Introduction to modern mathematical methods

For example, since the proposition group of college entrance examination drew attention to the application of mathematics in May, 1993, an application question about the prediction of freshwater fish breeding market appeared in the national college entrance examination in June. This is a good problem in the application of mathematics, especially in today's new system of socialist market economy, which has attracted everyone's attention.

The so-called economic mathematics problem is to study some problems in economics by mathematical methods, such as the problems of national economy in economic growth rate and population growth rate, banking problems, securities market problems, insurance calculation problems, consumption and market prediction problems, input-output problems and so on. Among the above problems, we should pay attention to some basic problems that can be solved by elementary mathematics methods acceptable to middle school students.

Here are a few examples.

Example 1: The market demand for a commodity is P (10,000 pieces)? , market supply q and market price x (yuan/piece) approximately satisfy the following relationship: p =-x+70; Q = 2x-20 When p = q, the market price is called the market equilibrium price, and the demand at this time is called the equilibrium demand.

(1) Balance price and demand;

(2) If every commodity is taxed in 3 yuan, seek a new equilibrium price;

(3) If the balanced demand increases by 40,000 pieces, how much subsidy should the government give to each commodity?

Solution: (1) The equilibrium price is 30 yuan per piece, and the equilibrium demand is 400,000 pieces.

(2) Let the new market equilibrium price be X yuan/piece, which is the price paid by consumers, while the price obtained by suppliers is (X-3) yuan/piece, which is -X+70 = 2 (X-3)-20 according to the meaning of the question, thus obtaining the new equilibrium price as 32 yuan/piece.

(3) Assuming that the government subsidizes T yuan/piece, the market equilibrium price at this time, that is, the price paid by consumers is X yuan/piece, then the price received by the provider is (X+T) yuan/piece, and the equation set is -X+70 = 44 according to the meaning of the question.

2 (x+t)-20 = 44 x = 26 t = 6。

Example 2: The daily output of a product is 20 sets, and the price of each set is 90 yuan. If the daily output increases by 1 set, the unit price will decrease by 3 yuan. How to design production to maximize total daily income?

Solution: If X units are produced every day, the total income is Y yuan, according to the meaning of the question, Y = (90-3x) (20+x). When the daily output is 25 sets, the total income is the largest.

Example 3: A factory borrowed RMB 6,543.8+0,000 at the beginning of this year, with compound interest at an annual interest rate of 654.38+00% (that is, the interest of this year is included in the interest of the principal of the next year). Calculated from the end of this year, the annual fixed amount will be paid off at the end of 654.38+02. How much is the annual repayment?

Solution: Assuming the annual repayment amount is X million yuan, according to the meaning of the question: X+X (1+10%)+X (1+10%) 2+…+X (1+/kloc-)

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For example:

Limit solution

1. Direct substitution

Applicable to numerator, the limit of denominator is not zero or different.

Example 1. Find it.

Analysis due to:

Therefore, the direct substitution method is adopted.

Solve the original formula =

2. Find the limit with four algorithms of limit.

For the convenience of narration, we omit a certain change process of the independent variable and mark the limit in a certain limit process, so the four algorithms of limit can be accurately described as follows:

Theorem in the same change process, assuming that all exist, then

( 1)

(2)

(3) When the denominator is

Generally speaking, the limit of sum, difference, product and quotient of functions is equal to the sum, difference, product and quotient of function limits.

Beg.

solve

3. Infinitely small separation method

Applicable to numerator, denominator tends to be simultaneous, that is, indefinite form.

Example 3.

In the analysis of a given function, the limit of numerator and denominator does not exist, so the law cannot be directly applied. Note that the numerator and denominator tend to be the same at the same time. Firstly, the function is deformed by elementary method, that is, the highest power of division of numerator and denominator, so that the infinitesimal quantity is separated, and then the limit is found according to the operation law.

Why do the numerator and denominator tend to be the same in a given function? To illustrate: because, but the speed of tendency is faster than that of tendency, so don't think it is static (because there are positive and negative points).

Solve the original formula (numerator and denominator divided by the same)

(algorithm)

When, everyone tends to. The reciprocal of infinity is infinitesimal.

4. Zero factor elimination method

Applicable to numerator, the limit of denominator is 0 at the same time, that is, the type is undetermined.

Example 4.

In the analysis of two functions, the limit of numerator and denominator is 0, so the fourth law cannot be used directly, so the zero factor elimination method is adopted.

Solve the original formula = (factorization)

= (reduction of zero factor elimination)

= (Apply Rule)

5. Using the properties of infinitesimal quantity

Example 5. Find the limit

Because the analysis does not exist, the algorithm cannot be used directly, so the function must be deformed and consistent first.

Solve the original formula = (constant deformation)

Because when, that is, infinitesimal, and ≤ 1, that is, bounded function, from the nature of infinitesimal, whether bounded function is multiplied by infinitesimal or infinitesimal,

Get =0。

6. Skills of using project splitting method

Example 6:

Analysis: Because =

Original formula =

7. Variable substitution

Example 7 Find the limit.

When analyzing time, the numerator and denominator tend to be the same, so the law cannot be directly applied and can be replaced by variables.

Solve the original formula =

Order, introduce new variables, and transform the original limit about into limit about. )

=. (Type, highest power on denominator)

8. Limit of piecewise function

Example 8 Suppose the limit of the discussion point exists.

The given functions are piecewise functions and piecewise points. To know whether it exists, we must start with the necessary and sufficient conditions for the existence of limit.

Solve the cause

So it does not exist.

Note: 1 tends to be from the left, so.

Note 2 because it tends to the right, then, therefore.

Jie Jun Hongzhi online schools

1, find the limit by definition.

2, using Cauchy criterion to find. Cauchy criterion: {xn** has a limit if and only if it is given ε >; 0, there is a natural number n, so when n >; When n, there is | xn-XM | < ε.

3. Find the limit by using the operational nature of the limit and the known limit. For example: lim (x+x0.5) 0.5/(x+1) 0.5 = lim (x0.5) (1+1/x0.5) 0.5/(x0.5).

4. Using inequality, that is, pinch theorem.

5. Use variable substitution to find the limit. For example, lim (x1/m-1)/(x1/n-1) can make x = y Mn: = n/m.

6. Find the limit with two important limits. ( 1)lim sinx/x = 1->； 0 (2) lim (1+1/n) n = e-> ∞ 7. Find the necessary limit by using monotone boundedness.

8. Use the property of function continuity to find the limit.

9. Use the Robida rule, which is used the most.

10, using Taylor formula, which is also very common.

A sequence of numbers arranged in a certain order is called a sequence. Every number in a series is called an item in this series. The number one is called the 1 item of this series (usually also called the first item), the number two is called the second item of this series ... and the number n is called the nth item of this series. Therefore, the general form of the sequence can be written as

a 1，a2，a3，…，an，…

The abbreviation is {an}, the sequence of finite items is "finite sequence", and the sequence of infinite items is "infinite sequence".

Starting from the second item, a series with an item greater than the previous item is called an increasing series;

Starting from the second item, the series with each item less than its previous item is called decreasing series;

Starting from the second item, some items are larger than the previous item, and some items are smaller than the previous item, which is called wobble sequence;

A series with periodic changes is called a periodic series (such as trigonometric function);

A series with equal terms is called a constant series.

General term formula: The relationship between the nth an of a series and the ordinal n of this series can be expressed by a formula, which is called the general term formula of this series.

The total number of numbers in a series is the number of items in the series. In particular, the sequence can be regarded as a function an=f(n) whose domain is a set of positive integers N* (or its finite subset {1, 2, ..., n}).

If it can be expressed by a formula, its general formula is a(n)=f(n).

[Edit this paragraph] Representation method

If the relationship between the nth term of the series {an} and the serial number n can be expressed by a formula, then this formula is called the general term formula of this series. For example, an = (-1) (n+1)+1.

If the relationship between the nth term of series {an} and its previous term or terms can be expressed by a formula, then this formula is called the recurrence formula of this series. For example, an = 2a (n-1)+1(n >; 1)

[Edit this paragraph] arithmetic progression

definition

Generally speaking, if a series starts from the second term, the difference between each term and its previous term is equal to the same constant. This series is called arithmetic progression, and this constant is called arithmetic progression's tolerance zone. The tolerance is usually expressed by the letter D.

abbreviate

A.p. (arithmetic progression can be abbreviated as A.P.).

arithmetic mean

Arithmetic progression, which consists of three numbers A, A and B, can be called the simplest arithmetic progression. At this time, a is called the arithmetic average of a and B.

This is very important: a = (a+b)/2

General term formula

an=a 1+(n- 1)d

an = Sn-S(n- 1)(n & gt； =2)

Sum of the first n terms

sn = n(a 1+an)/2 = n * a 1+n(n- 1)d/2

nature

The relationship between any two am and an is:

an=am+(n-m)d

It can be regarded as arithmetic progression's generalized general term formula.

From arithmetic progression's definition, general term formula and the first n terms formula, we can also deduce that:

a 1+an = a2+an- 1 = a3+an-2 =…= AK+an-k+ 1，k∈{ 1，2，…，n}

If m, n, p, q∈N*, m+n=p+q, then there is.

am+an=ap+aq

Sm- 1=(2n- 1)an，S2n+ 1 =(2n+ 1)an+ 1

Sk, S2k-Sk, S3k-S2k, …, Snk-S(n- 1)k… or arithmetic progression, and so on.

Sum = (first item+last item) × number of items ÷2

Number of items = (last item-first item) ÷ tolerance+1

First Item =2, Number of Items-Last Item

Last item =2, number of items-first item

Let A 1, A2 and A3 be arithmetic progression. Then a2 is the arithmetic average, then 2 times a2 equals a 1+a3, that is, 2a2=a 1+a3.

App application

In daily life, people often use arithmetic progression, for example, to grade the sizes of various products.

When there is little difference between the maximum size and the minimum size, it is often classified by arithmetic progression.

If it is arithmetic progression, and an = m and am = n, then a (m+n) = 0.

[Edit this paragraph] Geometric series

definition

Generally speaking, if a series starts from the second term and the ratio of each term to its previous term is equal to the same constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually represented by the letter Q.

abbreviate

Geometric series can be abbreviated as G.P. (geometric series).

geometric mean

If a number G is inserted between A and B to make A, G and B geometric series, then G is called the equal ratio median of A and B. ..

It matters: G2 = ab；; G= (ab)^( 1/2)

Note: two terms in the equal proportion of two nonzero real numbers with the same sign are opposite, so G 2 = AB is a necessary and sufficient condition for G and B to become geometric series.

General term formula

an=a 1q^(n- 1)

an=Sn-S(n- 1) (n≥2)

Sum of the first n terms

When q≠ 1, the formula of the sum of the first n terms of the geometric series is

sn=a 1( 1-q^n)/( 1-q)=(a 1-an*q)/( 1-q)(q≠ 1)

nature

The relationship between any two terms am and an is an = am q (n-m).

(3) A1an = a2an-1= a3an-2 = … = akan-k+1,k ∈ {1 can be deduced from the definition of geometric series, the general term formula, the first n terms and the formula.

(4) Equal ratio median term: aq ap = ar * 2, ar is ap, and aq is equal ratio median term.

If π n = A 1 A2 … an, then π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1.

In addition, each term is a geometric series with positive numbers, and the same base number is taken to form a arithmetic progression; On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic.

Nature:

(1) if m, n, p, q∈N*, and m+n = p+q, then am an = AP AQ；;

(2) In geometric series, every k term is added in turn and still becomes a geometric series.

G is the median term in the equal proportion of A and B, and G 2 = AB (G ≠ 0).

(5) The sum of the top n terms of geometric progression Sn = a1(1-q n)/(1-q).

In geometric series, the first term A 1 and the common ratio q are not zero.

Note: in the above formula, a n stands for the n power of a.

App application

Geometric series are often used in life.

For example, banks have a way of paying interest-compound interest.

That is, the previous interest and Hepburn gold price are counted as principal.

In calculating the interest of the next period, which is what people usually call rolling interest.

The formula for calculating the sum of principal and interest according to compound interest: sum of principal and interest = principal *( 1+ interest rate) deposit period.

If a series starts from the second term and the ratio of each term to the previous term is equal to the same constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually expressed by the letter q (q≠0).

The general formula of geometric series is: an = a 1 * q (n- 1).

If the general formula is transformed into an = a 1/q * q n (n ∈ n *), when q > 0, an can be regarded as a function of the independent variable n, and the point (n, an) is a set of isolated points on the curve y = a1/q * q X.

(2) Sum formula: Sn=nA 1(q= 1)

sn=a 1( 1-q^n)/( 1-q)

=(a 1-a 1q^n)/( 1-q)

= a1/(1-q)-a1/(1-q) * q n (that is, a-AQ n)

(Premise: Q is not equal to 1)

The relationship between any two terms am and an is an = am q (n-m).

(3) A1an = a2an-1= a3an-2 = … = akan-k+1,k ∈ {1 can be deduced from the definition of geometric series, the general term formula, the first n terms and the formula.

(4) Equal ratio mean term: AQAP = Ar 2, Ar is AP, and AQ is equal ratio mean term.

If π n = A 1 A2 … an, then π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1.

In addition, each term is a geometric series with positive numbers, and the same base is taken to form a arithmetic progression; On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic.

[Edit this paragraph] General sequence of general term solutions

Generally speaking, there are:

an=Sn-Sn- 1 (n≥2)

Cumulative sum (an-an-1= ... an-1-an-2 = ... a2-a1= ... add the above items to get one).

Quotient-by-quotient total multiplication (for a series with unknown numbers in the quotient of the latter term and the previous term).

Reduction method (transforming a sequence so that the reciprocal of the original sequence or the sum with a constant is equal to the difference or geometric series).

Special:

In arithmetic progression, there are always Sn S2n-Sn S3n-S2n.

2(S2n-Sn)=(S3n-S2n)+Sn

That is, the three are arithmetic progression and geometric progression. Tri-geometric series.

Fixed point method (often used in general fractional recurrence relation)

[Edit this paragraph] How to write the generic name of special series?

1,2,3,4,5,6,7,8.......- an=n

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8......- an= 1/n

2,4,6,8, 10, 12, 14.......- an=2n

1,3,5,7,9, 1 1, 13, 15.....- an=2n- 1

- 1, 1,- 1, 1,- 1, 1,- 1, 1......an=(- 1)^n

1,- 1, 1,- 1, 1,- 1, 1,- 1, 1......——an=(- 1)^(n+ 1)