:11
• Could you fill in the steps I'm jumping ahead so I can understand? – H5159 Mar 12 '15 at 3:25
• Since $x_n$ is monotone increasing and we know that $x_n \le X_N$ for $n < N$. So if $x_N < x + \varepsilon$ then $x_n \le x + \varepsilon$ when $n < N$ (we took care of the case $n \ge N$ earlier). – Ryan Sullivant Mar 12 '15 at 3:29

More generally, a convergent series is bounded; monotone is not needed.

Proof:

Suppose $(x_n)_{n=1}^{\infty}$ is convergent to limit $L$.

Then, for any $\epsilon > 0$, there is a $N(\epsilon)$ such that, for $n > N(\epsilon)$, $|x_n-L| \le \epsilon$.

Therefore, for $n > N(\epsilon)$, $L-\epsilon \le x_n \le L+ \epsilon$.

Now, let $u(\epsilon) =\min(x_n)_{n=1}^{N(\epsilon)}$ and $v(\epsilon) =\max(x_n)_{n=1}^{N(\epsilon)}$. Then, for all $n$, $\min(u(\epsilon), L-\epsilon) \le x_n \le \max(v(\epsilon), L+ \epsilon)$, so the sequence is bounded.