I believe that after reading the previous tutorial, everyone must have a certain understanding of calculus
PrevIntroduction to Calculus (1) - Definition of Limits
Then someone will definitely ask: Is this limit useful?
In fact, this limit is useful, for example, we can use it to do things . For example…
Question 1: A person sees a car. The car has traveled a total of 1 meter at the end of the first second, 4 meters at the end of the second second, and a total of 4 meters at the end of the third second. 9 meters, ... by the end of the xth second, a total of x² meters have been traveled.
This person wants to ask for the speed of the car in the first 3 seconds.
Some people: Isn't this simple? The distance in the 3rd second ÷ the time in the 3rd second = the speed in the 3rd second.
You: Is it that simple?
Question 1: How many meters did the car move in the 3rd second? How long did the car move in the 3rd second?
I'm sure someone has already found the problem. Theoretically speaking, the 3rd second is just a point, and the car has not moved. Because t=0, s=0. So v=?
But the car must have speed again in the third second, there is no doubt about it.
Yes, so now your expression must be the same as the picture below me.
The material comes from the Internet
Question 2: What is the relationship between average speed and instantaneous speed?
You have all calculated the speed, but have you ever thought about the difference between the average speed over a long period of time and the instantaneous speed at a point?
To find the average speed over a long period of time, the time and distance are not 0; but to find the instantaneous speed of a point, the time and distance are both 0.
So we can use the average speed to approach the instantaneous speed. For example, the instantaneous speed of 0.01 seconds is definitely closer to the real instantaneous speed than the instantaneous speed of 0.1 seconds. So we can calculate these relatively close, instantaneous speeds.
How about it? Got an idea?
So it can be calculated that these instantaneous speeds are:
’v=（x²-9）/（x-3）=x+3
3~3.1 seconds: 6.1 meters per second
3~3.01 seconds: 6.01 meters per second
......
3~3.0...01 seconds: 6.0...01 meters per second (114514 0s are omitted here.)
Someone has already seen that the instantaneous speed is 3 meters per second .
Question 3: Is the instantaneous speed the average speed limit?
indeed. The instantaneous speed is equivalent to the limit of the average speed. Because time infinitely approaches 0. This gives the instantaneous velocity.
6 meters per second.
Similarly, we can also find out the instantaneous speed of the car when X=X0.
Question 4: What is a derivative?
The question is simple. Just like what we asked for just now. The function increases the value lim(△x→0)△y/△x.
Of course, this is not accurate, it is more accurate to say this.
Derivative at x=x0.
or this
so
Of course, he also has many ways of writing, and the author will not list them one by one here.
The derivative is a function at every point x. This function is called the derivative function . Also called derivative . Generally, when we talk about finding the derivative of a function, we all default to the derivative function.
Derivative function writing: f'(x)
Knowing this derivative, we can do something . For example, if you know the distance, you can find the instantaneous speed. Acceleration can be calculated by knowing the instantaneous velocity. Of course, knowing the distance can also calculate the acceleration. This is the concept of the derivative of the derivative (two times, three times, n times)
Those who are interested can study this by themselves.