"Application of Functions (1)" Concept and Properties of Functions PPT

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"Application of Functions (1)" Concept and Properties of Functions PPT

Part One: Explanation of Curriculum Standards

1. Understand that function is an important mathematical language and tool for describing variable relationships and laws in the objective world.

2. In actual situations, appropriate function types can be selected to describe the changing patterns of real problems.

3. Be able to apply linear, quadratic and power function models to solve some simple practical problems.

Application of functions PPT, part 2: independent preview

Use concrete function models to solve practical problems

1. What are the common mathematical models?

Tip: The use of specific functions to solve practical problems is what we need to 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 linear functions, quadratic functions, and powers. Common functions such as functions and piecewise functions are used to solve problems. The following are several common function models:

(1) Linear function model: f(x)=kx+b(k, b are constants, k≠0);

(2) Inverse proportional function model: f(x)=k/x+b(k, b are constants, k≠0);

(3) Quadratic function model: f(x)=ax2+bx+c (a, b, c are constants, a≠0);

Note: The quadratic function model is the most widely used model in high school and is the most common in application questions in the college entrance examination.

(4) Power function model: f(x)=axn+b(a, b, n are constants, a≠0, n≠1);

(5) Piecewise function model: This model is actually a synthesis of the above two or more models, so it is also widely used.

2. The mathematical model can use the following chart to represent the solution process.

3. Do it

Assume that the revenue R from advertising sales of a certain product and the advertising fee A satisfy the relationship R=a√A, and the advertising effect D=R-A, then when A=_______, the maximum advertising effect is achieved.

parse

D=a√A-A=-(√A)2+a√A=-(√A "-" a/2)^2+a^2/4.

When √A=a/2, that is, A=a^2/4, D obtains the maximum value.

Function Application PPT, Part 3: Inquiry and Learning

Application of linear function model

Example 1 The relationship between the total daily production cost y (yuan) of a factory and the daily output x (sets) of stationery boxes is y=6x+30 000, and the ex-factory price is 12 yuan per set. To prevent the factory from losing money, At least daily production of stationery boxes ()

A.2 000 sets B.3 000 sets C.4 000 sets D.5 000 sets

Analysis: Because profit z=12x-(6x+30 000),

So z=6x-30 000, from z≥0 we get x≥5 000, so we can produce at least 5 000 sets of pencil boxes per day.

Answer:D

Reflection on the application of linear function model

Using a linear function to find the maximum value is often transformed into solving the inequality ax+b≥0 (or ≤0). When solving, pay attention to the sign of the coefficient a, and you can also combine the function graph or its monotonicity to find the maximum value.

Variation training 1. The store sells teapots and teacups. The teapots are priced at 20 yuan each and the teacups are 5 yuan each. The store offers two discounts:

(1) Buy a teapot and get a teacup for free;

(2) Payment is based on 92% of the total price.

A customer needs to purchase 4 teapots and a number of teacups (not less than 4). If he buys teacups x (units) and pays y (yuan), try to establish the functional analytical expressions between y and x in the two discount methods. , and discuss which of the two methods is more cost-effective when the customer buys the same number of tea cups?

Solution: According to the preferential method (1), the analytical formula of the function is y1=20×4+5(x-4)=5x+60(x≥4, and x∈N).

From the preferential method (2), we can get y2=(5x+20×4)×92%=4.6x+73.6 (x≥4, and x∈N).

y1-y2=0.4x-13.6(x≥4, and x∈N),

Let y1-y2=0, get x=34.

Therefore, when purchasing 34 tea cups, the payment is the same for both discount methods;

When 4≤x<34, y1

When x>34, y1>y2, preferential method (2) saves money.

Application of Quadratic Function Model

Example 2: A fruit wholesaler sells apples with a purchasing price of 40 yuan per box. Assume that the selling price per box should not be less than 50 yuan and not higher than 55 yuan. Market research found that if each box is sold at a price of 50 yuan, the average daily price 90 boxes are sold. Every time the price increases by 1 yuan, an average of 3 boxes less are sold every day.

(1) Find the functional relationship between the average daily sales volume y (box) and the sales unit price x (yuan/box);

(2) Find the functional relationship between the wholesaler’s average daily sales profit w (yuan) and the sales unit price x (yuan/box);

(3) When the selling price of each box of apples is how much, the maximum profit can be obtained? What is the maximum profit?

Solution: (1) According to the meaning of the question, we get y=90-3(x-50),

Simplify and get y=-3x+240 (50≤x≤55,x∈N).

(2) Because the wholesaler’s average daily sales profit = average daily sales volume × sales profit per box.

So w=(x-40)(-3x+240)=-3x2+360x-9 600(50≤x≤55,x∈N).

(3) Because w=-3x2+360x-9 600=-3(x-60)2+1 200, when x<60, w increases with the increase of x.

And 50≤x≤55, x∈N, so when x=55, w has a maximum value, and the maximum value is 1 125.

Therefore, when the selling price of each box of apples is 55 yuan, the maximum profit can be obtained, and the maximum profit is 1,125 yuan.

Application of functions PPT, part 4: thinking analysis

When finding the maximum value of a function, the actual restrictions on the domain of the function are ignored, resulting in an error.

A typical example is as shown in the figure. In the rectangle ABCD, it is known that AB=a, BC=b (b

Question: When x is what value, the area of ​​quadrilateral EFGH is the largest? And find the maximum area.

Wrong solution Suppose the area of ​​the quadrilateral EFGH is S,

Then S=ab-2[1/2 x^2+1/2 "(" a"-" x")(" b"-" x")" ]

=-2x2+(a+b)x=-2(x"-" (a+b)/4)^2+("(" a+b")" ^2)/8.

According to the properties of the quadratic function, we can know that

When x=(a+b)/4, S has the maximum value ("(" a+b")" ^2)/8.

What are the errors in the above problem-solving process? What are the reasons for the errors? How do you correct them? How to prevent them?

Tip: In the wrong solution, the domain of the obtained quadratic function was not considered, and the properties of the quadratic function were directly used to solve the problem, resulting in an error.

Application of functions PPT, part 5: practice in class

1. Under the premise of a fixed voltage difference (voltage is constant), when the current passes through a cylindrical wire, its current intensity I (unit: ampere) is proportional to the cube of the wire radius r (unit: millimeters). If it has It is known that when the current passes through a wire with a radius of 4 mm, the current intensity is 320 A. When the current passes through a wire with a radius of 3 mm, the current intensity is ()

A.60A B.240A C.75A D.135A

Analysis: Suppose the proportional coefficient is k, then the current intensity I=kr3. From what is known, when r=4, I=320, so 320=43k, the solution is k=320/64=5, so I=5r3, Then when r=3, I=5×33=135(A).

Answer:D

2. A car sales company sells the same brand of cars in places A and B. The sales profit in place A (unit: 10,000 yuan) is y1=4.1x-0.1x2, and the sales profit in place B (unit: 10,000 yuan) ) is y2=2x, where x is the sales volume (unit: vehicles). If the company sells a total of 16 vehicles of this brand in these two places, the maximum profit it can obtain is ()

A.105,000 yuan B.110,000 yuan

C. 430,000 yuan D. 430,250 yuan

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Update Time: 2024-10-05

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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