"Application of Functional Model" Exponential function and logarithmic function PPT

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"Application of Functional Model" Exponential function and logarithmic function PPT

Part One Content: Core Competency Cultivation Objectives

1. Be able to use known function models to solve practical problems.

2. Be able to establish functional models to solve practical problems.

3. Understand fitting function models and solve practical problems.

Application PPT of functional model, part 2: independent preview

1. Use specific function models to solve practical problems

1. In addition to the mathematical models mentioned in the previous chapter, what other common mathematical models are there?

Tip: The use of specific functions to solve practical problems is what we should pay attention to. The application of specific functions has many manifestations in life. After learning this part of functions, I hope students can focus on using the functions and exponents mentioned in the previous chapter. Common functions such as functions and logarithmic functions are used to solve problems. Here are several common function models:

(1) Exponential function model: f(x)=a·bx+c (a, b, c are constants, a≠0, b>0, and b≠1);

(2) Logarithmic function model: f(x)=mlogax+n (m, n, a are constants, m≠0, a>0, and a≠1);

2. Do it

When a certain cell divides, one cell divides into two cells, and two cells divide into four cells... There are two such cells, and the functional relationship between the number of cells y and x obtained after dividing x times is ()

A.y=2x B.y=2x-1

C.y=2x D.y=2x+1

Analysis: After splitting once, it becomes 2×2=22 (pieces) from 2, after splitting twice it becomes 4×2=23 (pieces),…, after splitting x times, it becomes 2x+1.

Answer:D

2. Fitting function model

1. The basic process of applying fitting function model to solve problems

2. What are the four steps generally required to solve practical application problems of functions?

Tips: The first step: analysis, association, transformation, abstraction;

Step 2: Establish a function model and convert practical application problems into mathematical problems;

Step 3: Solve the mathematical problems and obtain the results;

Step 4: Translate the mathematical results into conclusions about specific problems and provide answers.

Among these four steps, the most critical thing is to handle the second step well. As long as the function model is properly established, all problems can be easily solved on this basis.

3. Do it

"Red beans grow in the south, and spring will sprout a few branches." The figure shows a scatter plot of the red bean growth time t (month) and the number of branches y (branches). Then which of the following function models is used to model the relationship between the number of red bean branches and growth time? Best fit?()

A. Exponential function y=2t

B. Logarithmic function y=log2t

C. Power function y=t3

D. Quadratic function y=2t2

Analysis: According to the given scatter plot, it can be seen that the image is in the first quadrant, and the image is rising from left to right, and the growth rate is getting faster and faster. According to the growth trend of the function in the four options, we can get, The exponential function model is the best fit.

Answer:A

Application of function model PPT, part 3: inquiry learning

Application of exponential or logarithmic function models

Example 1 The original area of ​​a forest is a. It is planned to cut down some trees every year, and the percentage of the area cut down is equal each year. When half of the area is cut down, it will take 10 years.

In order to protect the ecological environment, the forest area must be retained at least 1/4 of the original area. It is known that as of this year, the remaining forest area is √2/2 of the original area.

(1) Find the percentage of area cut down each year;

(2) How many years has the forest been cleared as of this year?

(3) How many more years can it be cut down in the future?

Analysis: An exponential function model can be established to solve the problem.

Solution: (1) Suppose the annual percentage of felled area is x (0

That is, (1-x)10=1/2, the solution is x=1-(1/2)^(1/10).

(2) Assume that the remaining area after m years is the original √2/2,

Then a(1-x)m=√2/2a, that is, (1/2)^(m/10)=(1/2)^(1/2), m/10=1/2, the solution is m =5,

So as of this year, it has been cut down for 5 years.

(3) Assume that starting from this year, it can be cut down for n years at most,

Then the remaining area after n years is √2/2a(1-x)n. Let √2/2a(1-x)n≥1/4a,

That is, (1-x)n≥√2/4, (1/2)^(n/10)≥(1/2)^(3/2), n/10≤3/2,

The solution is n≤15. Therefore, it can be cut down for up to 15 years in the future.

Reflection 1. This question involves the issue of average growth rate. The solution can be expressed by an exponential function model, which can usually be expressed as y=N·(1+p)x (where N is the original basic number, p is the growth rate, and x is time )form.

2. In practical problems, exponential function models are commonly used in growth issues such as population growth, bank interest rates, and cell division.

Application PPT of functional model, part 4: practice in class

1. The image of the change of the driving distance s of a car in a certain distance with respect to time t is as shown in the figure, then the function model corresponding to the image is ()

A. Piecewise function B. Quadratic function

C. Exponential function D. Logarithmic function

Analysis: From the question picture, we know that in different time periods, the corresponding images are different, so the corresponding function model should be a piecewise function.

Answer:A

2. The vegetation in a certain area has been destroyed, and land desertification has become more and more serious. The measured added value of the desert in the past three years was 0.2 million hectares, 0.4 million hectares, and 0.76 million hectares respectively. The functional relationship between the increase in desert area y and the number of years x is: Approximately ()

A.y=0.2x B.y=1/10(x2+2x)

C.y=2^x/10 D.y=0.2+log16x

Analysis: When x=1, negate option B; when x=3, negate options A and D, and test that option C is closer.

Answer:C

3. It is known that there are two buckets A and B. There is a L of water in bucket A initially. The water in bucket A continues to flow into bucket B. After t min, the remaining water in bucket A follows the exponential decay curve y1=ae-nt. , then the water in bucket B is y2=a-ae-nt (n is a constant). Assume that at 5 minutes, the amount of water in bucket A and bucket B is equal, and after ____________ min, the water in bucket A is only a/ 8L.

Analysis: Because the amount of water in bucket A and bucket B is equal at 5 minutes,

So a·e-5n=a-a·e-5n,

So e-5n=1/2. Let a·e-nt=a/8,

Then e-nt=1/8=(1/2)^3=e-15n, so t=15.

So after 10 minutes, the water in bucket A will only be a/8 L.

Answer:10

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Update Time: 2024-10-03

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