2.6 Functional model and its application
Emphasis and difficulty: transform practical problems into function models, compare the growth differences of constant function, linear function, exponential function and logarithmic function models, and illustrate the significance of realizing the growth of different types of functions such as linear rise, exponential explosion and logarithmic growth.
Requirements: ① understand the growth characteristics of exponential function, logarithmic function and power function, and know the significance of growth of different function types such as linear rise, exponential growth and logarithmic growth;
② Understand the wide application of function model (such as exponential function, logarithmic function, power function, piecewise function, etc.).
Classic example: 1995, the total population of China is1200 million. If the annual natural population growth rate is controlled at 1.25%, in which year will the total population of China exceed1400 million?
Classroom exercises:
1. The temperature of an object in a day is a function of time t: T(t)=t3-3t+60, the time unit is hours, and the degree-day is. When t=0 means noon 12:00, and then the value of t is positive, the temperature at 8 am is ().
a . 8 b . 1 12 c . 58d . 18
2. A store sells two kinds of goods with different prices, A and B. Because the price of goods A has been increased by 20% for two consecutive times and the price of goods B has been reduced by 20% for two consecutive times, they are all sold at 23.04 yuan per piece. If the store sells one of these two commodities at the same time, the store will make a profit compared with the situation where the price does not rise or fall: ()
A. Earn 5.92 yuan more B. Earn 5.92 yuan less C. Earn 28.92 yuan more D. Earn the same profit.
3. Some parts needed in factory production can be purchased or produced by ourselves. If purchased, the price of each set is 1. 10 yuan; If you produce by yourself, the monthly fixed cost will increase by 800 yuan, and the materials and labor required to produce each accessory will be 0.60 yuan, so the turning point in deciding whether to outsource or produce this accessory is () pieces (that is, how many pieces are produced by yourself).
a 1000 b 1200 c 1400d 1600
4. In a mathematical experiment, the following set of data was collected by a graphic calculator.
x-2.0- 1.0
y 0.24 0.5 1. 1 2.02 3.98 8.02
So the functional relationship between x and y is closest to which of the following functions? (where a and b are undetermined coefficients) ()
a . y = a+bX b . y = a+bX c . y = a+log bX d . y = a+b/x
5. the functional relationship between the total cost y (ten thousand yuan) of a product and the output x (unit) is y = 3000+20x-0.1x 2 (0 < x <; 240, x∈N), if the price of each product is 250,000 yuan, the minimum output of the producer (sales income is not less than the total cost) is ().
100 b . 120 c . 150d . 180。
6. If you buy a GSM card of a mobile phone, you must pay a "basic monthly fee" (fixed monthly fee) to use it in 50 yuan, and the local call is 0.40 yuan per minute; When you buy the "Shenzhouxing" card, you don't charge the "basic monthly fee" when you use it, but the phone bill for local calls is 0.60 yuan per minute. If a user's monthly phone bill is 120 yuan, then it is cost-effective to buy a _ _ _ _ _ _ _ card.
7. A shopping mall bought a batch of daily necessities with the unit price of 6 yuan. After selling for a period of time, in order to get more profits, the mall decided to raise the price. Through experiments, it is found that at the price of each piece in 20 yuan, 360 pieces can be sold every month, and at the price of 25 yuan, 2 10 pieces can be sold every month. Assume that the number of pieces sold per month y (pieces) is a linear function of the price x (yuan/piece). Try to find out the relationship between y and X.
Under the condition that there is no backlog of goods and other factors are not considered, only by setting the sales price can we get the maximum profit every month.
The maximum monthly profit is.
8. To produce a new product, an enterprise must first rely on advertising to sell it. The advertising effect of a product should be the difference between the sales amount of the product and the advertising fee. If the sales amount is directly proportional to the arithmetic square root of the advertising fee, according to the sampling survey of the market, the sales amount is 1 1,000 yuan for each advertising fee paid. It is required that enterprises should invest in advertising fees in order to obtain the maximum advertising effect.
9. This shop sells teapots and teacups. Each teapot costs 20 yuan, and each teacup costs 5 yuan. The store has formulated two preferential measures: (1) buying teapots and sending teacups; (2) Pay 92% of the total price; A customer needs to buy four teapots and several teacups (not less than four). When purchasing the quantity of teacups, it is more economical to press (2).
10. A right-angled triangular iron sheet with right-angled sides of 40 cm and 60 cm respectively should be cut into a rectangle. The right angle of the triangle is an angle of the rectangle, so the maximum area of the rectangle is.
165438+
(1) Write the functional relationship between Y and T after taking medicine;
(2) It is effective to treat diseases when the drug content per milliliter of blood is not less than 4 micrograms. If the patient takes medicine for the first time at 7: 00 a.m., it's best to ask how the time of taking medicine (***4 times) is arranged.
12. There are frequent personnel exchanges between two similar and important cities in a province. In order to alleviate the traffic pressure, a special railway was specially built and trains were used as buses. It is known that a train can go back and forth 16 times if it drags four carriages at a time; If you tow seven cars at a time, you can go back and forth 10 times. The number of trains per day is a function of how many cars are towed at a time, and each car can carry 1 10 people at a time. Q: How many times does this train go back and forth every day and how many cars does it have to tow at a time to maximize the number of operators? And find out the maximum number of operators per day.
13. Marketers have made data analysis on the relationship between the price of a commodity and the sales volume in the past few years, and found the following laws: every time the price of a commodity increases by x% (x > 0), the sales volume decreases by kx% (where k is a normal number). At present, the price of this commodity is A yuan and the sales volume is B. 。
(1) When k=, the total sales volume is the largest by how much the commodity price rises.
(2) In the process of appropriate price increase, find the value range of K when the total sales amount keeps increasing.
14. The quantities of a product produced by a factory in June, February and March this year were 10000, 12000 and 13000 respectively. In order to estimate the future monthly output, according to the number of products in these three months, a function is used to simulate the relationship between the monthly output y and the monthly output x of this product. The simulation function may