1 Finding x- and y-intercepts algebraically. 2 y-intercepts For every point along the y-axis, the value of x will be zero (x=0) We can use this fact to find y-intercepts if we have an equation for a function.

3 Graph of f(x) = y = 2x – 3. 4 x-intercepts For every point along the x-axis, the value of y will be zero (x=0) We can use this fact to find x-intercepts if we have an equation for a function.

5 Graph of f(x) = y = -½x2 + 8.

In its essence, mathematics is the study of finding relationships. From planets to atoms.

It makes us capable of understanding, analyzing, and predicting how all the known phenomena in our universe occur. And it’s not like we use some advanced level science for this, we just use … graphs.

We graph the past and present of a phenomenon into simple figures and draw insights to predict future outcomes, and the branch of mathematics that makes this feat possible is known as Coordinate Geometry. Over the course of this article.

The Quadrants. In the cartesian system, the coordinate plane is divided into four equal parts by the intersection of the x-axis (the horizontal number line) and the y-axis (the vertical number line).

These four regions are called quadrants because they each represent one-quarter of the whole coordinate plane. They are denoted by Roman numerals and each of these quadrants have their own properties.

In this quadrant, the x-axis and the y-axis both have positive numbers. Quadrant II: The upper left quadrant is the second quadrant, denoted as Quadrant II.

Quadrant III: The bottom left quadrant is the third quadrant, denoted as Quadrant III. In this quadrant, both the x-axis and the y-axis have negative numbers.

In this quadrant, the x-axis has positive numbers and the y-axis has negative numbers. Note that the quadrants follow a counterclockwise order of naming.

The left and the bottom part of the plane have negative x-axis and negative y-axis for negative integers. The point where the number lines intersect is called the origin.

We already know that any point on the coordinate plane has two aspects: distance from the x-axis and distance from the y-axis. Let’s look at this through an example.

Now start from this point and draw a straight line on the x-axis and another straight line for the y-axis like this: So the location of P on the x-axis is 2 units and the location of P on the y-axis is 3 units.

Every point on the coordinate plane is in the form of the ordered pair (x,y), where x and y are numbers that denote the position of the point with respect to the x-axis and the y-axis respectively. The origin is denoted by (0,0).

By looking at this point, we can see that its x-coordinate is positive and y-coordinate is negative. So this point would lie in Quadrant IV.To plot this point on the coordinate plane, we will follow these steps:

In this case, it is 4.

Step 3: The y-coordinate in (4,-3) is -3, so we will start from this new point and move this point down until it faces -3 on the negative y-axis. That’s all we have to do to plot a point on the coordinate plane.

Identify the quadrants in which each of the following points lie. (i) (1,4).

(iii) (-4,-4). (iv) (-1,8).

(i) Quadrant I, because the x-coordinate and y-coordinate are both positive. (ii) Quadrant IV, because the x-coordinate is positive and y-coordinate is negative.

(iv) Quadrant II, because the x-coordinate is negative and y-coordinate is positive. Example 2.

Answer: In Quadrant III, the coordinates of the x-axis and y-axis are both negative.

Example 3. What quadrant is the origin in.

The x-axis and the y-axis intersect at the origin denoted by (0,0) since both these numbers are non-negative. the origin is said to be a part of Quadrant I.

Attend this Quiz & Test your knowledge. What is a quadrant.

A quadrant is the region formed by the intersection of the x-axis and the y-axis on the coordinate plane. What are the 4 quadrants.

The 4 quadrants are the regions formed by the intersection of the x-axis and the y-axis on the coordinate plane. Their characteristic features are given as follows:

We start from the upper right quadrant and mark that as Quadrant l and move anticlockwise, marking each quadrant with Roman numerals: Quadrant ll, Quadrant lll, Quadrant IV. Where do the four quadrants meet.

The four quadrants meet at the intersection of the x- and y-axis, called the origin. The origin is denoted by (0,0).

Mathematics in ancient days was divided into two branches ‘Algebra’ and ‘Geometry’. Algebraic equations were not used in geometry and geometrical figures were not used in algebra.

He introduced the concept of the Cartesian plane or coordinate system to explain geometry and algebra together. The number line is a straight line where the integers are placed at equal distances.

When two number lines are placed mutually perpendicular to each other it forms coordinate axes. A Cartesian coordinate system or Coordinate system is used to locate the position of any point and that point can be plotted as an ordered pair (x, y) known as Coordinates.

Note: 1.

X-axis is named as XX’ and Y-axis as YY’. Learn: Analytic geometry.

3 below:.

Accordingly, the distance measured along OX will be taken as positive and along OX’ will be negative. Similarly, the distance along OY will be taken as positive and along OY’ will be negative.

II-quadrant (-, +). III-quadrant (-, -).

Consider a point P in a plane.

Thus, for any given point, the abscissa and ordinate are the distance of a given point from the X-axis and Y-axis, respectively. The position of point P is given as (x, y).

Thus, the ordinate or Y-coordinate of every point on the X-axis is zero. Hence, the coordinate of a point on the X-axis is given as (x, 0).

Hence, the coordinates of a point on the Y-axis is given by (0, y). Note:

If x ≠ y, then (x, y) ≠ (y, x) and (x, y) = (y, x), only if x = y. 2.

The most commonly used coordinate systems are namely: As you have learned about the number line and Cartesian coordinate system in the previous section.

Polar Coordinate System: In this type of coordinate system, the points in the plane will be in the form of (r, θ). Learn more on polar coordinates here.

Learn more about the conversion of spherical coordinates into various forms here. Identify the quadrant in which the following points lie.

(-3, 8). 2.

(4, -2). 4.

To learn more maths concepts, please visit our website and download BYJU’S- The Learning App.

1 4-3: Trigonometric Functions of Any Angle What you’ll learn about ■ Trigonometric Functions of Any Angle ■ Trigonometric Functions of Real Numbers ■ Periodic Functions ■ The 16-point unit circle.. and why Extending trigonometric functions beyond triangle ratios.

2 Definitions of Trigonometric Functions of Any Angle. 3 (x, y) y x r opposite adjacent hypotenuse.

5 Let (-3, 4) be a point on the terminal side of Ѳ. Find the sin, cos and tan.

6 Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x – x + y + y – sin Ѳ + sin Ѳ – cos Ѳ – cos Ѳ + tan Ѳ – tan Ѳ +tan Ѳ – csc Ѳ + csc Ѳ – sec Ѳ – sec Ѳ + cot Ѳ – cot Ѳ + cot Ѳ -. 7.

One is positive and one is negative. 9 How do you get a negative.

10 How do you get a negative. One is positive and one is negative.

One is positive and one is negative cos Ѳ + cos Ѳ – cos Ѳ +. 12 How do you get a negative.

13 How do you get a negative. One is positive and one is negative cos Ѳ +.

15 Ranges of Trigonometric Functions We know that If the measure of increases toward 90 o, then y increases The value of y approaches r, and they are equal when So, y cannot be greater than r. Using the convenient point (0,1) y can never be greater than 1.

16 Ranges Continued Using a similar approach, we get:. 17 Determining if a Value is Within the Range Evaluate (calculator) (not possible) (not possible).

19 Definition of a Reference Angle Let Ѳ be an angle in standard position. Its reference angle is the acute angle α formed by the terminal side of Ѳ and the horizontal axis.

20 Find the reference angle for Ѳ=300⁰ Ѳ What quadrant is the terminal side in. α α=360⁰ – 300⁰ α=60⁰ α=360⁰ – Ѳ.

α α=3.14 – 2.3 α≈ 0.8416 α= π – Ѳ. 22 Find the reference angle for Ѳ=-135⁰ Ѳ What quadrant is the terminal side in.

23 Common Trigonometric Functions Ѳ(degrees) 0⁰30⁰45⁰60⁰90⁰180⁰270⁰ Ѳ(radians) sin Ѳ cos Ѳ tan Ѳ 0 π 0 1 0 1 1 0 und 0 0 0 und. 24 Positive Trig Function Values r r r r x-x y y -y ALL STUDENTS TAKE CALCULUS All functions are positive Sine and its reciprocal are positive Tangent and its reciprocal are positive Cosine and its reciprocal are positive.

26 Ѳ What quadrant is the terminal side in. α α= Ѳ – π Is cos positive or negative in quadrant III.

27 Positive Trig Function Values r r r r x-x y y -y ALL STUDENTS TAKE CALCULUS All functions are positive Sine and its reciprocal are positive Tangent and its reciprocal are positive Cosine and its reciprocal are positive.

α Is cos positive or negative in quadrant III.

α α= 180⁰ – 150⁰ α=30⁰ Is tan positive or negative in quadrant II. Find the coterminal angle for -210⁰ coterminal= -210⁰ + 360⁰ coterminal= 150⁰.

32 What quadrant is the terminal side in. Is tan positive or negative in quadrant II.

33 Ѳ What quadrant is the terminal side in. α Is csc positive or negative in quadrant II.

34 Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x – x + y + y – sin Ѳ + sin Ѳ – cos Ѳ – cos Ѳ + tan Ѳ – tan Ѳ +tan Ѳ – csc Ѳ + csc Ѳ – sec Ѳ – sec Ѳ + cot Ѳ – cot Ѳ + cot Ѳ -. 35 What quadrant is the terminal side in.

Ѳ α. 36 Finding Exact Measures of Angles Find all values of Sine is negative in QIII and QIV Using the 30-60-90 values we found earlier, we know.

Our reference angle is 60 o. We must be 60 o off of the closest x-axis in QIII and QIV.

38 Note: there is other way to remember special angle, radian and point of unit circle.

1) Show that the follwing limit does not exist. lim(x,y)-(0,0)(xy+x2)/(x^2+y2).

f(x,y,z)=x2tan-1(y+z). 3) Find the equation of thetangent plane to the surface.

4) Fin the linear approximation tothe function. f(x,y,z)=e-2xy+cos(2yz) at the point (1,-1,2).

z2=x2-y2 at the point (5,3,4). 6) You are standing on a surfacegiven by the equation.

Ifyou are standing at the poin (2,1,0), in which directio is thesteepest slope.

T(x,y,z)=105/(x2+y2+z2). If acomet travels along the path r(t)=(t, t2-4, 2t), howfast is the temperatur changing when t=2.

8) Fin the critical point of thefunction f(x,y)= e4x-e-y. Use the secondderivative test to classifyf them, if possible.

Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville.

The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at (7,0) in the xy-plane, Springfield is at (0,6), and Shelbyville is at (0,-6).

Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer.

1) f(x)= 2) We find that f(x) has a critical number at x=. 3) To verify that f(x) has a minimum at this critical number we compute the second derivative f.

(x) and find that its value at the critical number =_____ , a positive number. 4) Thus the minimum length of cable needed =.

#2 please.

1 Functions of Several Variables Copyright © Cengage Learning. All rights reserved.

2 Extrema of Functions of Two Variables Copyright © Cengage Learning. All rights reserved.

3 3 Find absolute and relative extrema of a function of two variables. Use the Second Partials Test to find relative extrema of a function of two variables.

4 4 Absolute Extrema and Relative Extrema. 5 5 Consider the continuous function f of two variables, defined on a closed bounded region R.

for all (x, y) in R are called the minimum and maximum of f in the region R, as shown in Figure 13.64. Figure 13.64 Absolute Extrema and Relative Extrema.

The Extreme Value Theorem deals with a region in the plane that is both closed and bounded. A region in the plane is called bounded if it is a subregion of a closed disk in the plane.

8 8 A minimum is also called an absolute minimum and a maximum is also called an absolute maximum. As in single-variable calculus, there is a distinction made between absolute extrema and relative extrema.

9 9 Absolute Extrema and Relative Extrema To say that f has a relative maximum at (x 0, y 0 ) means that the point (x 0, y 0, z 0 ) is at least as high as all nearby points on the graph of z = f(x, y). Similarly, f has a relative minimum at (x 0, y 0 ) if (x 0, y 0, z 0 ) is at least as low as all nearby points on the graph.

10 10 Absolute Extrema and Relative Extrema To locate relative extrema of f, you can investigate the points at which the gradient of f is 0 or the points at which one of the partial derivatives does not exist. Such points are called critical points of f.

11 11 Absolute Extrema and Relative Extrema If f is differentiable and then every directional derivative at (x 0, y 0 ) must be 0. This implies that the function has a horizontal tangent plane at the point (x 0, y 0 ), as shown in Figure 13.66.

12 12 Absolute Extrema and Relative Extrema It appears that such a point is a likely location of a relative extremum. This is confirmed by Theorem 13.16.

13 13 Example 1 – Finding a Relative Extremum Determine the relative extrema of f(x, y) = 2x 2 + y 2 + 8x – 6y + 20. Solution: Begin by finding the critical points of f.

14 14 Example 1 – Solution To locate these points, set f x (x, y) and f y (x, y) equal to 0, and solve the equations 4x + 8 = 0and2y – 6 = 0 to obtain the critical point (–2, 3). By completing the square, you can conclude that for all (x, y) ≠ (–2, 3) f(x, y) = 2(x + 2) 2 + (y – 3) 2 + 3 > 3.

cont’d. 15 15 Example 1 – Solution The value of the relative minimum is f(–2, 3) = 3, as shown in Figure 13.67.

16 16 The Second Partials Test. 17 17 The Second Partials Test To find relative extrema you need only examine values of f(x, y) at critical points.

Some critical points yield saddle points, which are neither relative maxima nor relative minima.

Figure 13.69. 19 19 The Second Partials Test At the point (0, 0), both partial derivatives are 0.

So, the point (0, 0, 0) is a saddle point of the surface.

21 21 Example 3 – Using the Second Partials Test Find the relative extrema of f(x, y) = –x 3 + 4xy – 2y 2 + 1. Solution: Begin by finding the critical points of f.

22 22 Example 3 – Solution To locate these points, set f x (x, y) and f y (x, y) equal to 0 to obtain –3x 2 + 4y = 0 and 4x – 4y = 0. From the second equation you know that x = y, and, by substitution into the first equation, you obtain two solutions: y = x = 0 and cont’d.

cont’d. 24 24 Furthermore, for the critical point and because you can conclude that f has a relative maximum at as shown in Figure 13.70.

25 25 The Second Partials Test Absolute extrema of a function can occur in two ways. First, some relative extrema also happen to be absolute extrema.

(On the other hand, the relative maximum found in Example 3 is not an absolute maximum of the function.) Second, absolute extrema can occur at a boundary point of the domain.

In statistics, the Pearson correlation coefficient (PCC)[a] is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations.

As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation).

It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s, and for which the mathematical formula was derived and published by Auguste Bravais in 1844.[b] The naming of the coefficient is thus an example of Stigler’s Law.

The form of the definition involves a “product moment”, that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables. hence the modifier product-moment in the name.

Pearson’s correlation coefficient, when applied to a population, is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. Given a pair of random variables ( X , Y ) {\displaystyle (X,Y)} (for example, Height and Weight), the formula for ρ is.

where. The formula for cov ⁡ ( X , Y ) {\displaystyle \operatorname {cov} (X,Y)} can be expressed in terms of mean and expectation.

the formula for ρ {\displaystyle \rho } can also be written as.

The formula for ρ {\displaystyle \rho } can be expressed in terms of uncentered moments. Since.

Pearson’s correlation coefficient, when applied to a sample, is commonly represented by r x y {\displaystyle r_{xy}} and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for r x y {\displaystyle r_{xy}} by substituting estimates of the covariances and variances based on a sample into the formula above.

where. Rearranging gives us this formula for r x y {\displaystyle r_{xy}} :.

This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be numerically unstable.

where n , x i , y i , x ¯ , y ¯ {\displaystyle n,x_{i},y_{i},{\bar {x}},{\bar {y}}} are defined as above.

where. Alternative formulae for r x y {\displaystyle r_{xy}} are also available.

where. If ( X , Y ) {\displaystyle (X,Y)} is jointly gaussian, with mean zero and variance Σ {\displaystyle \Sigma } , then Σ = [ σ X 2 ρ X , Y σ X σ Y ρ X , Y σ X σ Y σ Y 2 ] {\displaystyle \Sigma ={\begin{bmatrix}\sigma _{X}^{2}&\rho _{X,Y}\sigma _{X}\sigma _{Y}\\\rho _{X,Y}\sigma _{X}\sigma _{Y}&\sigma _{Y}^{2}\\\end{bmatrix}}}.

Under heavy noise conditions, extracting the correlation coefficient between two sets of stochastic variables is nontrivial, in particular where Canonical Correlation Analysis reports degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.

In case of missing data, Garren derived the maximum likelihood estimator.

The values of both the sample and population Pearson correlation coefficients are on or between −1 and 1. Correlations equal to +1 or −1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely supported on a line (in the case of the population correlation).

A key mathematical property of the Pearson correlation coefficient is that it is invariant under separate changes in location and scale in the two variables. That is, we may transform X to a + bX and transform Y to c + dY, where a, b, c, and d are constants with b, d > 0, without changing the correlation coefficient.

The correlation coefficient ranges from −1 to 1. An absolute value of exactly 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line.

A value of 0 implies that there is no linear dependency between the variables.

Thus the correlation coefficient is positive if Xi and Yi tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative (anti-correlation) if Xi and Yi tend to lie on opposite sides of their respective means.

Rodgers and Nicewander cataloged thirteen ways of interpreting correlation or simple functions of it:. For uncentered data, there is a relation between the correlation coefficient and the angle φ between the two regression lines, y = gX(x) and x = gY(y), obtained by regressing y on x and x on y respectively.

For centered data (i.e., data which have been shifted by the sample means of their respective variables so as to have an average of zero for each variable), the correlation coefficient can also be viewed as the cosine of the angle θ between the two observed vectors in N-dimensional space (for N observations of each variable).

Both the uncentered (non-Pearson-compliant) and centered correlation coefficients can be determined for a dataset. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively.

Then let x and y be ordered 5-element vectors containing the above data: x = (1, 2, 3, 5, 8) and y = (0.11, 0.12, 0.13, 0.15, 0.18).

This uncentered correlation coefficient is identical with the cosine similarity. The above data were deliberately chosen to be perfectly correlated: y = 0.10 + 0.01 x.

Centering the data (shifting x by ℰ(x) = 3.8 and y by ℰ(y) = 0.138) yields x = (−2.8, −1.8, −0.8, 1.2, 4.2) and y = (−0.028, −0.018, −0.008, 0.012, 0.042), from which. as expected.

Several authors have offered guidelines for the interpretation of a correlation coefficient. However, all such criteria are in some ways arbitrary.

A correlation of 0.8 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences, where there may be a greater contribution from complicating factors.

Methods of achieving one or both of these aims are discussed below.

A permutation test for Pearson’s correlation coefficient involves the following two steps:. To perform the permutation test, repeat steps (1) and (2) a large number of times.

Here “larger” can mean either that the value is larger in magnitude, or larger in signed value, depending on whether a two-sided or one-sided test is desired.

Free carrier is a trade term dictating that a seller of goods is responsible for the delivery of those goods to a destination specified by the buyer. When used in trade, the word “free” means the seller has an obligation to deliver goods to a named place for transfer to a carrier.

It might even be the seller’s business location.

At this point, the buyer assumes all responsibility.

Buyers and sellers engaged in economic trade requiring the shipment of goods can use FCA shipping terms to describe any transportation point, regardless of the number of transportation modes involved in the shipping process. The point must be a location within the seller’s home country, however.

The carrier can be any kind of transportation service, such as a truck, train, boat, or airplane.

The seller is only responsible for delivery to the specified destination as part of the liability transfer. It isn’t obligated to unload the goods, but the seller might be responsible for ensuring that the goods have been cleared for export out of the United States if the destination is the seller’s premises.

Under FCA shipping terms, the buyer doesn’t have to deal with export details and licenses because this is the responsibility of the seller. The buyer must arrange for transport, however.

Many experts recommend that any party involved in international trade consult with an appropriate legal professional—such as a trade attorney—before using any trade term within a contract. Contracts involving international transportation often contain abbreviated trade terms, or terms of sale, that describe shipment specifics.

To help facilitate the delivery of such items, the most commonly known trade terms are international commercial terms or Incoterms. Incoterms are internationally recognized standards published by the International Chamber of Commerce (ICC).

The term ‘free carrier’ or FCA is a typical and highly-used example of Incoterms. It has been internationally recognized as a standard set of instructions to designate delivery terms.

Established by the International Chamber of Commerce, the Incoterms rules may be purchased via the ICC’s website. Under FCA shipping terms, the seller delivers the goods to the destination named by the buyer.

The buyer would be responsible for loading the goods for transport.

Bob opts to use his shipper with whom he’s done business before. Joe agrees, and it’s his responsibility to deliver the goods to the shipper.

FCA and FOB are shipment terms used in different types of transportation. FOB delivery applies only to sea shipments and occurs when cargo is loaded onto a vessel.

Under FCA, many more types of transport are allowable. The supplier is usually obligated to issue an export declaration once goods have been placed onto a buyer’s vehicle.

In addition, the vendor usually holds all risks and responsibilities for the transportation of the goods until the buyer receives them. FCA shipping terms are usually paid for by the buyer since the carrier is nominated by the buyer.

Under FCA shipping terms, the seller is responsible for export duty, taxes, and custom clearance. The buyer is responsible for importing items.

The seller is also responsible for export packaging, licenses, and customs formalities. On the other hand, under FCA shipments, the buyer pays for the goods, is responsible for the main means of transportation, and pays for loading charges.

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1 8.5 Series Solutions Near a Regular Singular Point, Part IWe now consider the question of solving the general second order linear equation P(x)y” + Q(x)y’ + R(x)y = (1) in the neighborhood of a regular singular point x = x0. For convenience we assume that x0 = 0.

We assume that where a0 ≠ 0. In other words, r is the exponent of the first term in the series, and a0 is its coefficient.

2 How to proceed As part of the solution, we have to determine:1. The values of r for which Eq.

The recurrence relation for the coefficients an. 3.

3 Example Solve the differential equation 2×2 y’ − xy’ + (1 + x)y = 0.Answer: 1. Find the indicial equation for diff.

Find roots of the indicial equation called the exponents at the singularity for the regular singular point x = 0. 3.

4 8.6 Series Solutions Near a Regular Singular Point, Part IIConsider the general problem of determining a solution of the equation (1) Where (2) and both series converge in an interval |x| < ρ for some ρ > 0. The point x = 0 is a regular singular point, and the corresponding Cauchy– Euler equation (3) is.

We seek a solution of Eq. (1) for x > 0 and assume that it has the form (4) where a0 ≠ 0, and we have written y = φ(r, x) to emphasize that φ depends on r as well as x.

6 Indicial Equation The equation F(r ) = 0 is called the indicial equation. The roots r1 and r2 of F(r)=0 are called the exponents at the singularity.

Setting the coefficient of xr+n in Eq. (6) equal to zero gives the recurrence relation (8).

(1) in the form (4) We can also obtain a second solution Example. 8 Equal Roots We can always determine one solution of Eq.

9 Roots Differing by an IntegerTHEOREM 8.6.1 Consider the differential equation (1) x2y” + x[xp(x)]y’ + [x2q(x)]y = 0, where x = 0 is a regular singular point. Then xp(x) and x2q(x) are analytic at x = 0 with convergent power series expansions for |x| < ρ, where ρ > 0 is the minimum of the radii of convergence of the power series for xp(x) and x2q(x).

10 THEOREM (Ctd.) Then in either the interval −ρ < x < 0 or the interval 0 < x < ρ, there exists a solution of the form (21) where the an(r1) are given by the recurrence relation (8) with a0 = 1 and r = r1.

(21) and (22) converge at least for |x| < ρ.

The constant a may turn out to be zero, in which case there is no logarithmic term in the solution (24). Each of the series in Eqs.

In all three cases the two solutions y1(x) and y2(x) forma fundamental set of solutions of the given differential equation.

x = 0 is a regular singular point of Eq. (1).

We will consider the three cases ν = 0, ν = 1/2, and ν = 1 for the interval x > 0.

(1) reduces to L[ y] = x2y” + xy’ + x2 y = 0, (2) and the roots of the indicial equation are equal: r1 = r2 = 0. The first solution is The function in brackets is known as the Bessel function of the first kind of order zero and is denoted by J0(x).

14 Bessel Equation of Order ZeroThe second solution of the Bessel equation of order zero is Instead of y2, the second solution is usually taken to be a certain linear combination of J0 and y2. It is known as the Bessel function of the second kind of order zero and is denoted by Y0.

15 FIGURE 8.7.1 The Bessel functions J0 and Y0.

17 Bessel Equation of Order One-HalfIllustrates the situation in which the roots of the indicial equation differ by a positive integer but there is no logarithmic term in the second solution. Setting ν = ½ in Eq.

The roots of the indicial equation are r1 = ½, r2=− ½.

The Bessel function of the first kind of order one-half, J1/2, is defined as (2/π)1/2 y1.

where. 20 The Bessel functions J1/2 and J−1/2.

21 Bessel Equation of Order OneThis case illustrates the situation in which the roots of the indicial equation differ by a positive integer and the second solution involves a logarithmic term. Setting ν = 1 in Eq.

22 Bessel Equation of Order OneWe get The general solution for x > 0 is y = c1 J1(x) + c2Y1 (x).

24 Numerical Evaluation of Bessel Functionswe have shown how to obtain infinite series solutions of Bessel’s equation of orders zero, one-half, and one. In applications it is not unusual to require Bessel functions of other orders.

25 Chapter Summary. 26 Sections 8.2 and 8.3 Series Solutions Near an Ordinary Point, Parts I and II.

28 Sections 8.5 and 8.6 Series Solutions Near a Regular Singular Point, Parts I and II. 29 8.5, 8.6 (Ctd.).

Q: Question 11 For a random sample of 100 students the average textbook and supplies cost was $620 for one semester. Calculate a 99% confidence interval for the true mean cost for a semester’s materials.

Posted 3 days ago View Answer ► Q: Statistical Abstracts (117th edition) reports that the average amount spent annually for food by householders under 25 years of age is $2,690. A random sample of 16 people under 25 years of age who live in a university neighborhood were surveyed.

Posted 9 days ago View Answer ► Q: 1. The rates offered by a bank on deposits between $10,000 and $24,999 are shown in the following table: Term Rate 180 to 269 days 3.15% 270 to 364 days 3.45% How much more will an investor earn from a $10,000 investment in a 364-day GIC than from..

One variable in the table is the percentage of students at the university who responded strongly agree. The other variable in the table is the..

A data model is an abstract model that organizes elements of data and standardizes how they relate to one another and to the properties of real-world entities. Posted 17 days ago Q: The table below is from a regression analysis of y = number of children in family, x1 = mother’s educational level in years (MEDUC), and x2 = father’s socioeconomic status (FSES), for a random sample of 49 college students at Texas A&M University..

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Use the following information for this question: Women n =476 avg =33 sd =21.9. Men n =496 avg =19.9 sd =14.6.

Posted 9 days ago View Answer ►. Q: Question 11 For a random sample of 100 students the average textbook and supplies cost was $620 for one semester.

Assume a population standard deviation of $25..

Question 11 For a random sample of 100 students the average textbook and supplies cost was $620 for one semester. Calculate a 99% confidence interval for the true mean cost for a semester’s materials.

View Answer ►. Q: Statistical Abstracts (117th edition) reports that the average amount spent annually for food by householders under 25 years of age is $2,690.

The..

Statistical Abstracts (117th edition) reports that the average amount spent annually for food by householders under 25 years of age is $2,690. A random sample of 16 people under 25 years of age who live in a university neighborhood were surveyed.

View Answer ►. Q: 1.

The rates offered by a bank on deposits between $10,000 and $24,999 are shown in the following table: Term Rate 180 to 269 days 3.15% 270 to 364 days 3.45% How much more will an investor earn from a $10,000 investment in a 364-day GIC than from..

Q: Students at a number of universities were asked if they agreed that their education was worth the cost. One variable in the table is the percentage of students at the university who responded strongly agree.

Students at a number of universities were asked if they agreed that their education was worth the cost. One variable in the table is the percentage of students at the university who responded strongly agree.

View Answer ►. Q: What is a data model in math.

A data model is an abstract model that organizes elements of data and standardizes how they relate to one another and to the properties of real-world entities.

What is a data model in math.

Q: The table below is from a regression analysis of y = number of children in family, x1 = mother’s educational level in years (MEDUC), and x2 = father’s socioeconomic status (FSES), for a random sample of 49 college students at Texas A&M University..

The table below is from a regression analysis of y = number of children in family, x1 = mother’s educational level in years (MEDUC), and x2 = father’s socioeconomic status (FSES), for a random sample of 49 college students at Texas A&M University..

Q: For the next 2 questions Suppose the lead time for Part A is 2 weeks and the on-hand inventory is 28. The lot sizing rule is FOP with P = 2.

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For the next 2 questions Suppose the lead time for Part A is 2 weeks and the on-hand inventory is 28. The lot sizing rule is FOP with P = 2.

Part A 4 5 6 7 8..

Q: For a dental insurance policy, three types of claims are submitted: preventative care, minor, and major. The proportions of claims of each type are 0.50, 0.30, and 0.20 respectively.

For a dental insurance policy, three types of claims are submitted: preventative care, minor, and major. The proportions of claims of each type are 0.50, 0.30, and 0.20 respectively.

View Answer ►. Q: The mean daily sales are $2000 for a sample of 40 days at a fast-food restaurant.

a) What is the estimated mean daily sales of the population. What is this estimate called.

Adverse selection refers generally to a situation in which sellers have information that buyers do not have, or vice versa, about some aspect of product quality. In other words, it is a case where asymmetric information is exploited.

Typically, the more knowledgeable party is the seller. Symmetric information is when both parties have equal knowledge.

In the case of insurance, adverse selection is the tendency of those in dangerous jobs or high-risk lifestyles to purchase products like life insurance. In these cases, it is the buyer who actually has more knowledge (i.e., about their health).

Investopedia / Jiaqi Zhou. Adverse selection occurs when one party in a negotiation has relevant information the other party lacks.

In the case of insurance, avoiding adverse selection requires identifying groups of people more at risk than the general population and charging them more money. For example, life insurance companies go through underwriting when evaluating whether to give an applicant a policy and what premium to charge.

Underwriters typically evaluate an applicant’s height, weight, current health, medical history, family history, occupation, hobbies, driving record, and lifestyle risks such as smoking. all these issues impact an applicant’s health and the company’s potential for paying a claim.

A seller may have better information than a buyer about products and services being offered, putting the buyer at a disadvantage in the transaction. For example, a company’s managers may more willingly issue shares when they know the share price is overvalued compared to the real value.

In the secondhand car market, a seller may know about a vehicle’s defect and charge the buyer more without disclosing the issue.

This can also lower consumption as buyers may be wary of the quality of the products that are offered for sale. Or, it may exclude certain consumers that do not have access to or cannot afford to obtain information that could lead them to make better buying decisions.

One indirect effect of this is a negative impact on consumers’ health and well-being. If you buy a faulty product or dangerous medication because you don’t have good information, consuming these products can cause physical harm.

Because of adverse selection, insurers find that high-risk people are more willing to take out and pay greater premiums for policies. If the company charges an average price but only high-risk consumers buy, the company takes a financial loss by paying out more benefits or claims.

However, by increasing premiums for high-risk policyholders, the company has more money with which to pay those benefits. For example, a life insurance company charges higher premiums for race car drivers.

A health insurance company charges higher premiums for customers who smoke. In contrast, customers who do not engage in risky behaviors are less likely to pay for insurance due to increasing policy costs.

A prime example of adverse selection in regard to life or health insurance coverage is a smoker who successfully manages to obtain insurance coverage as a nonsmoker. Smoking is a key identified risk factor for life insurance or health insurance, so a smoker must pay higher premiums to obtain the same coverage level as a nonsmoker.

Another example of adverse selection in the case of auto insurance would be a situation where the applicant obtains insurance coverage based on providing a residence address in an area with a very low crime rate when the applicant actually lives in an area with a very high crime rate.

Adverse selection might occur on a smaller scale if an applicant states that the vehicle is parked in a garage every night when it is actually parked on a busy street. Adverse selection by increasing access to information, thus minimizing asymmetries.

Crowdsourced information in the form of user reviews along with more formal reviews by bloggers or specialist websites are often free and warn potential buyers about otherwise obscure issues around quality.

Laws and regulations can also help, such as Lemon Laws in the used car industry. Federal regulatory authorities such as the Food and Drug Administration (FDA) also help ensure that products are safe and effective for consumers.

Insurers reduce adverse selection by requesting medical information from applicants in the form of requiring paramedical examinations, querying doctors’ offices for medical records, and looking at one’s family history. This gives the insurance company more information that an applicant may fail to disclose on their own.

Like adverse selection, moral hazard occurs when there is asymmetric information between two parties, but where a change in the behavior of one party is exposed after a deal is struck. Adverse selection occurs when there’s a lack of symmetric information prior to a deal between a buyer and a seller.

Moral hazard is the risk that one party has not entered into the contract in good faith or has provided false details about its assets, liabilities, or credit capacity. For instance, in the investment banking sector, it may become known that government regulatory bodies will bail out failing banks.

The lemons problem refers to issues that arise regarding the value of an investment or product due to asymmetric information possessed by the buyer and the seller.

Akerlof, an economist and professor at the University of California, Berkeley. The tag phrase identifying the problem came from the example of used cars Akerlof used to illustrate the concept of asymmetric information, as defective used cars are commonly referred to as lemons.

The lemons problem exists in the marketplace for both consumer and business products, and also in the arena of investing, related to the disparity in the perceived value of an investment between buyers and sellers. The lemons problem is also prevalent in financial sector areas, including insurance and credit markets.

“Adverse” means unfavorable or harmful. Adverse selection is therefore when certain groups are at higher-risk because they lack full information.

Adverse selection arises from information asymmetries. In economic theory, markets are assumed to be efficient and that everybody has full and “perfect” information.

This creates market inefficiencies that can increase prices or prevent transactions from occurring. In stock markets, there are some natural information asymmetries.

This can lead to cases of insider trading, where those in-the-know profit from stock trades before public announcements are made (which is an illegal practice). Another asymmetry involves the inventories of market makers and some institutional traders.

This means that these players in the market may have a particular “axe to grind” – for example, a strong desire or need to buy or sell – that is not known by the investing public. Contrary to assumptions made by mainstream economic and financial models, information is not symmetrically accessible and available to all actors in a market.