Exercise:

1. According to market requirements, the impurity content of the solution produced by a factory should not exceed 0. 1%, and the initial impurity content of the solution is 2%. Now, it has been filtered. It is known that every time the impurity content is reduced by 13, the minimum filtration times for the product to meet the market requirements are (reference data: lg 2≈0.30 1.

A. 10

C.8 D.7

It is reported that in the past 50 years, global warming has reduced the snow area in the Arctic Ocean by 5% in winter. If the winter snow area in 20 10 is set as m at this rate, the functional relationship between the winter snow area y and x in the Arctic Ocean after x years from 20 10 is ().

In order to stimulate innovation, a company plans to increase its investment in R&D bonus year by year. If the company invested RMB 654.38+03 million in R&D dividends in 2065.438+06, and the annual investment in R&D dividends increased by 65.438+02% compared with the previous year, the year when the company began to invest more than RMB 2 million in R&D dividends in 2065 is 438+06 (reference data: LG 65.438)

20 18

C.2020 d.202 1 year

4. A virus multiplies twice after 30 minutes, and it is known that the spreading rule of the virus is y = ekt (where k is constant, t stands for time, unit: hours, and y stands for the number of viruses), then k = _ _ _ _ _ _ _ _ _ _, and after 5 hours, 1 virus can multiply to _ _ _ _ _ _ _ _.

5. The temperature change of an object at room temperature can be described by Newton's cooling law: let the initial temperature of the object be t and the temperature after a certain time t be t, then t-ta = (t-ta), where ta represents the ambient temperature and h is called the half-life. At present, a cup of instant coffee brewed with hot water at 88℃ is placed in a room at 24℃. If the coffee is cooled to 40℃, it will take 20 minutes.

Logarithmic function model of knowledge points

6. According to statistics, the relationship between the number y of cranes wintering in Poyang Lake National Wetland Park and the time x (year) approximately satisfies the relationship y = alog3 (x+2). It was observed that there were 3,000 wintering cranes in the winter of 2065,438+04 (65,438+0), and it is expected that there will be wintering cranes in the winter of 2020 ().

4000 years

C.6000，d . 7000

7. We live in a world of sound, and people have different requirements for the volume of sound on different occasions. The unit of volume is decibel (dB). For the sound wave with intensity I, its volume η can be calculated by the following formula: η = 10LG II0 (where I is the lowest sound intensity that human ears can hear). Let η 1 = 70 dB.

A.76 times B. 10 times

C. 1076 times D.ln 76 times

8. It is known that the relationship between the magnitude r of the earthquake and the energy e released by the source is r = 23 (LG E-C) (C is a constant), so the energy released by the earthquake of magnitude 9.0 is about _ _ _ _ _ _ _ _ times that released by the earthquake of magnitude 7. 1 (reference data:102.85 87700).

9. manned spaceship was launched by a rocket. It is known that the take-off weight of a rocket is the sum of the weight of the arrow (including the plane on board) and the fuel weight x t, and the maximum speed y km/s of the rocket is assumed to be y = k [ln (m+x)-ln (2m)]+4LN2 (where k) without considering the air resistance.

(1) Find the functional relationship between the maximum velocity y km/s of this rocket and the fuel weight x t;

(2) If the take-off weight of this type of rocket is 479.8 t, how many tons of fuel (accurate to 0. 1 t, e≈2.7 18) should be loaded to make the maximum flight speed of the rocket reach 8 km/s, and then the spacecraft can be successfully put into the predetermined elliptical orbit?

Courseware:

Teaching plan:

learning target

Understand the relationship between the root of the equation and the zero point of the function;

Understand the nature of function zero, master dichotomy, and use dichotomy to find the approximate solution of the equation;

Understand linear rise, exponential explosion and logarithmic growth, and compare the growth rates of exponential function, logarithmic function and power function; [Source: Subject Network ZXXK]

Can skillfully apply mathematical modeling to solve practical application problems of functions.

Cooperative learning [source: subject network ZXXK]

I. Review of knowledge

(A) the whole chapter knowledge points

The zero point of 1. function, the root of equation, the zero point of function and its properties.

2. Dichotomy, the step of finding the zero point of the function by dichotomy.

3. Comparison of several different growth function models (linear rise, exponential explosion, logarithmic growth), exponential function, logarithmic function and power function.

4. The basic process of applying function model to solve practical problems.

(2) Summary of methods

1. The function y=f(x) is the equation f(x)=, so the problem of finding the zero point of the function can usually be transformed into the problem of finding the root of the corresponding equation.

2. The discussion of the roots of quadratic equations in one variable is widely used in high school mathematics, and there are usually three ways to solve such problems:

(1) Use the root formula;

(2) imaging with quadratic function;

(3) Using the relationship between roots and coefficients.

No matter which method is adopted, the discriminant of roots can not be ignored, but because of the uninterrupted visualization of quadratic function, the discriminant of some problems has been implicit in the treatment of problems.

3. The general steps to find the zero point of a function by dichotomy:

It is known that the function y=f(x) is defined in the interval d, and the approximate value x of its sign-changing zero point x on d is found, so that the error between it and the zero point does not exceed a positive number ε, that is, |x-x|≤ε.

(1) take the closed interval [a, b] in d? D, making

Let a = a and b = B

(2) Take the midpoint of the interval [a, b], and the abscissa corresponding to this midpoint is

x=a+(b-a)=(a+b)。

Calculate f(x) and f(a).

Judgment: If f(x)=,;

If f (a) f (x) 0, the zero point is in the interval, so that A 1 = A, b1= x;

If f (a) f (x) >; The zero point is in the interval, so A 1 = X, b1= B.

(3) Take the midpoint of the interval [a 1, b 1], and the abscissa corresponding to this midpoint is

x 1 = a 1+(b 1-a 1)=(a 1+b 1)。

Calculate f(x 1) and f(a 1).

Judgment: if f(x 1)=, then x 1 is the zero point of f(x), and the calculation is terminated;

If f (a 1) f (x 1) 0, the zero point is in the interval [a 1, x 1], so that a2=a 1 and B2 = x1;

If f (a1) f (x1) >; , the zero point is in the interval [x 1, b 1], so a2=x 1, b2=b 1.

…

After the above steps are performed, the zero point of the function always lies in the interval [an, bn], which is the approximate zero point of the function y=f(x), and the calculation is terminated. At this time, the error between the approximate zero point of the function y=f(x) and the true zero point is less than ε.

4. for the straight line y=kx+b(k≥), the exponential function y = m ax(m >；; , a> 1), logarithmic function y = logbx(b >；; 1),

(1) Through examples and images, it is found that when the independent variable becomes very large, the exponential function grows faster than the linear function, and the linear function grows faster than the logarithmic function;

(2) Get multiple sets of data through calculator or computer, and combine with function images (images can be drawn with the help of modern information technology) to further understand:

Straight up, its growth is fixed;

Exponential growth, double growth, the growth rate is not linear growth. With the increasing of independent variables, the gap between linear growth and exponential growth is getting bigger and bigger. When the independent variable is large, the gap is staggering, so "exponential growth" can be described as "exponential explosion";

Logarithmic growth, its growth rate is gentle, and when the independent variable keeps increasing, its growth rate is less than linear increase.

5. On the interval (,+∞), although the function y = ax(a & gt；; 1)，y = logax(a & gt； 1)，y = xn(n & gt； ) are all increasing functions, but the growth rate is different, not in the same "grade". With the increase of x, y = ax (a >; 1) is growing faster and faster, and will far exceed y = xn(n & gt；; ) and y = logax(a & gt；; 1) will become slower and slower. Therefore, there will always be an x, when X >；; At x.

6. Modeling methods of practical problems.

(1) Carefully examine the questions and accurately understand the meaning of the questions;

(2) Starting from the problem, grasp the quantitative relationship, introduce variables appropriately or establish rectangular coordinate system, use existing mathematical knowledge and methods, express the quantitative relationship with mathematical symbols, and establish functional relationship;

(3) Study the domain of function relation, and give the answer according to the actual meaning of the question.

It must be pointed out that:

(1) Solve practical problems by establishing a function model, aiming at cultivating students' awareness of applying mathematics and their ability to analyze problems through examples;

(2) The mathematical model is a mathematical description of the actual problem by abstracting and summarizing the actual problem with mathematical language and reflecting or approximately reflecting the actual problem from a mathematical perspective.

7. Basic process of establishing function model and solving practical problems:

Second, give an example

Example 1 Draw the images of functions y=x3 and y=3x- 1, and write the approximate solution of equation x3=3x- 1 (accurate to. 1).

In example 2, the graphs of functions y=ax and y=logax are drawn for a=2, a= and a= respectively, and the number of solutions of equations y = ax and y = logax is obtained.

According to the government work report at the Third Session of the 11th Shanghai Municipal People's Congress, the gross domestic product (GDP) of Shanghai will reach 403.5 billion yuan in 20 13, and it is estimated that the GDP of Shanghai will increase by 9% in 20 14. The municipal party committee and the municipal government proposed that the annual natural growth rate of the permanent population in the city should be controlled at 0.08%. If both GDP and population increase at this rate, it will take at least years to make the per capita GDP of this city reach or exceed 20 13. (According to 20 13, the total resident population in this city is about 130,000. )

Example 4 Tomatoes from a certain place have been on the market since February 1. Through market research, the data of tomato planting cost Q (unit: yuan/102kg) and time to market T (unit: day) are as follows:

(1) According to the data in the above table, choose one of the following functions to describe the relationship between tomato planting cost q and time to market t 。

Q=at+b，Q=at2+bt+c，Q=a bt，Q=a logbt。

(2) Use the function you choose to find the number of days to market and the lowest planting cost of tomatoes.

Third, classroom exercises.

Textbook P 1 12 Review the reference questions A, 1, 2, 3, 4 and 5.

Fourth, class summary.

The close relationship between 1. function and equation is reflected in the connection between the zero point of function y=f(x) and the real root of corresponding equation f(x)=).

2. Dichotomy is a common method to find the approximate solution of the equation, and it is necessary to master the general steps to find the approximate solution of the equation by dichotomy;

3. Different function models can describe different changing laws in the real world. Exponential function, logarithmic function and power function are commonly used functional models with different growth laws in the real world.

4. The application of function model, on the one hand, uses the known function model to solve the problem; On the other hand, it is necessary to establish a suitable function model, and use the obtained function model to explain related phenomena and predict some development trends;

5. Pay attention to give full play to the role of information technology in function application research.

Verb (abbreviation for verb) assignment

Textbook p 1 12 A group review reference questions 7, 8 and 9; Group b, question 1, 2.

Reference answer

Second, give an example

Example 1 solution: display images with functions y=x3 and y=3x- 1 At the intersection of two function images, the function values are equal.

So the abscissa of these three intersections is the solution of equation x3=3x- 1

As can be seen from the figure, the solutions of equation x3=3x- 1 are in the intervals (-2,-1), (1) and (1, 2) respectively. Therefore, for the interval (-2,-1),

Solution of Example 2: Using Excel, graphic calculator or other drawing software, you can draw the image of the function, as shown in the following figure.

According to the image, we can know that when a=2, a= and a=, the number of solutions of the equation ax=logax is 2 1 respectively.

Example 3 Solution: Assuming that it takes N years, it is deduced from the meaning of the question.

[Source: Z&XX & ampk.Com]

Simplify to ≥2, and the solution is n>8.

A: It will take at least nine years.

Solution of Example 4: According to the data provided, the function describing the relationship between tomato planting cost Q and time to market T can't be a constant function, so any of the functions Q = at+B, Q = a Bt and Q = a logbt should have a≦. At this time, the above three functions are monotonous functions, which are inconsistent with the data provided in the table. So I chose the quadratic function.

Substitute the three groups of data provided in the table into Q=at2+bt+c respectively, and get

solve

Therefore, the function describing the relationship between tomato planting cost Q and time to market T is Q=t2-t+.

(2) When t=-= 150 days, the tomato planting cost is the lowest.

Third, classroom exercises.

1.[ Source: Xueke.com]C2

3. Let the running time of the train from place A to place B be T, and the distance from the train to place C after time T be Y, then

y=

The functional image is

4.( 1) cylindrical;

(2) a frustum with a small upper bottom and a large lower bottom;

(3) a frustum with a large upper bottom and a small lower bottom;

(4) It is cylindrical in two sections, the upper part is large and the lower part is small. (Figure omitted)

5. Let f(x)=2x3-4x2-3x+ 1, and the function image is as follows:

The function has a zero point in the interval (-1,), (1) and the interval (2,3) respectively, so the maximum root of the equation 2x3-4x2-3x+ 1= should be in the interval (2,3).

Take the midpoint of the interval (2,3) x1= 2.5, and use a calculator to calculate f(2.5)=-25.

Because f (2.5) f (3) 0, x∈(2.5, 3).

Then take the midpoint of (2.5,3 3) x2 =2.75, and use a calculator to calculate f(2.75)≈4.09.

Because f (2.5) f (2.75) 0, x∈(2.5, 2.75).

Similarly, we can get x ∈ (2.5,2.625), X ∈ (2.5,2.5625), X ∈ (2.5,2.53125), X ∈ (2.5/kloc-0).

From | 2.534375-2.515625 | = .00781250.01,

At this time, both endpoints of the interval (2.5 15625, 2.5234375) are accurate to the approximate value of .01,so the maximum root of the equation 2x3-4x2-3x+ 1= accurate to .01is about 2.52.

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