Mathematics of Economics and Finance:Analytic Geometry Tools

TOPICS COVERED

0. Numbers

1. Equations and Inequalities

2- Graph of a line, slope, intercept, inequality

3- Two Logarithms and Exponential: Logarithms ease computations

4- Intervals and Sets

5- Arithmetic and geometric progressions

6- Limits: Computations by Calculators.

7. Functional Relationship Interpolation Derivative

Extracted from:http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM

Numbers: The introduction of zero into the decimal system in the 13th century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. Without the notion of zero, the descriptive and prescriptive modeling processes in commerce, astronomy, physics, chemistry, and industry would have been unthinkable. The lack of such a symbol is one of the serious drawbacks in the Roman numeral system. In addition, the Roman numeral system is difficult to use in any arithmetic operations, such as multiplication. The purpose of this site is to raise students and teachers awareness of issues in working with zero and other numbers. Imprecise mathematical thinking is by no means unknown; however, we need to think more clearly if we are to keep out of confusions.

Counting is as old as prehistoric man; after he learned to count, man invented words for numbers and later still, symbolic numerals. The numeral system we use today originated with the Hindus. They were devised to go with the 10-based, or "decimal," method of counting, so named after the Latin word decima, meaning tenth, or tithe. The first popularizer of this notation was a Muslim mathematician, Al-Khwarizmi in the 9th Century; however it took the new numbers about two centuries to reach Spain and then to England in a book called Craft of Nombrynge.

Mathematics is a human endeavor which has spanned over four thousand years; it is part of our cultural heritage; it is a very useful, beautiful and prosperous subject. Mathematics is one of the oldest of sciences; it is also one of the most active; for its strength is the vigor of perpetual youth. Mathematics is also our native language. Numbers are cultural phenomena; humans invented them to quantify the external world around them. The external world is qualitative in its nature. However, human can understand, compare and manipulate numbers only. Therefore, we use some measurable and numerical scales to quantify the world. This enables us to understand the world by, for example finding any relationship, manipulating, comparing, calculating, etc. That is, to make an Analytical Structured Model for the external world. Then we use the same scale to qualify it back to the world. If you cannot measure it, you cannot manage it. This is the essence of human's understanding and decision-making process.

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Problem Solving:

Historically, Math Education systems focused on helping students to learn to carry out a number of different types of "step 2" using some combination of mental and written knowledge and skills. It takes a typical students hundreds of hours of study and practice to develop a reasonable level of speed and accuracy in performing addition, subtraction, multiplication, and division on integers, decimal fractions, and fractions. Even this amount of instructional time and practice -- spread out over years of schooling -- tends to produce modest results. Speed and accuracy decline relatively rapidly without continued practice of the skills.

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During the past 5,000 years there has been a steady increasing body of knowledge in mathematics, science, and engineering. The industrial age and our more recent information age have lead to a steady increase in the use of "higher" math in many different disciplines and on the job. Our Education System has moved steadily toward the idea that the basic computational aspects described above are insufficient. Students also need to know basic algebra, geometry, statistics, probability, and other higher math topics.

During the Renaissance era trigonometric and logarithmic functions, the idea of approximation, etc., were developed. In René Descartes' system there is a one-to-one correspondence between real numbers and points. This correspondence was postulated by an axiom of continuity by Hilbert later.

From Analytical-Geometry concepts introduced by Descartes in the 17th century, a real numbers is a point while an interval is the length between two points which is also the absolute value:

of the difference between two numbers.

Before the introduction of zero to Europe during the Renaissance era, there was no concept of a negative numbers. Even today, for many people, in particular for young pupils the concept of a negative number is hard to understand for strong ontological reasons.

René Descartes was the one who extended real numbers to include negative numbers. The way he accomplished this was by representing numbers on a real number axis as in above figure.

While the number zero had been accepted as a Natural number long ago, in became official much latter. For example, in Sweden school children were to learn that zero is a natural number in their 1960 textbooks.

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The positive numbers are on the right of zero (origin), and negative numbers are on the left of zero, which is arbitrary. The reason he chose the right side for positive numbers is because most people are right-handed--not because positive is better. The word "right" might have to do with the use of the English word "right", the Spanish "derecho", etc. to mean certain "positive" things.

Vectors and Numbers: Two Distinct Representations of Numbers: The word "vector" (literally "carrier", from "vectus", past passive participle of "veho" meaning "I carry", and related to the English word "vehicle") was invented by William Rowan Hamilton. However, the useful mixing of mathematics and physics by Isaac Newton's work in describing his three laws of motion is the first powerful tool we now call Analytical Modeling.

Any number represents a point on this real number axis, called O-X axis. For example, point B is +2. It could be represented also as a vector with its origin zero and end point B, with length 2 units, as depicted in the following Figure. Now a question for you: What -3 represents in the following figure? Is it a point?, a number?, a vector (what vector?)?

The four arithmetic operations are well defined by the vector representation of real numbers and appropriate kinds of movements on this real-number axis:

Addition and subtraction operations on numbers can be viewed as the results of movements in certain directions, either to the left or to the right. For example 2 - 3 means starting from the origin O going 2 steps to the right an then 5 steps to the left, you will end up at point -1. Therefore, by vector representation, addition and subtraction easily can be executed on this axis.

Multiplication of a two positive numbers can be considered as a vector multiplied by a scalar. For example, 1.5 time -3 means moving from the origin O toward the same direction (because the scalar 1.5 is positive) the vector -3 and continuing the one half more steps in the same direction, as shown in the following figure, 1.5B = C.

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Multiplication of a negative number by a positive number such as -2 times 3, means moving from the origin O toward the opposite direction (because the scalar -2 is negative) of vector +3 and continuing the same amount steps in the same direction. That is moving from the origin O toward the direction of vector -3 and continuing the same amount steps in the same direction.

Division of real numbers may be defined in terms multiplication. That is, dividing two numbers A by B is a number C such that A = B.C.

Dividing by Zero Can Get You into Trouble!

If we persist in retaining such errata in our educational texts, an unwitting or unscrupulous person could utilize the result to show that 1 = 2 as follows:

(a).(a) - a.a = a2 - a2

for any finite a. Now, factoring by a, and using the identity (a2 - b2) = (a - b)(a + b) for the other side, this can be written as:

a(a-a) = (a-a)(a+a)

dividing both sides by (a-a) gives

a = 2a

now, dividing by a gives

1 = 2, Voila!

This result follows directly from the assumption that it is a legal operation to divide by zero because a - a = 0. If one divides 2 by zero even on a simple, inexpensive calculator, the display will indicate an error condition.

Again, I do emphasis, the question in this Section goes beyond the fallacy that 2/0 is infinity or not. It demonstrates that one should never divide by zero [here (a-a)]. If one does allow oneself dividing by zero, then one ends up in a hell. That is all.

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Equation: Its Structure, Roots, and Solutions The word "Moaadeleh" meaning "equation" appeared in the writings of al-

Khwarizmi together with the common algebraic operations connected to the idea of an equation.

Later, the word "equation" introduced by Fibonacci's work to Europe. Like the Latin "aequatio" it is an abstract noun referring to any process of equalizing. However, as a technical term it means the mathematical expression that we now call an equation: namely, two separate combinations of various quantities, known and/or unknown, that are equal to each other.

Equations are any symbolic expression with equality sign, such as X2 = 4. It is good to know that, the word "root" is derived from Sanskrit word "Bija" for "seed" or "root", is the usual term for algebra, where the "root" is the unknown quantity (often called X) that then produces a definite result via the structure of equation.

The definite result (i.e., a fruit) of an equation is called its solution because it satisfies the equation. The above figure depicts the historical analogy developed during the second century between equations' elements and a tree structure. Since by plugging in the numerical value for the root (X), the equation resolves, i.e., it disappears. Hence the numerical value is called a solution, as for example when sugar is mixed with water in making lemonade sugar disappears, making a solution.

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= +2, and - 2 right?

Misplacement of the SignAnother common error is found in some textbooks which, announcing that the square root of 4 has two answers namely +2, and -2. When this writer confronted an author guilty of this practice observing that one number cannot be equal to two different numbers, the reply received was "check it for yourself by squaring both sides". He followed with self-satisfaction, "you see!". This writer advised that following his argument one could also demonstrate that one is equal to minus one. An observer witnessing this exchange jumped in volunteering the results of the computation performed with a calculator as producing a single result of plus 2 declaring "he is right."

Solving the equation X2 = 4 has two solutions: X = , - . The number, square root (Sqrt) of 4, is two, therefore, the solution is both X = 2, and X = -2. The symbol ± is plus OR minus (could be both, but not at the same time). This

correct result is distorted when one goes on to write X = and concludes that this later result is +2, and -2. This is the genealogy of this error. There is a clear distinction here and an important difference that a careful reader will note.

is a positive number that, when you square it, you get 4. While - must

first, be written as - ( ) and then interpreting the quantity inside the bracket. Do you see the difference?

Unfortunately, this distinction is not still recognized by many instructors. Many authors when taking, e.g., the square root of 9, commit this error. The authors profess to the students that "there are two possible numbers that square to 9, 3 and -3. So, when we take the square root of 9, we put a + and - in front of it." While the first part of this statement is correct, however the conclusion is wrong. When we take the square root of 9, we always get 3, NOT 3 and -3.

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What Is Infinity: While zero is a concept and a number, Infinity is not a number; it is the name for a concept.

To facilitate a visual understanding of the infinity concept, you may wish to use the following demonstration for your students: Draw to straight line segments parallel to each other on the board. Make one line segments clearly longer (say, twice) length than the other line segment, as shown in the following graph:

Now, pose the following question: Which line segments have the most points?

Higher Dimensional Analytical-Geometry:

One may extend the one dimensional analytical-geometry concepts in visualization of other algebraic elements,

such as equations. For example, the above figure is a graph (i.e., a picture) of equation Y = X + 1, which is a straight line. The slope of the line is the Tan (a), where the angle (a) is that of the horizontal axis making with the line, counterclockwise. For example, the slope of the above line is m = Tan (45) = 1. As another example, if a line has slope of m = -1, then the angle (a) is obtained by the inverse function aTan(-1) = 135 degrees.

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The convention of measuring angles in counterclockwise removed any existing ambiguities in communicating and in computations when angles are involved such as functional evaluations in trigonometry. For example, as shown clearly in the following figure:

By the above convention, the angle OB makes with OA is 45 degrees, while the angle OA makes with OB is 135 degrees (not -45 degrees).

The Two Numbers Nature Cares Most:Inventions or Discoveries

The two numbers that Nature loves most are denoted by and e. The first is relevant to planets movements around the sun while the second is related to the growth of population of different species.

What is ? Planets move around the Sun in ellipsoid shaped-path with major diameter and minor diameter denoted by 2a, and 2b respectively, then the areas they travel are .a.b. For a circle a = b = r the radius of a circle, therefore the area is .r2, and its circumference length is 2 .r . Therefore, is the ratio of the circumference length of ANY circle divided by the length of its diameter. That is, to have a notion for the numerical value of , take a robe of any size and make a circle, then circumference/diameter is the . Using such a geometric argument, Al-Biruni in 11th century suggested that must be an irrational number.

It is nice to notice that, the derivative of the area of a circle: A = r2, is the circumference C = 2 r. Similarly, for sphere the surface is S = 4 r2, which is the derivative of volume V = 4/3 r3.

Beside the fact that is a number it is also a measurement for an angle in terms of radians. A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius. Therefore:

180 degrees = radians

In both cases, is dimensionless; it is just a number, with two related applications.

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What is e? The growth of population for every species follows an Exponential law. The size of a population is, after length of time t years is P.e rt , where P is the initial population size and r is the rate of growth of a particular species. The growth rate of human population is about r = 0.019 since World War II.

What is the difference between the accumulations of $1000 invested at a given rate (r) if the interest is compounded daily versus annually?

Suppose you invest $1000, over a period if t-years, with an annual (fixed) interest rate of r, if the interest is added n times per year at the end of each period, then your compounded investment is $1000(1 + r/n)nt.

Now suppose the banker adds the interest at the end of each day then your investment growth faster 1000(1 + r/365) 365t which is very close to 1000e rt which is the compounded investment continuously.

In fact, increasing number of time intervals, e.g. days into half-days, this approximation gets much better, as shown by the following limiting result when the length of each period gets smaller and smaller:

The number e is discovered by John Napier , and it is the base for the so called natural-logarithm, because this number appears frequently in nature. Notice that, the explicit function y = Ln (x), x 0, is equivalent to the implicit function x = ey, by definition. Moreover, the first and the second function are generally called the logarithmic (Ln) and the exponential (Exp) functions, respectively.

Therefore, finally the algebra and geometry are unified by the means of analytical-geometry concepts in last couple of hundred years. This helps overcoming our human visual limitation to see with the eyes-mind and work within space of higher dimensions than the 3-dimensional space we live in.

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Intervals: An interval is a set of (real) numbers between, and possibly including, two numbers. The interval from a to b is denoted as follows:

[a, b] if a and b are included (i.e. [a, b] = {x: a  x  b}) (a, b) if neither a nor b is included (i.e. (a, b) = {x: a < x < b}) [a, b) if a is included but b is not (a, b] if b is included but a is not.

We use the special symbol "" ("infinity") in the notation for intervals that extend indefinitely in one or both directions, as illustrated in the following examples.

(a, ) is the interval {x: a < x}. (, a] is the interval {x: x  a}. (, ) is the set of all numbers.

Note that is not a number, but simply a symbol we use in the notation for intervals that have at most one endpoint.

The notation (a, b) is used also for an ordered pair of numbers. The fact that it has two meanings is unfortunate, but the intended meaning is usually clear from the context. (If it is not, complain to the author.)

The interior of an interval is the set of all numbers in the interval except the endpoints. Thus the interior of [a, b] is (a, b); the interval (a, b) is the interior also of the intervals (a, b], [a, b), and (a, b).

We say that (a, b), which does not contain its endpoints, is an open interval, whereas [a, b], which does contain its endpoints, is a closed interval. Notice that the intervals (a, b] and [a, b) are neither open nor closed.

Open and closed setsTo make precise statements about functions of many variables, we need to generalize the notions of open and closed interval to many dimensions.

We want to say that a set of points is "open" if it does not include its boundary. But how exactly can we define the boundary of an arbitrary set?

We say that a point x is a boundary point of a set of n-vectors if there are points in the set that are arbitrarily close to x, and also points outside the set that are arbitrarily close to x. A point x is an interior point of a set if we can find a (small) number such that all points within the distance of x are in the set.

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Example: Consider the following feasible region:

2 X1 + X2 40 labor constraint

X1 + 2 X2 50 material constraint

and both X1, X2 are non-negative.

Which is depicted in the following figure:

The point B in the following two dimensional figure, for example, is a boundary point of the feasible set because every small circle centered at the point B, however small, contains both points some in the set and some points outside the set. The point I is an interior point because the orange circle and all smaller circles, as well as some larger ones; contains exclusively points in the set. The collection of boundary points belonging to one set is called boundary line (segment), e.g. the line segment cd. The intersection of boundary line (segments) are called the vertices (or the corner points)

The set S is open if every point in S is an interior point of S.

The set S is closed if every boundary point of S is a member of S.

Example: The set of interior points of the set {(x, y): x + y  c, x  0, and y  0} is {(x, y): x + y < c, x > 0, and y > 0}, and the set of boundary points is {(x, y): x + y = c, x = 0, or y = 0}. Thus the set is closed. Page 12.

Arithmetic and geometric progressions: Many financial mathematics such as compound interest, discounting, annuities, etc are based on two kinds of series, the Arithmetic and Geometric Series

A "progression" is a series of numbers created by some rule. Two common kinds of progressions are "arithmetic" and "geometric".

Arithmetic Progression: An arithmetic progression is a sequence of the form ak + b where a and b are fixed and k runs through integer values. The goal of this section is to discover whatever we can about primes in arithmetic progression. An arithmetic progression is built using addition. For example, we could use the rule "add 2" and create the following progression starting with the number 1:1, 2, 4, 6, 8, 10... etc.

An arithmetic progression creates a straight line.

The sum of the numbers in (an initial segment of) an arithmetic progression is sometimes called an arithmetic series. A convenient formula for arithmetic series is available. The sum S of the first n values of a finite sequence is given by the formula:

S = n(a1 + an)/2

Where a1 is the first term and an the last.

A geometric progression is built using multiplication. For example, we could use the rule "multiply by 2" and create the following progression starting with the number 1:

1, 2, 4, 8, 16... etc.

Notice that a geometric progression will get big very quickly compared to an arithmetic progression. In fact, it keeps getting bigger faster.

A geometric progression creates a curved line. The sum of the numbers in a geometric progression is called a geometric series. A convenient formula for geometric series is:

Compare this with a arithmetic progression showing linear growth (or decline) such as 4, 15, 26, 37, 48, .... Note that the two kinds of progression are related: taking the logarithm of each term in a geometric progression yields arithmetic one.

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Functional Relationship Variables may exist independently or they may be related to each other according to a functional relationship, e.g., if the value taken by Y depends upon the value taken by X, then Y is called a function of X. This is denoted as:

Y = f(X)  

Independent Variable : A variable that is not affected by other variables. It can take on any value. In our example, X is the independent variable.

Dependent Variable : A variable whose value is determined by another variable or set of Variables. In our example, Y is the dependent variable.  

A function is an input/output rule: such that if you give it an input, it will associate with that input exactly one output

the set of all possible valid inputs is called the domain of the function the set of all possible outputs is called the range of the function

Example: Fahrenheit temperature is computed from the Centigrade temperature F = (9/5)C + 32

Linear Function

An equation that is represented by a straight-line graph

Y = 4 - 2 (X) Y = 6 - 2(X) Y = 4 + 2 (X) Y = 4 - 3 (X)

Y Intercept

The value of Y when X equals zero

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Slope : Slope provides the direction and steepness of a line on a graph. It shows the magnitude of change in Y that results from a unit change in X.

Slope = change in Rise / Change in Run Slope = change in Y / Change in X

Changes in Slope or Intercept

Increasing the slope makes the curve more steep and vice versa. Increasing the intercept shifts the curve to the right without changing the slope. Decreasing the slop makes the curve more flat

Rectangular coordinates in 2-D:

 A pair of numbers representing the coordinates of the point in a rectangular coordinate system can represent the location of a point in a two-dimensional plane. For example, the point A in the figure has coordinates (x, y) in the rectangular x-y coordinate system shown. To determine a coordinate one draws a perpendicular onto the coordinate axis. In this case, the arrows on the coordinate axis indicate that points to the right of the origin O on the x-axis are positive and to the left are negative. In a similar manner, points above the origin on the y-axis are positive and below it are negative.

Equation of a straight line: The equation of a straight line in a plane is given in the x-y coordinate system by the set of points (x, y) that satisfy the equation

 

where, as shown in the figure, a represents the slope of the line in terms of its rise divided

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by its run, and b is the y-coordinate of the point of intercept of the line and the y-axis.

One can evaluate the equation of a line from any two points on it. For example, consider points A and B shown in the figure with coordinates (x1,y1) and (x2,y2), respectively. Since ACD and ABE are similar triangles, we have

This can be put in the above format by selecting

 

 

If x1 = 0, then, as can be seen from the figure,

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 Equations of parabolas:

The basic equation of a parabola in a rectangular x-y coordinate system is given by

 

Depending on the sign of a, this equation will result in one of the two following graphs. 

 

Exponential Functions: Look at the following function definition:                 f(x) = 2x

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Here's the graph:

How about:                 f(x) = 2-x

Exponential functions with base e f(x) = ex

e is a real number constant (like pi) value = 2.7182818… frequently seen as the base for exponential functions called the natural base it arises "naturally" in many mathematical contexts

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Compound interest

The scenario:

you place $100 in the bank it earns 10% per year compound interest how much do you have after 2 years?

The key element: at the end of each year . . . the interest earned is added to the principal and we start earning interest on the interest! this property is the defining feature of compound interest

A Formula for Compound Interest

Make the following definitions:  

P principal amount invested  (present value) 

m number of periods in a year 

t number of years principal is invested 

r interest rate per year (nominal rate)

A total amount in account at end of t years (future value)

 Then:  

A = 

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Logarithmic Functions: Here's a formal definition of log2 x:

Definition of log2 xif:        2y = x

then:          y = log2 x

Common bases for logs:

Common bases for logs one common base is 10: log10

since e is a common base for the exponential it’s a common base for log function, as well:  loge

Conventions instead of writing log10 35, you can write log 35 instead of writing loge 35, you can write ln 35 log and ln are the names used by our calculators for these functions these are the only bases for logs that have keys to compute them directly

LOG FORM ==> EXPONENT FORM

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Just follow the arrows:

Example: log5 13 = x    ==>   5x = 13

EXPONENT FORM ==> LOG FORM

Example: y14 = x    ==>    logy x = 14Why LOG?

log MN = log M + log N Logs are exponents!  What do you do with exponents when you multiply?  e.g. axay =    ??     

Log M/N = log M - log N Logs are exponents!  What do you do with exponents when you divide?  e.g. ax/ay

=    ??     log Mp = p  log M

Think about this:  log x5 = log (x)(x)(x)(x)(x)= log x + log x + log x + log x + log x  (by log of a product rule)

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How to solve exponential equations: how do you solve equations like:         e3x - 2 =  5  

How to solve logarithmic equations: how do you solve equations like:     ln x = (2/3)ln 8 + (1/2)ln 9, OR log (x + 3) - log x = 1

Factorials:

Definition: 

        n! = n(n-1)(n-2)(n-3) ... (3)(2)(1)

 There is a group of 4 people what want 4 different officer positions in a club. The positions are: President, Vice president, Secretary and Treasurer. How many different combinations can you have of officers?

        4        X 3 X 2 X 1 = 24

The counting method we applied used Factorials:    4! = 24

Definition:     0! = 1

Applications:

Recall the formula for compound interest

A =

Suppose you have $10,000 to invest that is projected to accumulate earnings at 20% per year compounded quarterly. How many years will it take to double your investment?

SUPPLY, DEMAND AND EQUILIBRIUM

Demand : The amount of a good or service that an individual is willing and able to buy at each possible price.

Determinants of demand : Demand is affected by a number of factors. Price is the most important factor but there are other factors also that can influence demand such as prices of related goods.   

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The law of demand All else equal, if the price of a good goes up, quantity demanded goes down and vice versa.

Demand vs. Quantity Demanded

Demand is a set of number that lists the quantity demanded corresponding to each possible price whereas quantity demanded is the amount of a good or service that an individual is willing and able to buy at a given price. For instance, the information on price and quantity demanded presented in a table/demand schedule is collectively referred to as the demand. A change in price leads to a change in quantity demanded. A change in price does not lead to a change in demand.

Demand Curve

A graph illustrating demand, with prices on the vertical axis and quantity demanded on the horizontal axis. Demand curve slopes downward because of the negative relationship between price and quantity demanded.

Changes in Demand

As mentioned before, a change in price does not lead to a change in demand. But a change in a factor other than price such as income and prices of related goods etc. does lead to a change in demand. In that case the change in demand leads to a shift in the demand curve. A fall in demand shifts the demand curve to the left and a rise in demand shifts the curve to the right.

  System of Equations: y = x + 2, and y = -x + 4 It's called a system, because there's more than one equation. This is a system of two equations with two unknowns.

What do you do with a system of equations? Solve it.

To solve a system of two equations with two unknowns: find a point --- that is, a pair of numbers (x,y) --- that satisfies both equations

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The graphical method

If we graph these equations, we get:

recalling that the graph of an equation is the set of all points that satisfy the equation

Modeling validation:

Why it happened How to cope ExampleSystem consists of two parallel lines 

No solutions: inconsistent system

STOP!

Write as shown:

 x - y = 10 -x + y =5

0 + 0 = 15 no solution

system consists of two identical lines 

All points on either line are solutions: dependent system

STOP!

Write as shown:

  x - y = 10 -x + y = -10

  0 + 0 = 0

All points on the line  x - y = 10

 

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An application: supply and demand

price-demand equation:                  q = -1000p + 3000

the effect of price p on supply q of cards to be provided by Pokeyo this will be described by the price-supply equation (or just supply equation)o price-supply equation:                    q = 1200p - 800

Supply and demand equations graphed

the graph of the demand equation is a line with a negative slope . . .

expressing the fact that as the price increases, demand by the kids will drop this is the old "supply/demand" idea, a consumer market force

the graph of the supply equation is a line with a positive slope . . .

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Linear Interpolation:

You know the coordinates (x0, y0) and (x1, y1). You want to pick points on this line with a given x in the interval [x0, x1].

Linear interpolation approximation

A linear interpolation of certain points that are in reality values of some function f is typically used to approximate the function f.

Applications:1. To insert between or among others.2. To change by putting in new material.3. To estimate a missing value by taking an average of known values at neighboring points.

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Why Calculus?Why do we use calculus to determine the exact expressions for derivatives? For example, to determine the derivative of y = x2 we typically consider two points on this curve, (x, y) and (x+dx, y+dy), where dx and dy are initially regarded as small finite quantities. So we have the two equations

y = x2

(y+dy) = (x+dx)2

Subtracting the first from the second gives: dy = 2x dx + (dx) 2

Now, since dx is considered to be non-zero, it is legitimate to divide by dx, giving the result: dy/dx = 2x + dx

Now for the interesting part: "Let dx go to zero". Of course if dx actually EQUALS zero then we weren't justified in dividing the previous equation by zero. To avoid this little inconsistency the epsilon-delta limit device is used to establish that the LIMIT of the ratio dy/dx is 2x as dx APPROACHES zero.

I could be argued that the "limit" concept has been introduced here because we had reached the "limit" of our ability to algebraically solve equations. To illustrate, consider again the case : y = F(x) = x2

We want the slope of the line y = ax + b tangent to this curve. Equating y values, we have: x2 - ax -b = 0

Solving this quadratic for x gives the point(s) of intersection

a ± [ (-a) 2 + 4b) ] 1/2

x = ------------------------- 2

There is a single point of intersection (i.e., the line is tangent to the curve) iff the sqrt term vanishes, which leaves a = 2x, i.e., slope = 2x, for any given x value.

There are no divisions by "something approaching zero", no "limit", "differential", or "infinitesimal" involved in this derivation.

Of course, this purely algebraic approach to finding derivatives becomes unwieldy when dealing with more complicated functions. So it seems that we resort to calculus when algebra becomes difficult.

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Rules for Derivative:

The definition of a derivative implies the following formulas for the derivative of specific functions.

 f (x) = k  f '(x) = 0.  f (x) = kxn  f '(x) = knxn - 1.  f (x) = ln x  f '(x) = 1/x  f (x) = eX  f '(x) = eX

 f (x) = ax  f '(x) = aX ln a

Three general rules (very important!!): Sum rule

F (x) =  f (x) + g(x) F '(x) =  f '(x) + g'(x) Product rule

F (x) =  f (x)g(x) F '(x) =  f '(x)g(x) +  f (x)g'(x) Quotient rule

F (x) =  f (x)/g(x) F '(x) = [ f '(x)g(x)   f (x)g'(x)]/(g(x))2

Examples : Let F (x) = x2 + ln x. By the sum rule, F '(x) = 2x + 1/x.Let F (x) = x2ln x. By the product rule, F '(x) = 2xln x + x2/x = 2xln x + x.Let F (x) = x2/ln x. By the quotient rule, F '(x) = [2xln x  x2/x]/(ln x)2 = [2xln x  x]/(ln x)2.

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Applications: Change, Rate of change, Instantaneous Rate of Change, Slope of the tangent Line, Approximation of a complicated function by a straight line, Differentiation and its application. Marginal Value, Optimization (e.g., how to make an open-top box out of a square cardboard sheet?), Behavior of a function, Convexity and concavity, Shape of a function, to prove a variable is merely a dummy variable, L'Hopital rule.

Optimization: Decision-makers (e.g. consumers, firms, governments) in standard economic theory are assumed to be "rational". That is, each decision-maker is assumed to have a preference ordering over the outcomes to which her actions lead and to choose an action, among those feasible, that is most preferred according to this ordering. We usually make assumptions that guarantee that a decision-maker's preference ordering is represented by a payoff function (sometimes called utility function), so that we can present the decision-maker's problem as one of choosing an action, among those feasible, that maximizes the value of this function.

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