Give what equation is called the equation of this line. Line equation. Vector line equation

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ANALYTIC GEOMETRY

Lecture No. 7. Topic 1 : Lines on a plane and their equations

1.1. Lines and their equations in the Cartesian coordinate system

IN analytical geometry lines on a plane are considered as the geometric locus of points (g.m.t.) that have the same property common to all points of the line.

Definition. Line equation
is an equation with two variablesX And at, which is satisfied by the coordinates of any point on the line and is not satisfied by the coordinates of any other point not lying on this line.

The opposite is also true, i.e. any equationat

form, generally speaking, in Cartesian

coordinate system (DSC) defines the line

as g.m.t., whose coordinates satisfy

this equation. ABOUT X

Note 1. Not every equation of the form defines a line. For example, for the equation
there are no points whose coordinates would satisfy this equation. We will not consider such cases further.This is the case of so-called imaginary lines.

P example 1.Write an equation for a circle with radiusR centered at a point
.

For any point lyingatM

on a circle, by definitionR

circles as g.m.t., equidistant

from the point, we get the equationX

1.2. Parametric equations of lines

There is another way to define a line on a plane using equations calledparametric:

Example 1. The line is set parametric equations

It is required to obtain the equation of this line in DSC.

Let's exclude the parametert . To do this, let’s square both sides of these equations and add

Example 2. The line is given by parametric equations


A

It is required to obtain the equation

this line in DSK. —a a

Let's do the same thing, then we get

— A

Note 2. It should be noted that the parametert in mechanics is time.

1.3. Equation of a line in a polar coordinate system

DSC is not the only way determine the position of a point and, therefore, specify the equation of the line. On a plane it is often advisable to use the so-called polar coordinate system (PCS).

P The CS will be determined if you specify a point O – pole and beam OR emanating from this point, which is called the polar axis. Then the position of any point is determined by two numbers: the polar radius
and polar angle – angle between

polar axis and polar radius.

Positive reference direction

polar angle from the polar axis

counted counterclockwise.

For all points of the plane
, O R

and for unambiguity of the polar angle it is considered
.

If the beginning of DSC is combined with

pole, and the O axis X send by

polar axis, it is easy to verifyat

in connection between polar and

Cartesian coordinates:


ABOUT X R

Back,

(1)

If the equation of a line in DSC has the form , then in PSC - Then from this equation we can obtain an equation in the form

Example 3. Write an equation for a circle in UCS if the center of the circle is at the pole.

Using transition formulas (1) from DSC to PSC, we obtain

P example 4.Write the equation of a circle,

if the pole is on the circle and the polar axisat

passes through the diameter.

Let's do the same

O 2 R X

R

This equation can be obtained also

from geometric concepts (see figure).

P example 5.Plot a line

Let's move on to the PSK. The equation

will take the form
ABOUT

Let's build a line graph withA

taking into account its symmetry and ODZ

Features:

This line is calledBernoulli's lemniscate.

1.4. Coordinate system transformation.

Line equation in new system coordinates

1. Parallel transfer of DSC.at

Consider two DSCs havingM

the same direction of the axes, but

different origins.

In the coordinate system ABOUT xy dot

relative to the system
ABOUT X

has coordinates
. Then we have

And

In coordinate form, the resulting vector equality has the form

or
. (2)

Formulas (2) are formulas for the transition from the “old” coordinate system ABOUT xyto the “new” coordinate system and vice versa.

Example 5. Obtain the equation of a circle by performing a parallel translation of the coordinate systemto the center of the circle.

AND formulas (2) imply
at ABOUT

equations where endogenous variables are on the left and only exogenous ones are on the right

149. Indirect method least squares involves performing the following procedures:

The outgoing structure of the systems of equations is transformed to a system of reduced equations and, using least squares, we find unbiased estimates of the coefficients of the reduced system of equations. We use the relationship between the coefficients given in the system of equations and the structural system to find the coefficients of the structural system of equations.

150. The identified system of simultaneous equations has the number of coefficients:

the number of coefficients of the reduced system of equations is equal to the number of coefficients of the original structural system of equations

151. An unidentifiable system of simultaneous equations has the number of coefficients:

the number of coefficients of the reduced system of equations is less than the number of coefficients of the structural system of equations

152. An overidentified system of simultaneous equations has the number of coefficients:

number of coefficients of the reduced system of equations more number coefficients of the structural system of equations

In dynamic intersectoral balance model, a system of linear heterogeneous differential equations with i=1,2,3,….n(row numbers),j=1,2,3….n(column numbers) аij-technological coefficients, -incremental capital intensity coefficients has the form ..ANSWER: less Vol.

In dynamic intersectoral balance model system of linear inhomogeneous differential equations with ; technological coefficients incremental capital intensity coefficients The gross product of the industry The final product of the industry has the form: (
).

In dynamic interindustry balance model matrix column coefficients of incremental capital intensity shows for j industry: the size and structure of funds necessary to increase production capacity by 1 unit of its production capacity, i.e. product release.

In the classic model market economy money supply M=20,000, money manages to make 5 revolutions per year, the GDP value is 100,000. What is the established price of a unit of GDP? 1.



In classical market economy model, the supply of goods is determined

In classicalmarket economy model, the supply of goods is determined -employment level

in the labor market Y=Y(L), supply of goods = demand for goods.

In classicalmodel of a market economy with the same GDP, an increase in the supply of money will lead to - (price and GDP) -price increase, if for a given GDP the price p is less than p0 then there is an excess supply of money . In this case, it is believed that prices will increase to the level p0.

In classical model of a market economy, the production function has the form X t =K t 0.5 ´L t 0.5 K=200 units, L=50 units. What is the real wage at maximum profit? 1 or 2.

In classical market economy models with rising interest rates: consumer demand decreases and investment demand decreases.

In cross-industry balance sheet (Leontiev's static model) the statement is true. ANSWER: in the economic system are produced, consumed, invested. Each industry is pure, that is, it produces only 1 product; during the production process, the industry transforms some types of products into another type, and the ratio of products spent and produced by the industry into another type, and the ratio of products spent and products produced by the industry is constant, the final demand is from final consumption, exports and investment.

In cross-industrybalance sheet (Leontiev's static model) the statement is true.0

IntersectorallyIn the balance sheet as a whole for the economy, the value of internal production consumption = 5000 units, the total final product = 3000 units. ...3000What is the ORP?8000.

In cross-industryin the balance sheet as a whole for the economy, the value of internal production consumption = 7000 units, the total final product = 3000 units. Total volume of net production = 3000...What is the ORP?10000.

In cross-industry balance sheet sum of final products and sum of conditionally net products: equal to each other.

In cross-industryThe balance sheet of conditionally pure products includes:depreciation, wages, net income.

In Keynes's model demand for goods is determined by consumer demand and investment demand. Which statement, according to Keynes's model, will be true: When interest rates rise, consumer demand rises and investment demand falls.

In Keynes's model demand for goods is determined by consumer demand and investment demand. Which statement, according to Keynes' model, will be true... ANSWER: The demand for consumer goods grows linearly with an increase in the supply of goods, the demand for investment goods decreases linearly with an increase in the interest rate.

In the modelAccording to Keynes, the demand for goods is determined by consumer demand and investment demand.

In the model of R. Solow, expressed in relative units, the main macroeconomic indicators relate to: basic values, for example, the value of the indicator at the beginning of the period under study X(t), C(t),L(t), I(t), K(t).

In the Solow modelto reach a stationary development trajectory it is enoughso that stock-weapons.=0onst.

In the model Solow, the change in the number of people employed in production L(t) can be described by a differential equation of the form , where g is the growth rate of the number of employed. In this case, the magnitude of the number is equal to: answer : L(t)=L(0)*e g * t .

In the Solow model , where g is the growth rate of the number of employed. In this case, the number of employees L(t) is equal to: answer: L(t)= .

In the Solow modelthe change in the number of people employed in production can be described by a differential equation of the form , where g is the growth rate of the number of employed. In this case, the number of employees L(t) is equal to:L(t)=L(0)*e g * t .

In the Solow model the process changes in fixed production assets over time can be described by a differential equation using the notation: K(t) is the cost of fixed production assets at a point in time; m is the retirement rate of funds; I(t) is the volume of gross investments at time t: ANSWER: dK(t)/dt= -m*K(t)+I(t).

In the Solow model, expressed in relative units, the main macroeconomic indicators relate to... basic.

In the Sol modelou, written in relative units, the value of average per capita consumption depends on the rate of accumulation...at what value of phi the maximum is reached...α.

In the model Harrod-Domar labor costs for production are considered: constant over time, or output does not depend on labor inputs.

In the modelThe Harrord-Domar continuous growth rate of income is equal to if where B is the coefficient of incremental capital intensity; С(t) - volume of consumption; Y(t) - volume of income; In which case will it be maximum, and in which case will it be zero, if C(t)-const:the maximum is reached at

In model X-D labor costs for production are considered:Constant in time, or release.

In the model Evans, the demand for a product is dependent , and the product supply , where is the price of the product, parameters of the equations that are positive numbers. In this case:. (a= > < ).

In a cobweb is an increasing function of the price. In this case, the iterative process of searching for the equilibrium price can be displayed as a recurrence relation: lim f(p)=¥ pÞ0;Lim f(p)=0 pÞ¥;Limj(p)=0 pÞ0; Limj(p)= ¥; pÞ¥;.

In a cobweb models of the aggregate demand function a decreasing function of price, and the function of aggregate supply is an increasing function of price. In this case, the iterative process of searching for the equilibrium price can be displayed as a recurrence relation Ф(р t)=y(p t -1).

In productionfunctions of the form X=A*e*K*L, the time factor is a substitute variable reflecting influence...Scientific and technical progress.

In production functions of the form: X t =A 0 ´e pt ´K t a 1 ´L t a 2 , the time factor is a replacement variable reflecting the impact on gross output: scientific and technological progress .

In staticLeontiev model (input-industry balance) the statement is true...0

Magnitude where I is the consumer’s income, p1p2 is the price of the good, x2 is the quantity of the 2nd good. In this case there are benefits one and two:interchangeable.

Choose the right one statement consistent with Keynesian theory of market economics 1) the general case of equilibrium in a market economy in the presence of unemployment, and full employment is only a special case; 2) investment demand decreases with increasing interest rates.

Select rightsstrong statements, the implementation of which increases the reliability and accuracy of determining the parameters of the economic and mathematical model. 1. The accepted methodology for determining the parameters of the model must be correct from the point of view of ensuring reliability, 2. There must be a sufficient amount of initial information about the input and output indicators of the object to find the mathematical model, 3. the vector of input indicators must vary greatly over the studied interval, 4. Accepted a priori, the model must reflect in a significant way the actual patterns of the object being studied.

Sample equationie pairwise regression y=-3+2x, then the sample pairwise correlation coefficient can be equal to..(-3,2,0.6,-2,-0.6) ...0.7 or 0.6.

Selectively The pairwise regression equation has the form y=-3+2x. Then the sample pair correlation coefficient can be equal to: 0,7.

where in - coefficient of incremental capital intensity; C(t) - volume of consumption; Y(t) - volume of income; In which case will it be maximum and in which case will it be zero if C(t) = const: the maximum is reached at , and is equal to zero at Y(0)=C(0).

Hypotheses, used in deriving the labor demand function in the classical model of a market economy: Firms are completely competitive in supplying goods and hiring labor. Other things being equal, the marginal product of labor decreases as the use of labor increases.

Functions given demand and offers S=2p+1.5, where p is the price of the product. then the equilibrium price is . ANSWER: x1= 0.34+0.18+340.....x2=0;25+0.53+280.

Functions givendemand and offers S=2p+1.5, where p is the price of the product. then equilibrium price =1 .

Functions given demand and offers S=2p+1.5, where p is the price of the product. then equilibrium price = 5,5.

Functions given demand q=(p+6)/(p+2) and supply s=2p-2, where p is the price of the product. Then the equilibrium price is: 2.

Functions are givendemand q=p+6/p+2 and previous s=2p-2…..2.

If savedequal conditions, then with an increase in price the demand for Giffin goods: ...growing.

If in the modelSolow to take into account the investment lag in the form of a concentrated lag, then the connection between investments I(t) and the input of funds V(t) can be reflected in the form of the equation...V(t)= I(t-t)().

If from grossdomestic product subtract depreciation charges, we get:newly created value (N.D.).

If from gross domestic product subtract depreciation charges, we get: pure domestic product.

If cross price elasticity coefficient of demand >0, then...(I product replaces j).

If the production functiony=f(x 1;x 2), then the property means that with an increase in the use of one resource, the marginal efficiency¶ 2 f(x i)/¶x 1 ¶x 2 ³0.

If production function is a homogeneous function of degree p > 0, then with p = 2 and an increase in the scale of production by 3 times, how many times does the volume of output increase... 9.

If productionfunction is a homogeneous function of degree p > 0, then with p = 2 and an increase in the scale of production by 4 times, how many times does the volume of output increase...16.

If it happens an increase in consumer income, then demand moves (indicate the correct statement): from goods with low elasticity to goods with high elasticity. The volume of consumption of goods with low elasticity is reduced.

If the PF has view y=f(x 1 ;x 2), the property meaning that with an increase in the use of one resource, the marginal efficiency of another resource increases, expressed by the formula: ¶ 2 f(x i)/¶x 1 ¶x 2 ³0.

If saved equal conditions, then with an increase in price the demand for Giffin goods: growing.

Dependency between production costs and the volume of production is expressed by the function are equal: 3.

Dependency mbetween production costs and production volume is expressed by the function .Then the marginal cost of production is are equal:23.

Addictionbetween production costs C and production volume Q is expressed by the function . Then marginal costs for production volume Q = 10 are equal to: .. 3 .

Dependency between the cost of production C and the volume of production Q is expressed as C = 20-0.5*Q. Then the elasticity c/c with production volume Q=10 is equal to: -1/3.

Defined productionfunction of the form: Y=3 K 0.5 *L 0.5 then the average product of labor is equal at K=25,L=100……1.5.

The task of consumerchoice is:Find the number of goods from a given set at which the consumer's utility is maximal.

The task consumer choice is: The problem is to choose a consumer bundle (x, x) that maximizes the utility function under a given budget constraint.

The task of consumer choice is: find the number of goods from a given set that maximizes the consumer's utility function.

Law of Diminishing production efficiency is characterized by the fact that with an increase in the amount of resource used... ANSWER: min possible output volume .

Law of Diminishing production efficiency is characterized by the fact that with an increase in the amount of resource used: Each additional unit of resource gives a smaller and smaller increase in output.

Law of Diminishing production efficiency is characterized by the fact that with an increase in the amount of resource used.. ANSWER: the maximum possible volume of output (y) is increasing.

From Eq. Slutsky can be obtained ( quantity product, price goods). This corresponds to: (multiple answers possible): 1) Giffin product, 2) low-value product.

What are the hypotheses? are used to derive the labor demand function in the classical model of a market economy: firms are fully competitive when offering goods and hiring labor; other things being equal, the product of labor decreases as the slave force increases.

What additional sfalsities make it difficult to construct an EMM.... the difficulty of conducting an active experiment in economics. In addition, virtually every economic object or process is unique, which makes it impossible to simply replicate once constructed models.

What practicalproblems are solved using EMM. 1. Analysis of economic objects and processes 2. Economic forecasting and anticipation of the development of economic processes 3. Development of management decisions at all levels of the economy.

What a statementcorresponds to the solution to the gray translucent box problem: There is information about input and output indicators, and a model of a certain structure is known or accepted as a base. The identification task in this case is to find the parameters of this model.

What a statement corresponds to the solution to the gray box problem: In addition to the input and output parameters, the operating system of the converter is specified. be reduced to certain parm-th pages.

What a statement, according to Keynes's model, will be true:When interest rates rise, consumer demand rises and investment demand falls(Demand for consumer goods grows linearly with an increase in the supply of goods, Demand for investment goods decreases linearly with an increase in the interest rate).

Final product in a dynamic balance model compared to the final product in a static balance model does not include export.

Final product in a dynamic balance model compared to the final product in a static balance model does not include: intersectoral capital investments.

Coefficient price elasticity of demand E ii p<-1. Это соответствует товару с: high elasticity of demand.

Macroeconomic equilibrium models are considered to be those which describe a state of the economy when the resultant of all forces tending to bring the economy out of this state is equal to 0.

Leontiev model(static balance) includes an equation of the form: x i -Sa ij =y j .

Inter-industry modelbalance sheet for manufactured products of volume X1 and X2 with a matrix of direct cost coefficients and the final product in a volume of 340 and 280 units, respectively, has the form: x 1 =0.34x 1 +0.18x 2 +340; x 2 =0.25x 2 +0.53x 2 +280..

Törnqvist model n "demand-income" type (other letters): answer : luxury goods (group 2).

Törnqvist model, “demand-income” of the form Y=a 3 Z(Z-b 3)/Z+C 3:luxury items.

Harrod-Domar model in the form of a differential equation
has the following solution: ).

On an isoquant Cobb-Douglas production function:

On line

On line indifference consumer kits have: same values ANSWER: V(t)= I(t-τ).

At the productionCobb-Douglas functions on isoquant: combinations of values ​​of capital and labor are shown, providing the same output.

Along the linethe indifference consumer set has:the same level of satisfaction of the individual's needs.

As you increase income demand moves (indicate the correct statement): ANSWER: As income increases, demand moves from goods of the first and second groups to goods of the third and fourth groups, while the consumption of goods of the first group decreases in absolute terms.

As you increaseincome demand moves (indicate the correct statement): From goods with low elasticity to goods with high elasticity. The volume of consumption of goods with low elasticity is reduced.

Utility Limit1st product u = 8 and 2nd product u = 2. by how much should an individual increase the consumption of 2 products if he reduced the consumption of the first product by one unit...4.

Marginal utilities first product , and the second product . By how much should an individual increase the consumption of the 2nd product if he decreased the consumption of the first product by one unit? answer: not sure: 3.

Usingnotations: - share of gross investment in GDP, a - share of intermediate product in gross output, X (t) - gross output in the Solow model, the value of the non-productive consumption fund C (t) is determined by the formula:С(t)=(1- ) *(1-a)*X(t).

When analyzingLeontiev's model (statistical input balance) shows that the sum of final products and the sum of conditionally net products:…equal to each other.

Using notation: - share of gross investment in gross domestic product, a- the share of the intermediate product in the gross output, X(t) - gross output in the R. Solow model, the value of the non-productive consumption fund C(t) is found by the formula: C(t)=(1-j)*(1-a)*X(t) .

With little increasing production volume conditionally variable costs for 1 product: remain unchanged. (increase, perhaps)

When describing For the study of the process with the help of PFCD, the private effects were as follows: for funds E k = 2, for labor E l = 8. In this case, the generalized efficiency E is equal to: 16.

When describing Answer: 3 (2 to the power o.5 multiplied by 4.5 to the power o.5).

When describing 3 times. (2 not exactly)

When describingthe process under study using the Cobb-Douglas production function of the form private…..efficiencies were as follows: for funds Ek=2, for labor EL=4.5. In this case, the generalized efficiency indicator E is equal to. .. 3( 2 to the power o.5 multiplied by 4.5 to the power o.5).

When describingthe process under study using the Cobb-Douglas production function of the form private…..efficiencies were as follows: for funds Ek=2, for labor EL=8. In this case, the generalized efficiency indicator E is equal to:4 or 16.

When describing the process under study using the Cobb-Douglas production function of the form private…..efficiencies were as follows: for funds Ek=2, for labor EL=4.5. In this case, the generalized efficiency indicator E is equal to.

When describing of the process under study, using the Cobb-Douglas production function, it became known that the generalized production efficiency indicator is E = 1.5, and the production scale is M = 2. In this case, gross output increased 3 times.

When buildingEMM based on known input and output indicators of an object is most often used as a criterion for the closeness of the model’s reflection of control properties...minimum sum of squared differences.

When acceptednotation...Retirement of capital and the amount of gross investment.

When acceptednotation f(Kо) - labor productivity on a stationary trajectory, - capital-labor ratio on a stationary trajectory looks like...().

When accepted Notation in the Solow model, the condition for the economy to reach a stationary trajectory has the form answer: k(t)=k to the power 0=const.

With the accepted notation…one of the equations in R. Solow’s model in relative units will have the form: dk(t)/dt=(-(g+m)k(t)/(1)+j(1 -a)f/(2) In this equation, terms (1) and (2) reflect the impact on the change in capital-labor ratio.

Other than that equal conditions with rising prices demand for goods Giffin demand for everything is growing .

When deciding ;p1x1+p2x2=I where I=1000, p1=5, p2=10ed.. What is the quantity of the 1st product of the 2nd product….100 units - 1 product and 50 units - the second.

When decidingconsumer choice problems received a system of equations ;p1x1+p2x2=I where I=1000, p1=10, p2=5ed.. What is the quantity of the 1st product of the 2nd product? ….50, 100.

When increasingincome, the demand for a product at a constant price usually...Increases (changes according to the sinusoidal law).

Production I'm a function , then the marginal product at Kt=4, Lt=25 is equal to 2,5.

Production function , then the marginal product at Kt=4, Lt=25 is equal to...0.2.

Production Kt=1100, Lt=9900. What is the marginal return on capital?...1.5 (or 10)

Production function kind called: Linear, additive production function.

Production function is given as X t =K t 0.5 ´L t 0.5, where K t is capital, L t is labor. Then the marginal product of labor ¶У/¶L at K t =16, L t =25 is equal to: 0,4.

Cobb production function-Douglas has the appearance where Kt=4000, Lt=10. What is the marginal productivity of labor? Answer: 10.

ProductionThe Cobb-Douglas function has the form where Kt=9000, Lt=10. What is the marginal productivity of labor?...15.

Production The Cobb-Douglas function looks like the mathematical expectation of the correction factor is .. = 1.

Production function Cobb-Douglas has the form:X t =K t 0.5 ´L t 0.5;K t =900,L t =10. What is the marginal productivity of labor ¶Х/¶L: 15.

Production A function is called dynamic if: 1) time t appears as an independent variable affecting the volume of output 2) PF parameters depend on time 3) PF characteristics depend on time.

Production function This- such a function, the independent variable of which takes the values ​​of the volumes of the resource used (factor of production), and the dependent variable takes the values ​​of the volumes of output y=f(x).

Production-tion K-D has the form by what percentage will output Xt increase when capital Kt increases by 1 % (0,4).

Productiona function is called dynamic if:Time t appears..PF parameters depend on time …. The characteristics of the production function depend on time.

Intermediatethe product in the scheme reflecting the relationship of macroeconomic indicators in a closed economy of the country is:means of labor and consumer goods.

Establishment processequilibrium price in the cobweb model...Remain unchanged.

Let the function utility has the form , initial prices of goods and . The individual's income is 2000 units, and the optimal set of goods ; If the price has quadrupled, then what will be the compensated income of the individual and the values ​​of the optimal set of goods? :I k =2000; x 1 =50; x 2 =40.

Let the function utility has the form u(x1;x2)=x1*x2, the initial prices of goods P1 and P2. Individual income = 1000 units, and the optimal set of goods x1 = 100 units, x2 = 20 units. If the price has increased by 4 times, then what will the individual’s compensated income and the values ​​of the optimal set of goods (x1 x2) be equal to? 2000 50,40.

Equilibrium modelsare considered...Models that describe such a state of ek-ki, when the resultant of all forces. (answer equals 0)

Position in the correct order the stages of constructing the FUI: 1. Statement of the economic problem and its qualitative analysis 2. Construction of a mathematical model 3. Mathematical analysis of the model 4. Preparation of initial information 5. Numerical solution 6. Analysis of numerical results and their application.

Positionin the correct order the stages of constructing an EMM: 1. Statement of the economic problem and its qualitative analysis 2. Construction of a mathematical model 3. Mathematical analysis of the model 4. Preparation of initial information 5. Numerical solution 6. Analysis of numerical results and their application.

With the help of which model (in the form of one formula) it is possible to reflect gross output, intermediate product, gross domestic product at the level of the country’s economy: Leontiev's balance model.

By usingwhat model can reflect the dependence of gross output and used resources at the level of the country’s economy: ...Cobb-Douglas model (PFKD)

By usingwhat model (in the form of one formula).. the relationship between indicators of VP, intermediate product, GDP….Leontiev's balance model.

System of equations in the Leontiev model is called productive if it is solvable. answer: in non-negative Xi>0, with i=1÷n.

According to In the classical model of a market economy, the supply of goods is determined by: full employment level.

According toclassical model of a market economy, with the same GDP, an increase in the money supply will lead to...An increase in the price of a unit of GDP.

According to the modelSolow's “golden” rule of accumulation corresponds to a rate of accumulation equal to the α-elasticity coefficient for physical capital.phi=1.

According to the model Harrord-Domar at what…..r increase in consumption volume will it be equal to the rate of increase in income: ANSWER: r< 1/в, r=p .

According to the model Harrord-Domar, at what…..r increase in the volume of consumption it will be equal to the rate of increase in income: ANSWER: if r = р0, р0 = а0 /В, а0 is the rate of accumulation at the initial moment of time.

According to static Leontief model, if the final product of the first industry is y1 = 1000 units, and the gross output x1 = 2500 units, what is the volume of production of the first industry consumed by other industries? 1.5.(1500 or 3500).

According to static Leontief model, if the final product of the first industry is y1 = 1500 units, and the gross output x1 = 3500 units, what is the volume of production of the first industry consumed by other industries? 2000 units .

Static modelLeontiev includes equations of the form…. .

Conditionally pure pproduction in the inter-industry balance includes...Depreciation, labor and net income.

Utility function consumption has the form .The price for good x is equal to 10, for good y is equal to 5, consumer income is equal to 200. Then the optimal set of consumer goods has the form: 10,20.

Utility functionconsumption has the form .The price for good x is 5, for good y is 10, consumer income is 200. Then the optimal set of consumer goods has the form...20.10.(200or400)

Utility functionconsumer has properties... marginal utility decreases if consumption decreases; an increase in the consumption of one product leads to an increase in utility; (the marginal utility of each product increases if the quantity of another product increases).

Selling price one product is equal to 7 units. Fixed costs are equal to 8000 units. Variable costs are equal to 5 units. for 1 piece What is the break-even production volume? 4000 units

What is it equal to in the model Keynes demand for bonds if money supply = 1000 units. , the speed of money turnover in the real market is k=0.1, the price of a unit of GDP is p=0.5 units, the value of GDP is 10,000 units... 500.

What is equal to in Keynes's model, the demand for bonds if the supply of money = 1000 units. , the speed of money turnover in the real market is k=0.1, the price of a unit of GDP is p=0.2 units, the value of GDP is 10,000 units... 800.

Target: Consider the concept of a line on a plane, give examples. Based on the definition of a line, introduce the concept of an equation of a line on a plane. Consider the types of straight lines, give examples and methods of defining a straight line. Strengthen the ability to translate the equation of a straight line from general view into the equation of a straight line “in segments”, with an angular coefficient.

  1. Equation of a line on a plane.
  2. Equation of a straight line on a plane. Types of equations.
  3. Methods for specifying a straight line.

1. Let x and y be two arbitrary variables.

Definition: A relation of the form F(x,y)=0 is called equation , if it is not true for any pairs of numbers x and y.

Example: 2x + 7y – 1 = 0, x 2 + y 2 – 25 = 0.

If the equality F(x,y)=0 holds for any x, y, then, therefore, F(x,y) = 0 is an identity.

Example: (x + y) 2 - x 2 - 2xy - y 2 = 0

They say that the numbers x are 0 and y are 0 satisfy the equation , if when substituting them into this equation it turns into a true equality.

The most important concept Analytical geometry is the concept of the equation of line.

Definition: The equation of a given line is the equation F(x,y)=0, which is satisfied by the coordinates of all points lying on this line, and not satisfied by the coordinates of any of the points not lying on this line.

The line defined by the equation y = f(x) is called the graph of f(x). The variables x and y are called current coordinates, because they are the coordinates of a variable point.

Some examples line definitions.

1) x – y = 0 => x = y. This equation defines a straight line:

2) x 2 - y 2 = 0 => (x-y)(x+y) = 0 => points must satisfy either the equation x - y = 0, or the equation x + y = 0, which corresponds on the plane to a pair of intersecting straight lines that are bisectors of coordinate angles:

3) x 2 + y 2 = 0. This equation is satisfied by only one point O(0,0).

2. Definition: Any straight line on the plane can be specified by a first-order equation

Ax + Wu + C = 0,

Moreover, the constants A and B are not equal to zero at the same time, i.e. A 2 + B 2 ¹ 0. This first order equation is called general equation straight.

Depending on the values constant A, B and C the following special cases are possible:

C = 0, A ¹ 0, B ¹ 0 – the straight line passes through the origin

A = 0, B ¹ 0, C ¹ 0 (By + C = 0) - straight line parallel to the Ox axis

B = 0, A ¹ 0, C ¹ 0 (Ax + C = 0) – straight line parallel to the Oy axis

B = C = 0, A ¹ 0 – the straight line coincides with the Oy axis

A = C = 0, B ¹ 0 – the straight line coincides with the Ox axis

The equation of a straight line can be represented in in various forms depending on any given initial conditions.

Equation of a straight line with an angular coefficient.



If the general equation of the straight line Ax + By + C = 0 is reduced to the form:

and denote , then the resulting equation is called equation of a straight line with slope k.

Equation of a straight line in segments.

If in the general equation of the straight line Ах + Ву + С = 0 С ¹ 0, then, dividing by –С, we get: or , where

Geometric meaning coefficients is that the coefficient A is the coordinate of the point of intersection of the line with the Ox axis, and b– the coordinate of the point of intersection of the straight line with the Oy axis.

Normal equation of a line.

If both sides of the equation Ax + By + C = 0 are divided by a number called normalizing factor, then we get

xcosj + ysinj - p = 0 – normal equation straight.

The sign ± of the normalizing factor must be chosen so that m×С< 0.

p is the length of the perpendicular dropped from the origin to the straight line, and j is the angle formed by this perpendicular with the positive direction of the Ox axis.

3. Equation of a straight line using a point and slope.

Let the angular coefficient of the line be equal to k, the line passes through the point M(x 0, y 0). Then the equation of the straight line is found by the formula: y – y 0 = k(x – x 0)

Equation of a line passing through two points.

Let two points M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2) be given in space, then the equation of the line passing through these points is:

If any of the denominators is zero, the corresponding numerator should be set equal to zero.

On the plane, the equation of the straight line written above is simplified:

if x 1 ¹ x 2 and x = x 1, if x 1 = x 2.

The fraction = k is called slope straight.

Straight line on a plane and in space.

Studying properties geometric shapes using algebra is called analytical geometry , and we will use the so-called coordinate method .

A line on a plane is usually defined as a set of points that have properties unique to them. The fact that the x and y coordinates (numbers) of a point lying on this line are written analytically in the form of some equation.

Def.1 Equation of a line (equation of a curve) on the Oxy plane is called an equation (*), which is satisfied by the x and y coordinates of each point on a given line and is not satisfied by the coordinates of any other point not lying on this line.

From Definition 1 it follows that every line on the plane corresponds to some equation between the current coordinates ( x,y ) points of this line and vice versa, every equation corresponds, generally speaking, to a certain line.

This gives rise to two main problems of analytical geometry on the plane.

1. A line is given in the form of a set of points. We need to create an equation for this line.

2. The equation of the line is given. It is necessary to study its geometric properties (shape and location).

Example. Do the points lie A(-2;1) And IN (1;1) on line 2 X +at +3=0?

The problem of finding the intersection points of two lines, given by equations and, comes down to finding coordinates that satisfy the equation of both lines, i.e. to solving a system of two equations with two unknowns.

If this system has no real solutions, then the lines do not intersect.

The concept of a line is introduced in the UCS in a similar way.

A line on a plane can be defined by two equations

Where X And at – arbitrary point coordinates M(x;y), lying on this line, and t - a variable called parameter , the parameter determines the position of the point on the plane.

For example, if , then the value of the parameter t=2 corresponds to the point (3;4) on the plane.

If the parameter changes, the point on the plane moves, describing this line. This method of defining a line is called parametric, and equation (5.1) is a parametric equation of the line.

To move from parametric equations to a general equation (*), one must somehow eliminate the parameter from the two equations. However, we note that such a transition is not always advisable and not always possible.

A line on a plane can be specified vector equation , where t is a scalar variable parameter. Each parameter value corresponds to a specific plane vector. When changing the parameter, the end of the vector will describe a certain line.

Vector equation in DSC corresponds two scalar equations

(5.1), i.e. the equations of projections on the coordinate axes of the vector equation of a line are its



parametric equation.

Vector equation and parametric line equations have mechanical sense. If a point moves on a plane, then the indicated equations are called equations of motion , and the line is the trajectory of the point, the parameter t is time.

Conclusion: every line on the plane corresponds to an equation of the form.

In the general case, ANY EQUATION OF A VIEW corresponds to a certain line, the properties of which are determined by the given equation (with the exception that no geometric image corresponds to an equation on a plane).

Let a coordinate system on the plane be chosen.

Def. 5.1. Line equation this type of equation is calledF(x;y) =0, which is satisfied by the coordinates of every point lying on this line, and not satisfied by the coordinates of any point not lying on it.

Equation of the formF(x;y )=0 – called the general equation of a line or an equation in implicit form.

Thus, line Г is the locus of points satisfying this equation Г=((x, y): F(x;y)=0).

The line is also called crooked.


If a rule is specified according to which a certain number u is associated with each point M of the plane (or some part of the plane), then they say that on the plane (or on part of the plane) “a point function is specified”; the specification of the function is symbolically expressed by an equality of the form u=f(M). The number u associated with point M is called the value of this function at point M. For example, if A is a fixed point on the plane, M is an arbitrary point, then the distance from A to M is a function of point M. B in this case f(m)=AM.

Let some function u=f(M) be given and at the same time a coordinate system be introduced. Then an arbitrary point M is determined by the coordinates x, y. Accordingly, the value of this function at point M is determined by the coordinates x, y, or, as they also say, u=f(M) is function of two variables x and y. A function of two variables x and y is denoted by the symbol f(x; y): if f(M)=f(x;y), then the formula u=f(x; y) is called the expression of this function in the selected coordinate system. So, in the previous example f(M)=AM; if we introduce a Cartesian rectangular coordinate system with the origin at point A, we obtain the expression for this function:

u=sqrt(x^2 + y^2)

PROBLEM 3688 Given a function f (x, y)=x^2–y^2–16.

Given the function f (x, y)=x^2–y^2–16. Determine the expression of this function in the new coordinate system if the coordinate axes are rotated by an angle of –45 degrees.

Parametric line equations


Let us denote by the letters x and y the coordinates of a certain point M; Let's consider two functions of the argument t:

x=φ(t), y=ψ(t) (1)

When t changes, the values ​​x and y will, generally speaking, change, therefore, point M will move. Equalities (1) are called parametric line equations, which is the trajectory of point M; the argument t is called a parameter. If the parameter t can be excluded from equalities (1), then we obtain the equation of the trajectory of point M in the form