Advanced Mathematics (Part 1) [Basic Subjects, Extreme Parts]

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Advanced math basics

Higher mathematics is nothing more than divided into three parts: 极限, 导数（微分）and 积分- constitute the calculus

What you learn in higher mathematics iscalculus, the whole is actually just an idea + a formula (Newton-Leibniz formula)

Grasp the essence (derive the area of ​​a circle)

Assuming that there is such a circle currently, we all know that

• Area of ​​circle:π * r^2
• Circumference of circle:π * 直径

We know the circumference of the current circle, but how do we know it 圆的面积? !

Method 1: Cut into fan shape

We divide a circle into multiple sectors

Let's take out a small sector and take a look

If we cut it small enough, then we can approximately think of it as the area of ​​a triangle

This area is too small, and we have no way to ask for it, and it’s not easy to ask for it.

We can assemble this again, but instead of assembling it into a circle, assemble it according to this type.

If the sector we cut is thin enough, then we can approximately think of the figure as a rectangle.

• Width is approximately: R
• The long approximation is: πR (because it is the upper and lower halves, it is not 2 * π * r)
  

Therefore the area is:π * R ^ 2

We break it down into small elements that we can grasp - differentials

Method 2: Split into rings

We split it into small rings. If we split it into n points, then the width of this small ring is.

Take out one of the rings. How should we solve it?

Don’t use the formula for the area of ​​a circle to find it directly. Now we are proving the formula for calculating the area of ​​a circle.

When we split it, we can get a small rectangle

• Length: Circumference of the ring
• Width: R/n

After summarizing, you can get the following results

• The longest length is: circumference of the circle - 2 * π * R
• The inner length is: radius R (all widths spliced ​​together)

Area of ​​circle: 2 π R * R * 1 / 2 = π * R *Ｒ

But there still seems to be a problem here. When we cut fan-shaped or round shapes, how many portions should we cut?
If we cut to infinity, the width is decreasing,误差也在减小

Question 1: Why don’t we know the formula for the area of ​​a circle, but we can use the formula for the circumference of a circle?
The main purpose here is to understand the core ideas of calculus and use 重组面积the method.
During the reorganization process, the radius and circumference are unchanged, so they can be used (of course there will be actual errors here)

It seems that all advanced mathematics is about differential calculus and integral calculus, so why do we need to learn about limits?

——Because it belongs to both of them 基础!

Question 2: What is the difference between x tends to 0 and x = 0?
This is 动静之间the difference. When x approaches 0, it can keep getting smaller, which is dynamic . However, x = 0 is a constant, which is a static quantity.

The (infinitely small) limit can be expressed in one sentence:要多小有多小

How to solve for limits

Finding the limit can be briefly divided into three steps:
the first step: substitute the independent variable limit value into the limit expression, if the result cannot be obtained, continue to the next step; the second step: determine which type of limit type belongs to, the focus is to identify the formula Infinitely small and infinitely large components in Step 3: (transformation, deformation)代入


分类



求解

Remember in particular

• infinitesimal * bounded function = 0

Method 1: Bring it in directly (the simplest)

Solve

We can bring it in directly, you can get

infinitesimal * bounded function = 0

value sought

Bring it in directly and you will get 0

When an infinitesimal quantity is multiplied by a bounded function [-1,1], the result must still be an infinitesimal quantity.

Solve for the inverse function

value sought

What needs to be noted here is the infinite direction .
If yes 正无穷, then it is π / 2
If yes 负无穷, then it is - π / 2
However,若直接是无穷，则极限不存在

Method 2: Classification

This method is specifically used to handle situations that cannot be handled directly.

In fact, this is mainly due to the composition of 无穷小量and无穷大量

free of charge

example:

simplification

We found that in the above question, after we bring in the value, it is of type 0/0, so we need to simplify it.

• x - 2The first question is , when Then you can bring it in again to solve the problem.
  x-2

• For the second question , it is necessary to simplify x - 1such an expression; the denominator can be factored naturally, but for the first one, you need to 分子有理化use

substitution

The main thing is to remember the commonly used infinitesimal equivalent substitution ( 只有无穷小才可以换~！)

Here ln(cos x)is an example. When x approaches 0, of course the formula also approaches 0; but we need to simplify it to facilitate calculation ;
when x = 0, cos x = 1, so if we need to construct an infinitesimal We need -1 - that is, ln(1 + (cos x - 1 ) )
because ln(1 + x) ~ x; so ~ (cos x - 1)
because 1 - cosx ~ x^2 / 2, so ~（-x^2 / 2）

Add another question:

You cannot just take cosx ~ x for granted here, because the next step of solution cannot be performed here.

Instead, we need to construct

The formula needs to be used here

So it can be replaced with

Continue because sinx ~ xyou can change it

So the result is:

This is not something that can be seen at a glance, but to understand your purpose - to capture an infinitesimal amount (because it is a 0 / 0 type) - you can understand it by looking at the denominator. The infinitesimal amount should correspond tox - 1

Special note: absolutely cannot be replaced directly! ! !不能用于加减法

But it needs to be done拆解

Great size

Note here that for the first question, of course you can directly remove the square of x, but for the following example questions, you need to divide by x (x belongs to different ranges)

Because x here tends to 0, 只需要处理xso