"Applications of Functions" Exponential functions and logarithmic functions PPT (zero points of functions and solutions to equations in the first lesson)

Description

"Applications of Functions" Exponential functions and logarithmic functions PPT (zero points of functions and solutions to equations in the first lesson)

Part One: Learning Objectives

Understand the definition of the zero point of a function and be able to find the zero point of a function

Master the method of judging the zero points of a function, and be able to judge the number of zero points of a function and its range

The parameters will be calculated based on the zero point of the function.

Application of functions PPT, part 2: independent learning

Problem guide

Preview the textbook P142-P144 and think about the following questions:

1. What is the concept of zero point of function?

2. How to determine the zero point of a function?

3. What is the connection between the roots of the equation, the intersection of the graph of the function and the x-axis, and the zero points of the function?

A preliminary exploration of new knowledge

1. zero point of function

(1) Concept: For the general function f(x), we call the real number x that makes f(x)=0 the zero point of the function y=f(x).

(2) The connection between the roots of the equation, the intersection of the graph of the function and the x-axis, and the zero point of the function

■Instructions from famous teachers

The zero point of a function is not a point, but a real number. When the independent variable takes this value, its function value is equal to zero.

2. Determining the zero point of a function

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) The zero point of a function is a point. ()

(2) Any function has zero points. ()

(3) If the function y=f(x) has zero point in the interval (a, b), then f(a)•f(b)<0.()

The zero point of function f(x)=log2(2x-1) is ()

A． 1B. 2

C. (1,0) D． (2,1)

The interval where function f(x)=x3-3x-3 has zero point is ()

A． (-1,0) B． (0,1)

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

It is known that the zero point of the function f(x)=-2x+m is 4, then the value of the real number m is ________.

It is known that the domain of the function y=f(x) is R and the image is continuous. If f(1)<0, f(1.25)<0, f(1.5)>0 is calculated, the interval where the zero point is located can be determined. for________.

Application of functions PPT, the third part: lecture and practice interaction

Find the zeros of a function

Determine whether the following functions have zero points, and if so, find out.

(1)f(x)=x+3x;

(2)f(x)=x2+2x+4;

(3)f(x)=2x-3;

(4)f(x)=1-log3x.

[Solution] (1) Let x+3x=0, and the solution is x=-3,

So the zero point of the function f(x)=x+3x is -3.

(2) Let x2＋2x+4＝0,

Since Δ=22-4×4=-12<0,

Therefore, the equation x2+2x+4=0 has no solution,

Therefore, the function f(x)=x2+2x+4 does not have zero points.

(3) Let 2x-3=0,

Solve to get x=log23,

So the zero point of the function f(x)=2x-3 is log23.

(4) Let 1-log3x=0,

Solve to get x=3,

So the zero point of the function f(x)=1-log3x is 3.

regular method

How to find the zero point of a function

There are usually two methods to find the zero point of the function y=f(x): one is to set f(x)=0 and find the zero point of the function according to the root of the equation f(x)=0; the other is to draw the function y= For the graph of f(x), the abscissa of the intersection point of the graph and the x-axis is the zero point of the function.

Determine the interval or number of zero points of the function

(1) The number of zero points of function f(x)=x2+2x-3, x≤0,-2+lnx, x>0 is ()

A． 3B． 2C. 1D． 0

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

A． (1,2)B. (2,3) C. (3,4) D. (e, +∞)

[Analysis] (1) When x≤0, from f(x)=x2+2x-3=0, we get x1=-3, x2=1 (dropped);

When x>0, from f(x)=-2+lnx=0, we get x=e2.

So the number of zeros of the function is 2.

(2) Because f(1)=-2<0, f(2)=ln2-1<0,

Therefore, f(x) has no zero point in (1, 2), and A is wrong;

And f(3)=ln3-23>0,

So f(2)·f(3)<0,

So f(x) has zero point in (2, 3).

regular method

(1) Three steps to determine the interval where the zero point of the function is located

① Substitution: Substitute the interval endpoint value into the function analytical expression to obtain the corresponding function value.

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

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

(2) Two methods to determine the existence of zero points in a function

①Equation method: If the solution to the equation f(x)=0 can be found or the number of solutions can be determined, the solution of the equation can be used to determine whether there are zero points in the function or determine the number of zero points.

②Image method: From f(x)=g(x)-h(x)=0, we get g(x)=h(x), and draw y1=g(x) and y2 in the same plane rectangular coordinate system =h(x) image, the number of zero points of the function is determined based on the number of intersection points of the two images.

Function Application PPT, Part 4: Feedback on Compliance

1. The zero point of function f(x)=2x2-3x+1 is ()

A． -12,-1　B.12,1

C.12,-1 D. -12,1

2. The function y=x2-bx+1 has a zero point, then the value of b is ()

A． 2B. -2

C. ±2 D. 3

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

A． (-2,-1) B. (-1,0)

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

4. The function f(x)=2x+x-2 has ________ zeros.

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Update Time: 2024-07-04

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