总结微积分中的所有重要概念

Posted 2023-02-04 Xurtle

Introduction

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

更多详情请参考 ： http://www.wikihow.com/Understand-Calculus

Reviewing the Basics of Calculus

Know that calculus is the study of how things are changing

Calculus is a branch of mathematics that looks at numbers and lines, usually from the real world, and maps out how they are changing

Remember that functions are relationships between two numbers, and are used to map real-world relationships

Functions are rules for how numbers relate to one another, and mathematicians use them to make graphs. In a function, every input has exactly one output. All calculus studies functions to see how they change, using functions to map real-world relationships.

Think about the concept of infinity

Infinity is important to study change: you might want to know how fast your car is moving at any given time, but does that mean how fast you were at that current second? Millisecond? Nanosecond? You could find infinitely smaller amounts of time to be extra precise, and that is where calculus comes in.

Understand the concept of limits

A limit tells you what happens when something is near infinity. 比如，当 x 无限接近于2时，函数发生了什么

• Limits are easiest to see on a graph – are the points that a graph almost touches, for example, but never does?
• Limits can be a number, infinity, or not even exist.

Understanding Derivatives

Know that calculus is used to study “instantaneous change.”

Knowing why something is changing at an exact moment is the heart of calculus. For example, calculus tells you not only the speed of your car, but how much that speed is changing at any given moment.

Finding instantaneous change is called differentiation (微分). Differential calculus is the first of two major branches of calculus.

Use derivatives to understand how things change instantaneously

A “derivative” just means “how fast is something changing.” 比如，速度的导数是加速度，即 how the speed is changing.

Know that the rate of change is the slope between two points

这是微积分中一个关键的发现。The slope is the same thing as the rate of change.

• The slope of a line is the change in y divided by the change in x.
• The bigger the slope, the steeper a line. Steep lines can be said to change very quickly.

Know that you can find the slope of curved lines

找一条直线的 slope 很容易，但是，想找出一条曲线的 slope 是相对来说要更加困难。当然了，你也可以找到曲线上的2个点，然后用1条直线把它们连接起来，但是这样做求出的 rate of change 会很不准确。

Make your points closer together for a more accurate rate of change

The closer your two points, the more accurate your answer.

Use infinitely small lines to find the “instantaneous rate of change,” or the derivative

1. First, you know that the slope of a line equals how quickly it is changing
2. Second, you know that closer the points of your line are, the more accurate the reading will be

But how can you find the rate of change at one point if slope is the relationship of two points? The answer: you pick two points infinitely close to one another.

学会怎么找出不同函数的导数

There are a lot of different techniques to find a derivative depending on the equation, but most of them make sense if you remember the basic principles of derivatives outlined above. All derivatives are is a way to find the slope of your “infinitely small” line. Now that your know the theory of derivatives, a large part of the work is finding the answers.

Find derivative equations to predict the rate of change at any point

Using derivatives to find the rate of change at one point is helpful, but the beauty of calculus is that it allows you to create a new model for every function, 即可以为每个函数求出一个导数模型，这样我们就可以利用这个新模型找出每个点的变化率

There are different notations for derivatives.

• Lagrange’s notation: derivative of $y$ , you would write $y\'$
• Leibniz’s notation: the derivative of $y$ with respect to x, you would write $\\fracdydx$

Understanding Integrals

Know that you use calculus to find complex areas and volumes

比如，让你求出下图中湖里面水的体积，那么对于这个不规则形状的湖来说，你应该怎么求呢？Making geographic models and studying volume is using integration. Integration is the second major branch of calculus.

Know that integration finds the area underneath a graph

Integration is used to measure the space underneath any line, which allows you to find the area of odd or irregular shapes. 比如我们可以求出下图中（$y = 4 - x^2$） 的面积。While this may seem useless, think of the uses in manufacturing – you can make a function that looks like a new part and use integration to find out the area of that part, helping you order the right amount of material.

Know that you have to select an area to integrate

You cannot just integrate an entire function. For example, $y = x$ is a diagonal line that goes on forever, and you cannot integrate the whole thing because it would never end. When integrating functions, you need to choose an area, such as all points between $x = 2$ and $x = 5$.

Know that integration adds up many small rectangles to find area

如果让我们去算出一个矩形的面积，我相信没有人不会算吧。但是，对于下图中阴影部分的面积来说，它就并没有那么好求了！通过把阴影部分的面积划分成无限个小矩形，然后我们把它加在一起就可以很容易地求出面积了。 This happens every day – you cannot see the curve of the earth because we are so close to its surface. Integration makes an infinite number of little rectangles under a curve that are so small they are basically flat, which allows you to measure them. Add all of these together to get the area under a curve.

Know how to correctly read and right integrals

Integrals come with 4 parts. A typical integral looks like this:

$$\\int f(x) dx$$

• The first symbol, $\\int$ is the symbol for integration
• The second part, $f(x)$ is your function. When it is inside the integral, it is called the integrand
• Finally, the $dx$ at the end tells you what variable you are integrating with respect to
• Remember, the variable you are integrating is not always going to be x <script type="math/tex" id="MathJax-Element-13">x</script>, so be careful what you write down.

Learn how to find integrals

Integration comes in many forms, and you will need to learn a lot of different formulas to integrate every function. However, they all follow the principles outlined above: integration sums up an infinite number of things. 下面是找出积分的3种方法：

• Integrate by substitution
• Calculate indefinite integrals
• Integrate by parts

Know that integration reverses differentiation, and vice versa

This is an ironclad rule of calculus that is so important, it has its own name: the Fundamental Theorem of Calculus.

Know that integration can also find the volume of 3D objects

原文中只提到了一种方法求体积，但是并没有细节上的描述。于是，我找到了一篇文章是关于求体积的一个例子，读过以后感觉讲的很详细，文章中对积分的每一步都给出了具体的步骤，推荐给大家。

Solid of Revolution - Finding Volume by Rotation

KHan Academy 中也讲解了其它的几个方法求体积，讲解地非常好。

Volume using calculus