"Applications of Functions" Exponential functions and logarithmic functions PPT (Using the bisection method to find approximate solutions to equations in the second lesson)

Description

"Applications of Functions" Exponential functions and logarithmic functions PPT (Using the bisection method to find approximate solutions to equations in the second lesson)

Part One: Learning Objectives

Understand the concept of dichotomy and its applicable conditions through specific examples, and understand that dichotomy is a common method for finding approximate solutions to equations.

Can use the bisection method to find the zero-point approximation of a function in a given interval, thereby obtaining an approximate solution to the equation

Application of functions PPT, part 2: independent learning

Problem guide

Preview the textbook P144-P146 and think about the following questions:

(1) What is the concept of dichotomy?

(2) What are the steps to find the zero-point approximation of a function using the bisection method?

A preliminary exploration of new knowledge

1. dichotomy

Condition (1) The graph of function y=f(x) is on the interval [a, b]__________.

(2) The function value at the endpoint of the interval satisfies __________

Method: Continuously __________ the interval where the zero point of the function y=f(x) is located, so that the two endpoints of the interval are gradually __________, and then obtain the approximate value of the zero point

■Instructions from famous teachers

Bisection is to divide the given interval into two parts equally, find a small enough interval near the zero point through continuous approximation, and use a certain value in this interval to approximately represent the real zero point according to the required accuracy.

2. Steps to find the zero-point approximation of a function using the bisection method

self-test

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

(1) The zero points of all functions can be found using the bisection method. ()

(2) The accuracy ε is an approximation. ()

(3) When using the bisection method to find an approximate solution to an equation, it can be accurate to any digit after the decimal point. ()

Observe the graph of the following functions and determine which one can use the bisection method to find its zero point ()

The initial interval that can be selected to find the zero point of the function f(x)=x3+5 using the bisection method is ()

A． [-2,1]B． [-1,0]

C. [0,1] D. [1,2]

When using the dichotomy method to study the zero points of the function f(x)=x3+lnx+12, after calculating f(0)<0, f12>0 for the first time, one of the zero points x0∈________ can be obtained, and _________ should be calculated for the second time.

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

The concept of dichotomy

(1) Among the following functions, the zero point cannot be found using the bisection method ()

A． f(x)=3x-1B． f(x)＝x3

C. f(x)＝|x| D. f(x)＝lnx

(2) Use the bisection method to find the roots of the equation 2x+3x-7=0 in the interval [1,3]. Take the midpoint of the interval as x0=2, then the next interval with roots is ________.

[Analysis] (1) For option C, let |x|=0, we get x=0, that is, the function f(x)=|x| has a zero point, but when x >0, f(x)>0; when x<0, f(x)>0. Therefore, the function value of f(x)=|x| is non-negative, that is, the function f(x) =|x|has a zero point, but the function values ​​on both sides of the zero point have the same sign, so the bisection method cannot be used to find the zero point.

(2) Suppose f(x)=2x+3x-7, f(1)=2+3-7=-2<0, f(3)=10>0, f(2)=3>0, f(x) is zero The interval it is in is (1, 2), so the interval where the equation 2x+3x-7=0 has roots is (1, 2).

regular method

Conditions that should be met to find the zero point of a function using the bisection method

(1) The graph of the function is continuous near the zero point.

(2) The function values ​​have different signs around the zero point.

Only when the above two conditions are met, the bisection method can be used to find the zero point of the function.

Track training

1. Regarding the "dichotomy method" to find approximate solutions to equations, which of the following statements is correct ()

A． The "dichotomy method" to find the approximate solution of the equation must be obtained from all the zero points of y=f(x) in [a, b]

B． The "dichotomy method" to find the approximate solution of the equation may not get the zero point of y=f(x) within [a, b]

C. Apply the "dichotomy method" to find the approximate solution of the equation. y=f(x) may have no zero points in [a, b]

D. The "dichotomy method" to find the approximate solution of the equation may obtain the exact solution of f(x)=0 within [a,b]

2. When using the bisection method to find the zero point of the function f(x) as shown in the figure, the impossible zero point is ()

A． x1B. x2

C. x3 D． x4

Analysis: Choose C. From the idea of ​​dichotomy, it can be seen that the function values ​​​​on the left and right sides of the zero point x1, x2, and x4 have opposite signs, that is, there is an interval [a, b], such that f(a)·f(b )<0, so x1, x2, and x4 can be solved by the dichotomy method, but when x3∈[a, b], f(a)•f(b)≥0, so the dichotomy method cannot be used Find the zero point.

Function Application PPT, Part 4: Feedback on Compliance

1. Approximate solutions can be obtained using the "dichotomy method". The correct statement about accuracy ε is ()

A． The larger ε is, the higher the accuracy of the zero point is.

B． The larger ε is, the lower the accuracy of the zero point is.

C. The number of repeated calculations is ε

D. The number of repeated calculations has nothing to do with ε

2. When using the "dichotomy method" to find the approximate value of the zero point of function f(x), the interval taken for the first time is [-2, 4], then the interval taken for the third time may be ()

A． [1,4]B． [-2,1]

C.-2,52 D.-12,1

3. If the graph of function f(x) on [a,b] is a continuous curve and satisfies f(a)f(b)<0, f(a)fa+b2>0, but()

A． f(x) has zero points on a, a+b2

B． f(x) has zero points on a+b2 and b

C. f(x) has no zero points on a, a+b2

D. f(x) has no zero points on a+b2 and b

4. When using the bisection method to find a positive real zero point of the function f(x), after calculation, f(0.64)＜0, f(0.72)＞0, f(0.68)＜0, then the accuracy of the function is 0.1 The approximate value of the zero point of a positive real number is ()

A． 0.6 b. 0.75

C. 0.7D. 0.8

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Update Time: 2024-11-18

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