The comparative size is 4.25 4 and 1 in 100, 15 4 and 1 in 8

The comparative size is 4.25 4 and 1 in 100, 15 4 and 1 in 8

four point two five
4 and 15 / 100 = 4.15
4 and 1 / 8 = 4.125
So four and one in eight

The x power of Ln (LNX)
How to seek derivation?

{[In(In)]^x}'
=x*[In(In(x)]^(x-1)*[In(Inx)]'
=x*[In(In(x)]^(x-1)*[1/Inx]*[Inx]'
=x*[In(In(x)]^(x-1)*[1/Inx]*1/x
=[In(In(x)]^(x-1)/Inx.

The following functions are expanded into Taylor series at point x0: ln (2 + 2x + x ^ 2) ^ (- 1) x0 = - 1; LNX x0 = 2; ln (2 + 2x + x ^ 2) ^ (- 1) x0 = - 1;
Expand the following functions into Taylor series at point x0: ln [(2 + 2x + x ^ 2) ^ (- 1)] x0 = - 1;
lnx x0=2;

Not all functions can clearly write out all terms of Taylor series
Let's take a look at the Taylor series. However, some Taylor series expansions are easy to use. See reference
Is there a problem with the first question? X0 = - 1 - > F (x) = 1 / 0?
Is it ln (2 + 2x + x ^ 2)? Or ln [(2 + 2x + x ^ 2) ^ (- 1)]
ln(1+x)=x-x^2/2+x^3/3-...
ln(1+(x+1)^2)=(x+1)^2-(x+1)^2/2+(x+1)^3/3+...
Ln (1 / (1 + (x + 1) ^ 2)) = - ln (1 + (1 + x) ^ 2) = - above
Second question:
lnx=ln(2+(x-2)) =ln(2(1+(x-2)/2))=ln2+ln(1+(x-2)/2)
Now you know how to do it!
Generally speaking, Taylor series can be solved directly, which is very troublesome
Sometimes it can be solved indirectly, such as substitution, differentiation and integration
So it's important to remember some common expansions

The power series expansion of F (x) = 1 / (x + x ^ 2) at x = 1

Let t = X-1, x = t + 1
Original formula = 1 / (T ^ 2 + 3T + 2)
=1/(t+1)-1/(t+2)
The following self will be right
I can't teach you all

The power series expansion of F (x) = (x-1) 2 ^ x at x = 1

f(x)=(x-1)2^x=2(x-1)2^(x-1)=2(x-1)2^(x-1)=2(x-1)e^[(x-1)ln2]
= 2(x-1)∑[(x-1)ln2]^n/n!= ∑[2(ln2)^n/n!](x-1)^(n+1) .

What should be paid attention to in ascending and descending power arrangement?

You need to pay attention to the number of times you order by that letter, by that letter, always by that letter

I don't know how to arrange, for example:
4x+9y-4x-7
3y-6x+3+9
The ascending and descending powers of X and Y
Also tell me what definition 4x + 9y-4x-7 should be changed to 4x + 9y-9y-5 + 7
Don't merge the same class

The increase or decrease of the exponent of the function of ascending and descending power exponent
For example, 4x + 9y-4x-7
Merge similar items first
9y-7
The decreasing power of X
9y-7
The decreasing power of Y
9y-7
4x+9y-9z-7
The decreasing power of X
4x+9y-9z-7
The decreasing power of Y
9y+4x-9z-7
It's very simple
It doesn't matter which variable you want to find the power of birth and fall, no matter which variable is in the order of other variables, but the constant should be put at the end and get the beginning
Ascending and descending power permutation of polynomials
Author: anonymous source: Qilu education network hit number: 1292 update time: December 21, 2005
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Sun Xiaorong E-mail:[email protected]
instructional objective
Master the concept of integral
2. The polynomial will be arranged according to the ascending or descending power of a letter
Capability objectives:
Cultivate students' ability of observation, induction and generalization
Teaching emphasis and difficulty
Key points: arrange polynomials according to descending power or ascending power
Difficulty: symbol problem
Teaching process:
Question 1:
Please have a look. I have four small trees, a, B, C and D. how do you put them in a straight line?
The student answers: (the student answers many, the teacher gives certain prompt)
Exploration:
What do you think are the characteristics of the two good-looking ways?
The first type: B, a, D, C from low to high
The second: C, D, a, B from high to low
Question 2:
Teacher: let's take a look at the polynomial x2-x3-1 + X. how many times and terms is this?
Student: (review the last lesson here)
Teacher: very good. It's a binomial. It's made up of four monomials. Please answer, what are the four monomials?
Student: (pay attention to the sign of monomial)
Teacher: what do you think of this polynomial? Is it a bit messy? Can you rearrange it by imitating the four little trees above?
Teacher: (ask more students to answer and write on the blackboard, then let the whole class discuss these answers, so as to lead to the problem of "polynomial ascending and descending power arrangement". After guiding the students to come to - X3 + x2 + X-1 and - 1 + X + x2-x3, give the definition.)
Definition: to arrange a polynomial according to the exponent of a letter from large to small is called to arrange the polynomial according to the descending power of the letter. To arrange a polynomial according to the exponent of a letter from small to large is called to arrange the polynomial according to the ascending power of the letter
Ask the question: after rearranging the polynomials, whether the resulting polynomials are equal to the original polynomials. Here, what is our basis?
Lead the students to answer "according to the commutative law of addition"
Exercise 1:
The polynomial: X3 + 5x-6-4x2 is arranged in ascending and descending order
Exploration:
Teacher: what do you think should be paid attention to when you arrange the ascending and descending powers of polynomials?
Student: when you rearrange polynomials, each term moves with its symbol
Teacher: very good. How many letters does the polynomial 3x2y-4xy2 + x3-5y3 contain?
student:
Exploration: when there are two or more letters in a polynomial, it is necessary to specify which letter is arranged according to its exponent
For example, rearrange the polynomial 3x2y-4xy2 + x3-5y3
(1) It is arranged according to the ascending power of X;
(3) They are arranged according to the ascending power of Y and (4) according to the descending power of Y
(1) The original formula = - 5y3-4xy2 + 3x2y + X3;
(2) The original formula is X3 + 3x2y-4xy2-5y3;
(3) The original formula is X3 + 3x2y-4xy2-5y3;
(4) The original formula = - 5y3-4xy2 + 3x2y + X3
Question: what do you think should be paid attention to when you arrange the ascending and descending powers of polynomials?
(1) The term in a polynomial includes the property symbol in front of it. Therefore, when arranging, we still need to move each property symbol together as a part of this term. If the original first term omits the property symbol "+", we need to add the "+" sign when moving to the back. If the original middle term moves to the first term and the property symbol is positive, we can also omit the "+" sign, But the character sign "-" cannot be omitted
(2) Polynomials with two (or more) letters are arranged according to the exponent of the letter when they are arranged according to a certain letter. Items without this letter are ranked last when they are arranged by descending power, and first when they are arranged by ascending power
Exercise 2. Arrange (1), (2) of the following polynomials according to the descending power of the letter X, (3), (4) according to the ascending power of the letter Y: (find a board for four, and then correct by the whole class)
(1)2xy+y2+x2； (2)3x2y-5xy2+y3-2x3；
(3)2xy2-x2y+x3y3-7； (4)xy3-5x2y2+4x4-3x3y-y4．
Summary:
What is the basis for rearranging polynomials? What are the problems that need to be paid attention to? Students are required to arrange the items of polynomials according to the requirements. Special attention should be paid to the arrangement of descending power or ascending power according to a certain letter
Editor's note: the capacity of this class is very large. We should pay attention to the time. Because I have not worked long, I have prepared detailed cases. If we want to arouse the atmosphere of the class, we can change question 1 to four students, But I can't think of a good method for the moment, and it can't reflect the mathematics thought of the new textbook of China Normal University. If you have a good opinion, please give me some advice, because I need to use this teaching plan to complete a lecture in our county

On the problem of ascending and descending power
The power of x-8x2 + the third power of 9x-1. How to arrange it according to the descending and ascending power of X? (he said that it is arranged according to x, what about the constant term? What about the formula of X
He's in X-1, there's no X

The constant term is zero
therefore
Descending power of X
9x³-8x²+x-1
By the ascending power of X
-1+x-8x²+9x³

What is the operation of polynomial ascending (descending) power permutation

Let's arrange polynomials in ascending order,
It's the commutative law and associative law of addition
The arrangement of ascending power or descending power is just to make the formula more orderly in form,
It's just a skill to facilitate the later calculation
Don't take it too seriously
Later you learn something else,
When you need to deal with polynomials (for example, derivation)
You will find it more convenient to reduce the power of polynomials
That's why I want to tell you what is ascending power permutation and descending power permutation
Don't take it seriously.

How to raise or lower the power of polynomials with the same exponent
Question 1 (title) example: 3a-5a-6-b
If 3a-5a-6b-6 is sorted by a, is 6B regarded as a constant term?
There are also three polynomials in which there are various letters. Are these letters sorted by ABCD?
All the time

First of all, can't your 3a-5a-6-b be merged into - 2A - (6 + b)
Secondly, ascending and descending powers are actually arranged by a certain letter
For example, a ^ 2 + 2Ab + B + 1
This is the descending power of a, the degree of a only depends on the degree of a, the others are ignored, that is, B can be regarded as a part of the constant term