"Finding Approximate Solutions to Equations Using the Bisection Method" Exponential Functions and Logarithmic Functions PPT

Description

"Finding Approximate Solutions to Equations Using the Bisection Method" Exponential Functions and Logarithmic Functions PPT

Part One Content: Core Competency Cultivation Objectives

1. Understand the operating steps, ideas and theoretical basis of the dichotomy.

2. Be able to use the bisection method to find approximate solutions to equations or find approximate zeros of functions with the help of a calculator.

Use the bisection method to find approximate solutions to equations PPT, part 2 content: independent preview

1. The concept of dichotomy

1. In an entertainment program, the host asked the contestants to guess the price of an item within a specified time. If the guess was correct, the item would be awarded to the contestant. The items in a certain guess were priced between 800 yuan and 1,200 yuan. For a mobile phone, the contestants began to quote:

Players:1 000.

Host: It’s low.

Players:1 100.

Host: It’s high.

Players:1 050.

Moderator: Congratulations, you got the answer right.

(1) When the host said "low", what range does the price of mobile phones imply?

Tip: (1 000,1 200].

(2) What is the relationship between the player’s bid value each time and the price range of the mobile phone before the betting?

Tip: The quoted value is the middle value of the range of mobile phone prices before betting.

2. Fill in the blanks

For the function y=f(x) whose image is continuous on the interval [a,b] and f(a)f(b)<0, by continuously dividing the interval where its zero point is located into one Second, the method of gradually approaching the zero point by making the two endpoints of the obtained interval gradually approach the zero point, and then obtaining the zero point approximation is called the bisection method.

3. Determine right or wrong

Function f(x)=|x| can use the bisection method to find its zero point. ()

Answer: ×

4. Do it

The graphs of the following functions all have common points with the x-axis. Among them, the zero point of the function in the graph cannot be found using the bisection method ()

Analysis: Using the bisection method to find the zero point of a function must satisfy the fact that the function values ​​on both sides of the zero point have different signs. In option B, f(a)·f(b)<0 is not satisfied, and the bisection method cannot be used to find the zero point of the function. Because of option A, The function values ​​on both sides of the zero point in C and D have different signs, so the bisection method can be used to find the function zero point.

Answer:B

2. Steps to find the zero-point approximation of f(x) using the bisection method

1. In the above example of guessing the price of an item, is there a pattern to the guessing process?

Tip: The guessing process boils down to: assuming the original price is If c>x, then guess within the interval (a, c); if c < x, then guess within the interval (c, b); (4) and so on until the original price x is guessed.

2. Fill in the blanks

Given the accuracy ε, the general steps for finding the approximate value of the zero point x0 of f(x) using the bisection method are as follows:

(1) Determine the initial interval [a,b] of zero point x0, and verify f(a)f(b)<0;

(2) Find the midpoint c of the interval (a, b);

(3) Calculate f(c) and further determine the interval where the zero point is located;

If f(c)=0 (x0=c at this time), then c is the zero point of the function;

If f(a)f(c)<0 (at this time the zero point x0∈(a,c)), then let b=c;

If f(c)f(b)<0 (at this time the zero point x0∈(c,b)), then let a=c.

(4) Determine whether the accuracy ε is reached: if |a-b|<ε, then obtain the zero-point approximation value a (or b), otherwise repeat (2)~(4).

Use the dichotomy method to find approximate solutions to equations PPT, part three content: inquiry learning

The concept of dichotomy

Example 1 Among the functions represented by the following graphics, the zero point can be found using the bisection method ()

Analysis: The condition for using the bisection method to find the zero point of a function is: the function values ​​on the left and right sides of the zero point have opposite signs, that is, they pass through the x-axis. The answer can be obtained by analyzing the options.

Analysis: A function that can use the bisection method to find the zero point of a function. The function values ​​on the left and right sides of the zero point have opposite signs. It can be seen from the image that A, B, and D cannot meet this condition.

Answer:C

Reflection and insights (1) The dichotomy method is to gradually approach the zero point by continuously dividing the selected interval into two, find a small enough interval near the zero point, and use a certain value in this interval to approximate the zero point according to the required accuracy. represents the true zero point.

(2) Only when the function graph is continuous near the zero point and the function values ​​have different signs around the zero point can the "dichotomy method" be applied to find the zero point of the function.

Variation training 1. If the quadratic function f(x)=2x2+3x+m has a zero point, and the zero point can be obtained by using the bisection method, then the value range of the real number m is ________.

Find the zero points of a function using the bisection method

Example 2 Find the approximate value of the negative zero point of the function f(x)=x2-5 (accuracy 0.1).

Analysis: First determine the signs of f(-2) and f(-3), and then solve the problem by following the steps of finding the zero point approximation of the function using the bisection method.

Solution: Since f(-2)=-1<0, f(-3)=4>0, the interval [-3,-2] is taken as the initial interval for calculation. Use the dichotomy method to calculate successively , the list is as follows:

Reflection on the principles and solution flow chart that should be followed to find the approximate value of the zero point of a function using the bisection method

1. The principles that should be followed when using the bisection method to find the approximate value of the zero point of a function:

(1) Based on the image, estimate the initial interval [m,n] where the zero point is located (this interval must not only contain the roots sought, but also make its length as small as possible, and the endpoints of the interval should be integers as much as possible) .

(2) Take the average c of the endpoints of the interval, calculate f(c), determine whether the solution interval is (m, c) or (c, n), and gradually reduce the "length" of the interval until the length of the interval meets the accuracy requirements (In this process, it should be checked in time whether the absolute value of the difference between the end points of the obtained interval reaches the given accuracy) before the calculation is terminated and the approximate value of the zero point of the function is obtained (in order to express the calculation process and the interval where the zero point of the function is located more clearly, the list method is often used. ).

2. Flow chart for finding the approximate zero point of a function using the bisection method:

Extended exploration: What if the accuracy in this example is changed to 0.2?

Solution: According to the table of [Example 2], the length of the interval (-2.25,-2) is |-2-(-2.25)|=0.25>0.2;

The length of the interval (-2.25,-2.125) is |-2.125-(-2.25)|=0.125<0.2, so the two endpoint values ​​of this interval can be used as its approximate value, so its approximate value can be - 2.125.

Use the bisection method to find approximate solutions to equations PPT, part 4 content: thinking methods

Application of Transformation and Reduction Thoughts in Dichotomy

Typical example: Find the approximate value of ∛2 (accuracy 0.01).

[Question Review Perspective] Suppose x=∛2→∛2 is the root of the equation x3-2=0→∛2 is the zero point of the function y=x3-2

Solution: Assume x=∛2, then x3-2=0. Let f(x)=x3-2,

Then the approximate value of the zero point of the function f(x) is the approximate value of ∛2.

The following uses the bisection method to find the approximate value of its zero point.

Since f(1)=-1<0, f(2)=6>0, the interval [1,2] can be taken as the initial interval for calculation. Calculate step by step using the dichotomy method, and the list is as follows:

Use the bisection method to find approximate solutions to equations PPT, part 5: in-class drills

1. The graph of the known function f(x) is as shown in the figure. The number of zero points and the number of zero points that can be found using the bisection method are ()

A.4,4

B.3,4

C.5,4

D.4,3

Analysis: From the picture in question, we know that function f(x) has 4 common points with the x-axis, so the number of zero points is 4. The function values ​​on the left and right sides of the abscissa of the fourth common point from left to right have the same sign, so The zero point cannot be found using the bisection method, but the other three can be found using the bisection method. Therefore, D is selected.

Answer:D

2. The initial interval when using the bisection method to find the approximate zero point of the function f(x)=-x3-3x+5 is ()

A.(1,3) B.(1,2) C.(-2,-1) D.(-3,-2)

Analysis: This question tests the understanding of using the bisection method to find the zero-point approximation of a function and the selection of the initial interval. ∵f(1)=1,f(2)=-9,f(-1)=9,f(-2)= 19,f(3)=-31,∴f(1)f(2)<0.

And the domain of function f(x)=-x3-3x+5 is R,

Therefore, the initial interval where the approximate value of a zero point of f(x) lies is (1,2).

Answer:B

3. When using the bisection method to find the approximate solution of the equation f(x)=0 in the interval (0,1), after calculation, f(0.425)<0, f(0.532)>0, f(0.605)<0, That is, an approximate solution to the equation is obtained as _______________. (Accuracy 0.1)

Analysis: ∵0.605-0.532=0.073<0.1, the values ​​within ∴(0.532,0.605) can be used as an approximate solution to the equation with an accuracy of 0.1.

Answer: 0.532 (the answer is not unique)

Keywords: Free download of PPT courseware for compulsory course 1 of Mathematics of High School People's Education A version, PPT download of approximate solution to equation using bisection method, PPT download of exponential function and logarithmic function, .PPT format;

For more information about the PPT courseware "Exponential functions and logarithmic functions use the bisection method to find approximate solutions to equations", please click on the exponential function and logarithmic functions ppt use the bisection method to find approximate solutions to equations ppt tag.

"End of Chapter Review Lesson" Exponential Functions and Logarithmic Functions PPT:

"End of Chapter Review Lesson" Exponential and Logarithmic Functions PPT reminds you to explore the operations of exponential and logarithms [Example 1] Calculation: (1) 2log32-log3329+log38-5log53; (2)1.5-13-760+80.2542+(323)6 －－2323. Regular method index, pair...

"End-of-Chapter Review Improvement Course" Exponential Function and Logarithmic Function PPT:

"End of Chapter Review and Improvement Course" Exponential function and logarithmic function PPT Comprehensive improvement of exponential and logarithmic operations to find the values ​​of the following formulas: (1)827-23-3ee23+(2-e)2+10lg 2; (2)lg25+lg 2lg 500-12lg125-log29log32. [Solution]..

"Application of Functions" Exponential Function and Logarithmic Function PPT Courseware (Application of Function Model in Lesson 3):

"Applications of Functions" Exponential Functions and Logarithmic Functions PPT Courseware (Application of Function Models in Lesson 3) Part One: Learning Objectives 1. Be able to use known function models to solve practical problems. (Key points) 2. Ability to build function models to solve practical problems. (Key point, difficulty...

File Info

Update Time: 2024-09-03

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

Preview