Nowadays, mathematical knowledge and ideas are widely used in industrial and agricultural production and people's daily life. For example, after shopping, people should keep an account for year-end statistical inquiry; Go to the bank to handle savings business; Check the water and electricity charges of each household. These facilitate the use of knowledge of arithmetic and statistics. In addition, the "push-pull automatic telescopic door" at the entrance of the community and government compound; Smooth connection between straight runway and curved runway in sports field; Calculation of the height of the building whose bottom cannot be closed: determination of the starting point of two-way operation of the tunnel; The design of folding fan and golden section is the property of straight line in plane geometry, and it is about the application of solving Rt triangle knowledge. Because these contents don't involve a lot of high school mathematics knowledge, I won't go into details here.
It can be seen that throughout the ages, human society has been developing and progressing in the process of constantly understanding and exploring mathematics. Mathematics has played a decisive role in the development of human civilization.
Next, I will briefly talk about the application of mathematical knowledge in production and life from five aspects: function, inequality, sequence, solid geometry and analytic geometry.
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The first part is the application of functions.
We have learned eight kinds of functions: linear function, quadratic function, fractional function, irrational function, power, exponent, logarithmic function and piecewise function. These functions reflect the dependence of variables in nature from different angles, so the knowledge of functions in algebra is closely related to production practice and life practice. Here we focus on the application of the first two types of functions.
Application of unary linear function
One-dimensional linear function is widely used in our daily life. When people are engaged in business, especially in consumer activities in social life, if the linear correlation of variables is involved, one-dimensional linear function can be used to solve the problem.
For example, when we shop, rent a car or stay in a hotel, operators often provide us with two or more payment schemes or preferential measures for publicity, promotion or other purposes. At this time, we should think twice, dig deep into the mathematical knowledge in our minds and make wise choices. As the saying goes: "From Nanjing to Beijing, it is better to buy than to sell." Never follow blindly, lest you fall into the small trap set by the merchants and suffer immediate losses.
Next, I will tell you one thing I experienced personally.
With the diversification of preferential forms, "selective preferential treatment" is gradually adopted by more and more operators. Once, I went shopping in a supermarket in Wu Mei, and an eye-catching sign attracted me, which said that buying teapots and teacups could be discounted, which seemed rare. What's even more strange is that there are actually two preferential ways: (1) sell one for one (that is, buy a teapot and get a teacup); (2) 10% discount (that is, 90% of the total purchase price). There is also a prerequisite: buy more than three teapots (teapot 20 yuan/one, teacup 5 yuan/one). From this, I can't help thinking: Is there a difference between these two preferential measures? Which is cheaper? I naturally thought of the functional relationship, and decided to apply the knowledge of functions I learned to solve this problem by analytical methods.
I wrote on the paper:
Suppose a customer bought X cups and paid Y yuan (x>3 and x∈N), then
Pay y1= 4× 20+(x-4 )× 5 = 5x+60 according to the first method;
Use the second method to pay y2=(20×4+5x)×90%=4.5x+72.
Then compare the relative sizes of y 1y2.
Let d = y1-y2 = 5x+60-(4.5x+72) = 0.5x-12.
Then there will be a discussion:
When d>0, 0.5x-12 >; 0, namely x & gt24;
When d=0, x = 24.
When d < 0, x
To sum up, when buying more than 24 teacups, method (2) saves money; When only 24 pieces are purchased, the prices of the two methods are equal; When the purchase number is only between 4 and 23, the method (1) is cheap.
Visible, using a function to guide shopping, that is, exercise the mathematical mind, divergent thinking, but also save money, put an end to waste, really kill two birds with one stone!
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Second, the application of univariate quadratic function
When enterprises engage in large-scale production such as construction, breeding, afforestation and product manufacturing,
The relationship between profit and investment can generally be expressed by quadratic function. Business managers often predict the prospects of enterprise development and project development based on this knowledge. They can predict the future benefits of enterprises through the quadratic function relationship between investment and profit, so as to judge whether the economic benefits of enterprises have been improved, whether enterprises are in danger of being merged, and whether the project has development prospects. Common methods include: finding the maximum value of a function, the maximum value in a monotonous interval and the corresponding function value of an independent variable.
Third, the application of trigonometric functions
Trigonometric functions are widely used. Here only the simplest and most common type-the application of acute trigonometric function: the problem of "forest greening".
In forest greening, trees must be planted at equal distance on the hillside, and the distance between two trees on the hillside should be consistent with the distance between trees on the flat ground when projected on the flat ground. (as shown on the left) Therefore, foresters should calculate the distance between two trees on the hillside before planting trees. This requires a keen knowledge of trigonometric functions.
As shown in the figure on the right, let c = 90, B=α, flat distance d and hillside distance r, then the problem of secα=secB =AB/CB=r/d. ∴r=secα×d is solved.
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The second part is the application of inequality.
Inequalities commonly used in daily life are: one-dimensional linear inequality, one-dimensional quadratic inequality and average inequality. The application of the first two types of inequalities is exactly the same as that of their corresponding functions and equations, and the average inequality plays an important role in production and life. Below, I mainly talk about the application of mean inequality and mean value theorem.
In production and construction, many practical problems related to optimization design can usually be solved by applying mean inequality. Although the author has not personally experienced the application of mean inequality knowledge in daily life, it is not difficult to find that mean inequality and extreme value theorem can usually have the following extremely important applications from TV, newspapers and other news media and the application problems we have done: (focus on the "packaging can design" after the table)
Solution of optimal scheme with known conditions in practical activities
Theorem 1 Design the maximum value of perimeter or hypotenuse area of flower bed green space
Minimum function and extreme theorem of unit price and sales cost of operating cost II
Calculate the design voyage mileage, the number of people with limited load and the lowest fare by extreme value theorem 2.
Speed, various expenses and the corresponding minimum cost, and then from this
Proportional relationship to calculate the lowest fare.
(Fare = lowest fare+average profit)
Design of packaging tank (see table) (see table) (see table)
Design problems of packaging cans
1, "White Cat" washing powder bucket
The shape of the "White Cat" washing powder bucket is an equilateral cylinder (as shown on the right).
If the volume is constant and the thickness of the bottom surface and the side surface is the same, the height and radius of the bottom surface are
What is the relationship between the minimum material consumption (that is, the minimum surface area)?
Analysis: the volume must be = & gtлr h=V (fixed value).
= & gts = 2лr+2лRH = 2л(r+RH)= 2л(r+RH/2+RH/2)
≥2л3 (r h) /4 =3 2лV (if and only if r = RH/2 =>；; Take an equal sign when h=2r),
It should be designed as an equilateral cylinder with h = d.
2. The problem of "canned food"
The upper and lower radius of the cylinder is R and the height is H. If the volume is constant V, the upper and lower bottoms
The thickness is twice that of the side. What is the relationship between the height and the radius of the bottom?
Province (i.e. minimum surface area)?
Analysis: h=2d can also be obtained by applying the mean value theorem.
Write, this article is omitted) ∴ should be designed as a cylinder with h=2d.
In fact, the application of inequality, especially mean inequality, in production practice is far more than that, so I won't list them here.
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The third part is the application of sequence.
In real life and economic activities, many problems are closely related to series. For example, installment payment, personal investment and financial management, population problems, resource problems and so on. , you can use the knowledge of series to analyze and solve them.
This paper focuses on the application of arithmetic progression and geometric progression in real life and economic activities.
(a) A series of mortgage payments.
With the implementation of the central government's proactive fiscal policy, the introduction of mortgage loan system (provident fund loan) has greatly stimulated people's desire for consumption, expanded domestic demand and effectively stimulated economic growth.
As we all know, mortgage loans (provident fund loans) have equal monthly principal and interest. It is often difficult for people to know how this equal amount is obtained and how much principal should be returned to the bank in a few months. Let's find a solution to this problem.
If the loan amount is a0 yuan and the monthly interest rate of the loan is P, the repayment method is equal to the monthly repayment of the principal and interest of A yuan. Let the principal after repayment in the nth month be an, and then there are:
a 1=a0( 1+p)-a，
a2=a 1( 1+p)-a，
a3=a2( 1+p)-a，
......
an+ 1=an( 1+p)-a，.........................(*)
Convert (*) into (an+1-a/p)/(an-a/p) =1+p.
It can be seen that {an-a/p} is a geometric series with a 1-a/p as the first term and 1+p as the common ratio. All problems related to mortgage payment in daily life can be calculated according to this formula.
(2) Other application problems related to sequence.
Sequential knowledge is not only widely used in personal investment and financial management, but also indispensable in enterprise management. Readers and friends must have done a lot of application problems! Although these practical problems are slightly higher than those in real life, they are the kind of problems that best reflect the close connection between mathematical knowledge and real life. Therefore, answering application questions is helpful for us to understand and understand the wide application of mathematics in daily life. Let's take a look at an application problem in the sampling test in Xicheng District, Beijing in 2003-Senior Two Mathematics Test Paper.
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