What is the role of mathematical functions in life?

It is used to solve some scientific problems.

Theoretical Basis of Architectural Engineering.

A factory produces a certain product, the ex-factory price of each product is 50 yuan, and its cost is 25 yuan. Because in the production process, on average, there are cubic meters of sewage discharged for each product produced, so in order to purify the environment, the factory designed two plans to treat the sewage and prepare for implementation.

Scheme 1: The factory wastewater is purified and then discharged. The cost of raw materials used for each cubic meter of sewage treatment is 2 yuan, and the monthly loss fee of sewage equipment is 30,000 yuan;

Scheme 2: The factory discharges the sewage to the sewage treatment plant for unified treatment. For every 1 cubic meter treated, a sewage fee of 14 yuan is charged.

Question: 1 Set up a factory to produce x products per month, and the monthly profit is y yuan, and find the functional relationship between y and x when treating sewage according to scheme 1 and scheme 2 respectively; (Profit, Total Revenue, Total Expenditure).

2 When setting up a factory with a monthly production capacity of 6,000 products, if you, as the factory manager, should choose which sewage treatment plan should be selected without polluting the environment and saving money, please explain it through calculations.

Solution: (1) Set the selection scheme 1 with a monthly profit of y1 yuan; The monthly profit of option 2 is Y2 yuan.

According to option 1, it is available.

y1＝（50－25）x－2×

25x－x－30000

24x－30000.

y1＝24x－30000.

According to option 2, it is available.

y2＝（50－25）x－14×

25x－7x

18x. y2＝18x.

2) When x 6000, y1 24x 30000 24 6000 30000 114000 (yuan), y2 18x 18 6000 108000 (yuan), y1 y2

It is not enough to form ability by reading books, but more importantly, participating in practice. For example, if you learn to swim alone, you can't learn to swim if you are familiar with books such as "swimming instruction" and listen to the instructor talk about how to ventilate and how to master the movements of the whole body, especially the limbs. You don't have to get off the water, and combined with the guidance, you can really learn to swim in practice.

The research study we are carrying out now is precisely to enable students to form the knowledge and abilities needed for future study and work through various "learning to swim" practices. Through our research work some time ago, the students have basically mastered the methods and means of research problems, if the students are interested, we can also pay attention to other aspects of the things that can be studied, through our research, we deeply realize that there is mathematics around us, mathematics is around, in the future learning process, as long as we have the courage to explore, some students may become real inventors, creators, our current research makes it a foundation, Through our research, we will open up ideas and create good conditions for becoming a mathematician and inventor in the future.

Life is inseparable from mathematics, mathematics is inseparable from life, mathematical knowledge comes from life and is higher than life, so what is the role of mathematics in life?

1. Mathematics is a discipline that studies concepts such as quantity, structure, change, and spatial models. Through the use of abstraction and logical reasoning, it is generated from counting, calculating, measuring, and observing the shape and motion of objects. Mathematicians have expanded these concepts in order to formulate new conjectures and to establish rigorously deduced truths from appropriately selected axioms and definitions.

2. Mathematics is a highly logical subject. Learning mathematics and doing math problems can help to exercise divergent thinking and logical ability, and mathematics can also make people learn to think about problems and make people wise.

3. Learn mathematics. Grocery shopping, accounting, finance, statistics, construction, ......The various uses speak for themselves.

That's all for the role of mathematics in life.

In fact, the function comes from life, mainly to prepare for the plan, and after accurately drawing the function image, it can clearly show the implementability of the plan and the accuracy of the plan. However, the function value takes into account the problem of numbers, and does not involve economic politics, so it is not very convenient in large problems.

In mathematics, a function is a relation that coincidentally causes each element in one set to correspond to the unique element in another (possibly identical) set. The concept of function is fundamental to every branch of mathematics and quantification.

The terms function, mapping, correspondence, and transformation usually mean the same thing.

Put simply, a function is a "law" that assigns a unique output value to each inputThis "rule" can be expressed in terms of function expressions, mathematical relationships, or a simple ** that lists the input values in line with the output values. The most important property of a function is its determinism, i.e. the same input should always have the same output for the key (note that the reverse may not be true).

From this perspective, a function can be thought of as a "machine" or a "black box" that transforms a valid input value into a unique output value. The input value is usually called the number of parameters of the function, and the output value is called the value of the function.

Definition of function: Given a set of numbers a, suppose the element in it is x. Now apply the corresponding rule f to the element x in a, denoted as f(x), to get another set b.

Suppose the element in b is y. Then the equivalence relationship between y and x can be expressed by y=f(x). We call this relation a function relation, or a function for short.

The concept of a function has three elements: the definition domain a, the value range c, and the corresponding law f. The core of this is the correspondence law f, which is the essential feature of functional relations.

The first thing to understand about extended representation is that a function is a correspondence that occurs between sets. Then, it is necessary to understand that there is more than one function relationship between a and b. Finally, it is important to understand the three elements of a function.

The corresponding law of the function is usually expressed in analytical, but a large number of functional relations cannot be expressed analytically, and can be expressed in images, ** and other forms. Concept: In a process of change, the amount of change is called a variable (in mathematics, it is often x, and y changes with the change of x value), and some values do not change with the variable, we call them constants. Independent Variables (Functions):

A variable that is associated with another quantity, and any value of this quantity can find a fixed value in that quantity. Dependent variable (function.

Application of unary primary functions.

Unary linear functions are widely used in our daily life. When engaging in buying and selling, especially consumption activities, in social life, if the linear dependence of variables is involved, the unary function can be used to solve the problem.

For example, when shopping, renting a car, or staying in a hotel, operators often offer two or more payment options or concessions for promotional, ** or other purposes. Applications of unary quadratic functions.

When an enterprise carries out large-scale production such as construction, breeding, afforestation and greening, product manufacturing and other large-scale production, the relationship between its profit and investment can generally be expressed by a quadratic function. Business managers often rely on this knowledge to anticipate the prospects for business development and project development. Through the quadratic function relationship between investment and profit, the future benefits of the enterprise can be judged whether the economic benefits of the enterprise have been improved, whether the enterprise is in danger of being merged, and whether the project has development prospects.

Common methods are: finding the maximum value of a function, the maximum value on a monotonic interval, and the value of a function corresponding to an independent variable.

Application of trigonometric functions.

The application of trigonometric functions is extremely extensive, and here we will only talk about the simplest and most common category - the application of acute trigonometric functions: the problem of "mountain and forest greening".

If you don't know math, how do you know how much to find if you give someone 100.

Everything in life, everything can be expressed in terms of functions. But this function is very, very complex. And these very, very complex functions are made up of these very, very simple functions that you're learning right now.

For example: parabolic function without resistance + drag function = the motion process of throwing an object in reality. And what you're learning now is the little scum function: the parabolic function without resistance, the friction relationship function, these little scum functions.

Another example: uniform linear motion function a + uniform linear motion function b+.Uniform linear motion function n = variable velocity motion in reality.

Application of unary primary functions.

Unary linear functions are widely used in our daily life. When people are engaged in buying and selling, especially consumption activities, in social life, if the linear dependence of variables is involved, the unary one-dimensional function can be used to solve the problem.

For example, when we make a purchase, rent a car, or stay in a hotel, the operator often offers us two or more payment options or offers for promotional,** or other purposes. This is when we should think twice and dig deeper into the math in our minds to make an informed choice. As the saying goes:

From Nanjing to Beijing, what is bought is not sold fine. "We must not blindly follow, so as not to fall into the small trap set by the merchant and suffer immediate losses.

Now, I will tell you about one of my personal experiences.

With the diversification of preferential forms, "selective preferential treatment" has gradually been adopted by more and more operators. Once, I went shopping at the "Wumart" supermarket, and an eye-catching sign attracted me, which said that the purchase of teapots and teacups can be discounted, which seems to be rare. What's even stranger is that there are actually two ways to get a discount:

1) Sell one get one free (i.e. buy a teapot and get a teacup free); (2) 10% discount (i.e. payment of 90% of the total purchase price). There are also prerequisites for the purchase of more than 3 teapots (20 yuan for teapots and 5 yuan for teacups).

From this, I can't help but think: is there a difference between the two preferential methods? Which is cheaper?

I naturally thought of functional relations, and I was determined to apply the knowledge of functions I had learned to solve this problem analytically.

I wrote on paper:

Suppose a customer buys x teacups and pays y yuan, (x>3 and x n), then.

Pay y1=4 20+(x-4) 5=5x+60 with the first method;

Pay y2=(20 4+5x) 90%=

Next, compare the relative size of y1y2.

Let d=y1-y2=5x+60-(

Then it's time to discuss:

When d>0, >0, i.e., x>24;

when d=0, x=24;

when d=0, x=24;

To sum up, when there are more than 24 teacups purchased, method (2) saves money; When buying exactly 24 of them, the two methods** are equal; When the number of purchases is between 4 and 23, method (1) is cheaper.

It can be seen that the use of a one-dimensional one-time function to guide shopping not only exercises the mathematical mind, divergent thinking, but also saves money and eliminates waste, which is really a double win!

1.It has to do with bank compounding.

Let the principal a, the annual interest rate r, calculated by compound interest, how many years after the sum of principal and interest be b?

n=(lnb-lna)/ln(1+r).

2.Logarithmic growth.

That is, the relationship between the change of a variable y and time x is approximately a logarithmic function.

As x increases, y grows more and more slowly.

The opposite of it is exponential growth. Exponential growth is more commonly used a little.

3.Find tangents, areas, volumes, etc.

This requires knowledge of advanced mathematics.

For example, the curves y=1 x,x=a,x=b(a,b>0), and the area they enclose is lnb-lna=ln(b a)

To put it bluntly, for you, there is a bird.

A function is that there are two variables x and y in a certain change process, and the variable y changes with the variable x, and it depends on x. If the variable x takes a specific value and y takes the corresponding value according to a definite relationship, then y is said to be a function of x. This principle was put forward by the French mathematician Riemann in the 19th century, but it was first produced by the German mathematician Caibunitz.

He and Newton were the inventors of calculus. At the end of the 17th century, in his article, the phrase "function" was first used"word. Translated into Chinese, it means "function."

However, it doesn't have the same connotation as the term function we use today, it denotes concepts such as "power", "coordinates", "tangent length", etc.

It was not until the 18th century that the French mathematician D'Alembert redefined the function in his research, and he believed that the so-called function of variables refers to the analytic expression composed of these variables and constants, that is, the analytic expression of the functional relationship. Later, the Swiss mathematician Euler further standardized the definition of a function, and he believed that a function is a curve that can be traced. The images of primary functions, images of quadratic functions, images of proportional functions, and images of inverse proportions are all represented by the image method.

If D'Alembert and Euler's methods are used to express functional relations, each has its advantages, but if it is used as a definition of function, it still has shortcomings. Because both methods are still superficial, and do not indicate the essence of the function.

In the mid-19th century, the French mathematician Li Jian absorbed the results of Leibniz, D'Alembert and Euler, and for the first time accurately proposed the definition of a function: if a certain quantity depends on another quantity, so that when the latter quantity changes, the previous quantity also changes, then the former quantity is called the function of the latter quantity. The most important feature of Riemann's definition is that it highlights the relationship between dependencies and changes, and reflects the essential properties of the concept of functions.