One of the Breakthroughs —— Learning the Steps of Mathematical Modeling and Analysis
Steps of mathematical modeling and analysis:
1. Look at the topic. It should include the overall understanding and partial understanding of the meaning of the problem, as well as the analysis of the relationship and the understanding of the essence.
"Overall understanding" is to find out the events and research objects mentioned in the topic;
"Partial understanding" refers to grasping the key words in the topic and correctly grasping their meanings;
"Relationship analysis" is to find out the quantitative relationship between the related quantities in the question according to the meaning of the question;
"Understanding the essence" refers to grasping the main problems in the topic and correctly identifying their types.
2. Establish a mathematical model. The actual problem is abstracted as a mathematical problem, and the direct preparation of modeling is to find out the most critical quantitative relationship from various relationships in the final stage of investigation, and express this relationship with related quantities, numbers and symbols, so as to obtain the mathematical model for solving the problem.
3. Solve the mathematical model. According to the established mathematical model, choose the appropriate mathematical method, design a reasonable and simple operation mode, find the solution of mathematical problems, and pay special attention to the constraints such as the limitation of variable range in practical problems.
Step 4 check. It is necessary to check whether the obtained results are suitable for the mathematical model and judge whether the obtained results meet the requirements of practical problems, so as to give practical answers to the original problems.
2. Breakthrough 2-Mastering the specific methods of mathematical modeling and analysis
1. Relationship analysis. That is, the mathematical model of the problem is established by finding the quantitative relationship between keywords and key quantities.
Example 1. (Water supply for water tower) A factory has a water tower with a capacity of 300 tons, which provides domestic and production water for the factory from 6 am to afternoon 10 every day. It is known that the domestic water consumption of this plant is 10 ton/hour, and the functional relationship between industrial water consumption w (ton) and time (unit: hour, defined as 6: 00 a.m.) is: water intake from water tower 10, water intake in the first stage 10 ton/hour, and the water intake per hour will increase with each level increase.
(1) Set the water inflow to the first gear, and write down the water storage at this moment;
(2) Ask which level of water intake to choose, which can not only ensure the water consumption of plants (the water in the water tower is not empty) but also prevent water from overflowing.
Understanding the topic: There are many keywords involved in the topic: domestic water consumption, industrial water consumption, water storage capacity, water inflow, and original quantity. Its quantitative relationship is: storage = incoming water-water consumption+original quantity, water consumption = domestic water consumption+industrial water consumption. The focus of the first question is to find the "first file of water inflow". The focus of the second question is that "the water in the water tower is not empty or overflowing" is transformed into "storage".
Establish a mathematical model: storage capacity = inflow-water consumption+original quantity, water consumption = domestic water consumption+industrial water consumption = 10. When selecting the water inflow in the first stage, the water storage capacity at any moment is, if the water in the trap is not empty or overflowing, then the problem will be transformed into certainty, which will be established forever in (). Solving mathematical model