"Applications of Functions" Exponential functions and logarithmic functions PPT courseware (zero points of functions and solutions to equations in Lesson 1)

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"Applications of Functions" Exponential functions and logarithmic functions PPT courseware (zero points of functions and solutions to equations in Lesson 1)

Part One: Learning Objectives

1. Understand the concept of function zeros and the relationship between function zeros and equation roots. (easy to mix)

2. Can find the zeros of a function. (emphasis)

3. Master the existence theorem of function zero points and be able to determine the number of function zero points. (difficulty)

core competencies

1. Cultivate the literacy of mathematical operations and logical reasoning with the help of zero-point methods.

2. With the help of the relationship between the zero points of functions and the roots of equations, cultivate the mathematical literacy of intuitive imagination.

Application of functions PPT, part 2: independent preview and exploration of new knowledge

A preliminary exploration of new knowledge

1. zero point of function

For the function y=f(x), let _______________ be called the zero point of the function y=f(x).

Thinking 1: Is the zero point of the function the intersection of the function and the x-axis?

Tip: No. The zero point of a function is not a point, but a number. This number is the abscissa of the intersection of the function graph and the x-axis.

2. The relationship between equations, functions, and function graphs

The equation f(x)=0 has real roots⇔The graph of function y=f(x) has an intersection with ______⇔The function y=f(x) has ______.

3. Theorem about the existence of function zeros

If the graph of the function y=f(x) on the interval [a,b] is a ______ curve, and there is ______, then the function y=f(x) is on the interval (a , there is at least one zero point in b), that is, there is c∈(a, b), such that ______, this c is the solution of equation f(x)=0.

Thinking 2: What conditions does this theorem meet?

Tip: The theorem requires two things: ① The graph of the function on the interval "a, b" is a continuous curve; ② f(a)·f(b)<0.

First try

1. The function represented by the following graphs does not have zero points ()

2. The zero point of function y=2x-1 is ()

A.12B.12,0

C.0,12　D． 2

3. The zero point of the function f(x)=3x-4 is in the interval ()

A． (0,1) B. (-1,0)

C. (2,3) D． (1,2)

4. In the quadratic function y=ax2+bx+c, a·c<0, then the function has ________ zeros.

Application of functions PPT, the third part: cooperative exploration to improve literacy

Find the zeros of a function

[Example 1] (1) Find the zero point of function f(x)=x2+2x-3, x≤0, -2+ln x, x>0;

(2) It is known that the zero point of the function f(x)=ax-b(a≠0) is 3, find the zero point of the function g(x)=bx2+ax.

[Solution] (1) When x≤0, let x2+2x-3=0, and the solution is x=-3;

When x>0, let -2+ln x=0, and the solution is x=e2.

Therefore, the zero points of function f(x)=x2+2x-3, x≤0-2+ln x, and x>0 are -3 and e2.

(2) From what is known, f(3)=0, that is, 3a-b=0, that is, b=3a.

Therefore, g(x)=3ax2+ax=ax(3x+1).

Let g(x)=0, that is, ax(3x+1)=0,

Solve to get x=0 or x=-13.

So the zeros of function g(x) are 0 and -13.

regular method

How to find the zero point of a function

1 Algebra method: Find the real roots of the equation fx=0.

2 Geometric method: For the equation fx=0 that cannot use the root formula, it can be related to the graph of the function y=fx. The abscissa of the intersection of the graph and the x-axis is the zero point of the function.

Determine the interval where the zero point of the function lies

[Example 2] (1) The approximate interval where the zero point of function f(x)=ln(x+1)-2x is located is ()

A． (3,4)B． (2,e)

C. (1,2) D. (0,1)

(2) According to the data in the table, it can be concluded that the interval where a root of the equation ex-x-3=0 is located is ()

x-1 0 1 2 3

ex 0.37 1 2.72 7.39 20.08

x+3 2 3 4 5 6

A.(-1,0) B. (0,1)

C. (1,2) D. (2,3)

(2) Constructor function f(x)=ex-x-3, from the above table we can get f(-1)=0.37-2=-1.63<0,

f(0)＝1－3＝－2<0,

f(1)＝2.72-4＝-1.28<0,

f(2)=7.39-5=2.39>0,

f(3)＝20.08－6＝14.08>0，

f(1)•f(2)<0, so the interval where a root of the equation is located is (1,2), so choose C.]

regular method

Three steps to determine the interval where the zero point of a function lies

1 Substitution: Substitute the endpoint values ​​of the interval into the function to find the value of the function.

2 Judgment: Multiply the obtained function values ​​and perform sign judgment.

3 Conclusion: If the sign is positive and the function is monotonic in the interval, there is no zero point in the interval. If the sign is negative and the function is continuous, there is at least one zero point in the interval.

Class summary

1. In the theorem about the existence of function zero points, we should pay attention to three points: (1) the function is continuous; (2) the theorem is irreversible; (3) there is at least one zero point.

2. The root of the equation f(x)=g(x) is the abscissa of the intersection of the graphs of functions f(x) and g(x), and is also the graph of the function y=f(x)-g(x) and the x-axis. The abscissa of the intersection point.

3. Functions and equations are closely related. Some equation problems can be converted into function problems to solve. Similarly, function problems can sometimes be converted into equation problems. This is the basis of the idea of ​​functions and equations.

PPT on the application of functions, part 4: meeting the standards in class and solidifying the base

1. Thinking and analysis

(1) The zero point of f(x)=x2 is 0.()

(2) If f(a)·f(b)>0, then f(x) has no zero point in [a, b]. ()

(3) If f(x) is a monotonic function on [a, b] and f(a)·f(b)<0, then f(x) has And there is only one zero point. ()

(4) If f(x) has and has only one zero point in (a, b), then f(a)·f(b)<0.()

2. The interval where the zero point of function f(x)=2x-3 is located is ()

A． (0,1)

B． (1,2)

C. (2,3)

D. (3,4)

3. For function f(x), if f(-1)•f(3)<0, then ()

A． The equation f(x)=0 must have a real solution

B． The equation f(x)=0 must have no real solution

C. The equation f(x)=0 must have two real roots

D. The equation f(x)=0 may have no real solution

4. It is known that the function f(x)=x2-x-2a.

(1) If a=1, find the zero point of function f(x);

(2) If f(x) has zero point, find the value range of the real number a.

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Update Time: 2024-09-11

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

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