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indifference curve approach show the combination of two goods that an individual would be willing to buy, and which would make the buyer equally satisfied (or different). indi…fference curve assume that more is preferred to less. thay are convex as seen from the origin. the indifference curve form an entire map of various level of satisfaction.. (MORE)

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In Economics

Indifference curves are the graphical representation of the preferences of an individual. In microeconomics there are three assumptions about preferences in order for them to …make sense in terms of the graphical representation through indifference curves. One assumption is the notion of completeless. This means that there is no ambiguity between what the individual prefers. The individual prefers one over the other or the individual does not care which one he gets. Another assumption is that preferences are consistent. For example if an individual prefers good A to good B and good B over good C. Then the individual prefers good A to good C. The last assumption is that an individual will always prefer more goods than less of a good. An indifference curve depicts all the combinations of goods that provide the individual the same amount of satisfaction. If two indifference curves intersect, this violates the last assumption of individual preferences that more is preferred over less of a good. In this case the individual is indifferent to less when the individual can have more. The problem with this violating these assumptions is that they are the basis of utility or profit maximization. If a firm or individual prefers less than the tools of utility or profit maximization will not make accurate predictions. (MORE)

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In Economics

Indifference curve is convex to the origin.This means that the slope of indifference curve decreases as we move the curve from left to right.This can be explained in terms of …Marginal rate of substitution of good X for good Y. The marginal rate of substitution is the maximum amount of Y the consumer is willing to give up to get an additional unit of X.This specifies the terms of trade-off between bundles of goods among which the consumer is indifferent. As the consumer moves down the curve he acquires more X and is left with less Y.So the amount of Y he would be willing to give up to get an additional unit of X becomes progressively smaller as is natural. So, the MRSxy diminishes as he moves from left to right.The convexity of the indifference curve illustrate the diminishing rate of substitution of X for Y associated with the movement down the curve from left to right. (MORE)

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indifferent curves are convex to their origin, they do not intersect, and have a negative slope

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Properties/Characteristics of Indifference Curve: Definition, Explanation and Diagram:An indifference curve shows combination of goods between which a person is ind…ifferent. The main attributes or properties or characteristics of indifference curves are as follows: (1) Indifference Curves are Negatively Sloped:The indifference curves must slope down from left to right. This means that an indifference curve is negatively sloped. It slopes downward because as the consumer increases the consumption of X commodity, he has to give up certain units of Y commodity in order to maintain the same level of satisfaction.In fig. 3.4 the two combinations of commodity cooking oil and commodity wheat is shown by the points a and b on the same indifference curve. The consumer is indifferent towards points a and b as they represent equal level of satisfaction.At point (a) on the indifference curve, the consumer is satisfied with OE units of ghee and OD units of wheat. He is equally satisfied with OF units of ghee and OK units of wheat shown by point b on the indifference curve. It is only on the negatively sloped curve that different points representing different combinations of goods X and Y give the same level of satisfaction to make the consumer indifferent. (2) Higher Indifference Curve Represents Higher Level:A higher indifference curve that lies above and to the right of another indifference curve represents a higher level of satisfaction and combination on a lower indifference curve yields a lower satisfaction.In other words, we can say that the combination of goods which lies on a higher indifference curve will be preferred by a consumer to the combination which lies on a lower indifference curve.In this diagram (3.5) there are three indifference curves, IC1, IC2 and IC3 which represents different levels of satisfaction. The indifference curve IC3 shows greater amount of satisfaction and it contains more of both goods than IC2 and IC1 (IC3 > IC2 > IC1). (3) Indifference Curve are Convex to the Origin:This is an important property of indifference curves. They are convex to the origin (bowed inward). This is equivalent to saying that as the consumer substitutes commodity X for commodity Y, the marginal rate of substitution diminishes of X for Y along an indifference curve.In this figure (3.6) as the consumer moves from A to B to C to D, the willingness to substitute good X for good Y diminishes. This means that as the amount of good X is increased by equal amounts, that of good Y diminishes by smaller amounts. The marginal rate of substitution of X for Y is the quantity of Y good that the consumer is willing to give up to gain a marginal unit of good X. The slope of IC is negative. It is convex to the origin. (4) Indifference Curve Cannot Intersect Each Other:Given the definition of indifference curve and the assumptions behind it, the indifference curves cannot intersect each other. It is because at the point of tangency, the higher curve will give as much as of the two commodities as is given by the lower indifference curve. This is absurd and impossible.In fig 3.7, two indifference curves are showing cutting each other at point B. The combinations represented by points B and F given equal satisfaction to the consumer because both lie on the same indifference curve IC2. Similarly the combinations shows by points B and E on indifference curve IC1 give equal satisfaction top the consumer.If combination F is equal to combination B in terms of satisfaction and combination E is equal to combination B in satisfaction. It follows that the combination F will be equivalent to E in terms of satisfaction. This conclusion looks quite funny because combination F on IC2 contains more of good Y (wheat) than combination which gives more satisfaction to the consumer. We, therefore, conclude that indifference curves cannot cut each other. (5) Indifference Curves do not Touch the Horizontal or Vertical Axis:One of the basic assumptions of indifference curves is that the consumer purchases combinations of different commodities. He is not supposed to purchase only one commodity. In that case indifference curve will touch one axis. This violates the basic assumption of indifference curves.In fig. 3.8, it is shown that the in difference IC touches Y axis at point C and X axis at point E. At point C, the consumer purchase only OC commodity of rice and no commodity of wheat, similarly at point E, he buys OE quantity of wheat and no amount of rice. Such indifference curves are against our basic assumption. Our basic assumption is that the consumer buys two goods in combination. (MORE)

The interest rate on bonds can vary tremendously due to varying maturity dates. The yield curve is a graphical representation of how interest rates will vary for bonds with s…imilar credit ratings but different maturity dates. (MORE)

In Statistics

The bell curve is arguably one of the most important curves in the field of statistics. Simply by glancing at this curve, one can tell a great deal about how a certain data se…t has been arranged. More complex information can also be derived from the basic data that is presented in the graph. Being able to derive this information requires a bit of expertise. This article will explore some of the kinds of data that can be extracted from a bell curve as well as the means for extracting it.The mean value is the easiest value derived from the bell curve. On any bell curve, the peak of the bell curve's X coordinates correspond to the value of the mean. This value on the bell curve also corresponds to the median and the mode, since it is not only the average value but the most recurring one, as it is the highest point on the Y axis as well.Extracting the standard deviation explicitly from the bell curve is a relatively difficult undertaking. However, general information regarding the standard deviation can be told simply by glancing at the bell curve. The standard deviation is indicated on the bell curve by how narrow or wide the distribution is. A narrow distribution indicates that there is a low standard deviation value. A wide curve indicates that there is a high standard deviation value.Probabilities are also another value that can be found using the bell curve. In statistics, bell curves are usually used to find probabilities by finding the area underneath the curve. Instead of using the curve itself and trying to roughly estimate the area under the curve, you can instead use a table that contains values for the areas under the curve from the mean until a certain point.A bell curve can also be used to find the range of data, which is to say the difference between the smallest value and the largest value in the data set. To do this, you simply need to look at the X axis and find the lowest value, which will correspond to the leftmost side of the bell curve. The highest value will correspond to the rightmost side. By using these two values, you can find what the range is.In statistics, skewness is a term given to a nearly normal distribution with data that is slightly offset. This offset is indicated by a peak that is not perfectly centered in the middle of the graph. When a data set is skewed, it is considered to be not normally distributed. If this happens, you are presented with two choices. You can either use nonparametric inferential statistical analysis in order to analyze the data, or you can transform this data set into a normal distribution by using a reciprocal, square root, or logarithmic transformation.There is plenty of information that can be extracted from a normal distribution curve, otherwise known as a bell curve. For example, you can extract information regarding the size of the data set simply by looking at the X axis and finding the lowest value and the highest value and subtracting them from one another. Additionally, you can infer data about the standard deviation of the data set by seeing how narrow or wide the peak is. Not all data can be modeled using a bell curve, but it is still incredibly useful at modeling plenty of real-world distributions.It is a good idea to learn other kinds of curves beside the bell curve, as not all data can be modeled using it. Some other curves that are also worth looking into include the chi-square curve and the T-Curve. (MORE)

Having to find the derivative of a certain function is one of the basics upon which calculus in particular, and computational mathematics in general, is based. Unfortunately, …there is no one surefire way to find the derivative of all functions. Depending on the expressions within the function, the means for obtaining the derivative of this function varies from one function to the next. In many cases, you may have to apply several different rules simultaneously in order to find the derivative of a function. This article lists some of the formulas associated with finding the derivatives of various functions.Usually, when you are first introduced to finding the derivative, the first kind of derivative that you are asked to find is the derivative of a polynomial. This polynomial usually consists of several expressions that contain a certain variable that is raised to varying powers. In order to find the derivative of a polynomial, you need to subtract the power of the variable by one, and multiply the coefficient of the polynomial term by the original power.The exponential function e the power of x is one of the most singular expressions in mathematics. The unique thing about this particular expression is that its derivative is itself. This means that whenever you obtain the derivative of e-power-x, the derivative itself will be e-power-x. It is this particular property that makes the value e, Euler's constant, a universal constant in mathematics, as well as a transcendental number.Trigonometric functions are a special class of functions in mathematics. This is because they are periodical and interrelated. This can be illustrated by the fact that the derivative of the sine function is another trigonometric function: the cosine function. It is important to note that there are no sign changes when finding the derivative of the sine function. The derivative of the function is a cosine function with the exact same sign and the exact same expression.The derivative of the cosine function is another trigonometric derivative. Unlike the derivative of the sine function, the sign is not preserved when you obtain the derivative of cosine. The derivative of cosine is negative sine. This means that when you evaluate the derivative of a cosine function, you need to flip the sign of the resulting sine function. If the sign was originally positive, it becomes negative. Likewise, if the sign was originally negative, it becomes positive.Ln(X) is one final function of interest for which you might want to find the derivative. The derivative of this function is none other than 1/X. Evaluating the value which results in 1/X using conventional means is not possible, since the number must give negative 1 when the power of the variable has 1 subtracted from it. When you multiply the original value by the coefficient, the expression evaluates to be 0, which does not compute in the traditional sense.There are almost as many different kinds of derivatives as there are functions. Some of these are very commonly used, such as the derivatives of trigonometric functions and the derivatives of exponential functions. On the other hand, there are other derivatives that are much less likely to be used, such as the derivatives of Ln(X). The most important derivatives by far that you must know if you are to derive functions correctly in mathematics are the derivatives of polynomials. These are the most commonly recurring ones, both on exams and in real-life applications.If you have time to confirm your answer, you can integrate the function for which you found the derivative. If you find that the resulting integration is the original function, then you have carried out the derivation correctly. (MORE)

In Pitches

The curve ball is not an optical illusion. It often appears that way to a batter however. It is one of the more difficult pitches in baseball to hit when properly thrown. A cu…rve ball will curve in two directions. It will curve downward due to the force of gravity. It will curve to one side or the other due to the interaction of the stitches on the baseball with the flow of air around the baseball.Throughout much of the history of baseball, most were convinced that the curve that could be achieved when pitching a ball was simply an optical illusion. Pitchers were shown how to grip and release a baseball so it would curve as it approached the plate. Most pitchers were nevertheless convinced that they were taught to throw the ball in a manner that would create an optical illusion. The baseball after all was not designed to be thrown as anything other than as a fastball. The fact that it could be thrown as a curve ball came about either by experiment or by accident. It took slow-motion photography to demonstrate that a thrown baseball can actually be made to curve.The answers lie in the stitches. There are 216 stitches in a regulation baseball. Those stitches were not put there to allow a pitcher to throw fancy pitches. They were put there to keep the cover from falling off. One might think that just by looking at the stitches, which are slightly raised, they would help the pitcher get a better grip on the ball, and make it easier to throw more accurately than one could throw a billiard ball for example. That's true, but it's the presence of the stitches that makes it possible to throw a curve ball.Stitches accomplish the same thing that the surfaces on an airplane wing do. An airplane wing, or an airfoil, has an upper surface that is slightly curved, and a lower surface that is mostly flat. As it moves through the air, the air flows differently over the curved surface than the flat surface, creating a difference in air pressure between the two surfaces. The air pressure on the upper surface is less, creating a slight vacuum. This causes the wing to move upwards or lift. On a baseball, the stitches create a disturbance in the way the air flows past the ball. If the ball is thrown a certain way, the stitches can cause the ball to deviate from an otherwise straight path. It will move off to one side or the other, or curve.Any baseball that is thrown or hit will curve downward due to the force of gravity. What makes a curve ball appear to drop when observed on TV, which it does, is because it is thrown more slowly than is a fastball, so its drop due to gravity will be much more pronounced. It also curves slightly to one side. The curving in a downward direction plus the curving to the side is what makes the curve ball, at least one thrown at the major league level, difficult to hit. It's not only the stitches however that cause this movement, but the manner in which the ball is gripped and thrown as well.A curve ball is a thrown ball that happens to have a great deal of spin on it. The spin causes the stitches to significantly affect the flow of air past the baseball. The grip from which the ball is released determines the relative position of the stitches with respect to the ground as the ball travels through the air. The relative position of the stitches, and the speed at which the ball is rotating, is what influences the flow of air past it. As is the case with the airfoil, if the air is flowing in a manner that creates a difference in pressure between one side of the ball and the other, the ball will be forced to move to one side or the other, depending upon the direction of its rotation.Theoretically, almost anyone can learn to throw a curve ball. All one has to do is grip the stitches or seams correctly, give the ball the right amount of spin when it is released, and throw it fast enough so that the spinning stitches have a chance to interact with the airflow before the ball reaches the plate. The ball doesn't have to be thrown super-fast, but at least half the speed of the fastball, if not a little more.On a baseball, the stitches create a disturbance in the way the air flows. If the ball is thrown a certain way, the presence of the stitches can cause the ball to deviate from an otherwise straight path. It will move off to one side or the other, or curve. (MORE)

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In Economics

Because if they intersected, they would not be "indiffernece" curves. Imagine two intersecting ICs, call them x and y. That means that all points on y have the same utility. A…lso, all the points on x have the same utility, but that number is different from x. (From this we know that X1 = X2 = X3 = X4 etc etc and the same thing with Y1 = Y2 etc etc where the subscripts are points on the curves). We can refer to each curve by its utility value, call them x and y. So being that they are different curves, we expect them to have different utilities. So, U(x) != U(y). In words, the utility of x is never equal to the utility of y. Now imagine a point at which they cross. We would obviously have a point, let's call it A, where this does not hold true. More importantly though, we need to remember that ALL POINTS ON AN INDIFFERENCE CURVE HAVE THE SAME UTILITY. So back to this interesection at point A, we get X1 = X2 = XA. Also, Y1 = Y2=YA where A is the intersection. From this, we transitively know that X1 = X2 = Y1 = Y2. This creates on obvious problem. It would mean that every point on the two separate indifference curves would have the same utility, which is the complete opposite of the first rule of indifference curves, that all points on them have the same utility. (MORE)

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In Economics

The three major characteristics of an indifference curve are: 1. They are negatively sloped 2. They are convex to the origin 3. Indifference curve cannot be intersected

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indifference curve is the graphical representation of the bundles of commodities for a given income level or budget that yields equal satisfaction at all the points.

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o Indifference curves are curves that have a negative slope and are bowed inward. Each point on the line has the same exact util value. In other words, a person would be the… same amount of "happy" at each point on the indifference curve. There are an infinite amount of indifference curves on every graph. G2 (MORE)

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In Economics

Demand curve describes the relationship between the product price and the number of the product demanded through the use of graph. This is also an illustration of demand s…chedule. (MORE)