It should be said that why differential and integral are inverse operations, and why can the area be calculated by inverse derivatives? This is probably the most difficult point for many people to understand when learning calculus.

Even a long time ago, everyone regarded differential and integral as two unrelated and irrelevant things, until Newton and Leibniz appeared later.

Considering that the proof process is difficult to understand intuitively, Li Zong gave an example that may not be too rigorous, but unexpectedly easy to understand, treating the integral figure as a graph of instantaneous velocity change.

Then find out how many distances have been taken in the time from a to b? Is this the reverse derivative? After using capital f to represent the original function, the area of ​​the yellow area is equal to f(b)-f(a).

This is a very important formula for calculating integrals. The process of continuously summing plumb lines into a value that only needs to be substituted into the boundary, and the area can be calculated by subtracting it.

Seeing that the two were still hesitating, Li Zong also wrote about the process of multiplication of the distance equal to speed by time and the area equal to the base edge by height. Both were multiplication and said: "In fact, we don't have to worry about why the distance can be regarded as area."

"We just need to know that they are all multiplication operations, and that they are all functions in units of drops and drops, and they will get a certain value."

"And, if you understand it in reverse, and use differential to express this graph of integral, it can be the rate of change of area within a drop of time."

Seeing that the two were still pondering, Li Zong continued: "So, assuming this idea is correct, we have learned that there is a reversal relationship between these two operations, so how can we use this relationship?"

"Can you find the integral? Integration originally required to add up the areas of many, many plumb lines. Normally, we humans cannot do it, but if we can convert it into the original function of differential time, can the integral be calculated?"

"When directly substituting the points at the two boundaries, the answer will not come out. The mileage of point b, such as 15 miles, subtracting the mileage of point a, such as 10 miles, and subtracting the 5 miles in the middle, is the journey we have traveled."

"Then the question is! What kind of relationship does this integral function have with its original function when it is differentiated?"

"Or it is now known that the function of integral is equal to y=2x. So, what is the original function of differential? So is it a process of inversely deducing the beginning of differential from the result of differential?"

"Then we will try to take an example to find the differential."

"For example, the original function y=x2, according to the definition of differentiation just now, can there be the following formula:"

picture.

"How to understand this formula? We just used the t-a method, but this obviously cannot be calculated. So we replaced t with xΔx, which means that t has more increments than a, but this increment is infinitely small. We define infinitely small as not equal to 0, but it infinitely approaches 0."

"Then you can perform calculations on the equations."

picture.

"Just as we said before, let t equal a, so in a very short time, there is no dispute. We regard this Δx as no increment, so there is no dispute when the formula is divided into 2x at the end."

"Of course, the premise is that we define infinite small, which tends to 0."

"This is exactly the result of the differential and the original function."

"Next, we can substitute some numbers to test it."

"First of all, it is clear that y=x2 is a function of distance about time, and y=2x is a rate of change of distance, that is, a function of speed about time."

"Now I require y=2x to travel over a certain period of time, that is, the area of ​​this function in a given boundary range."

"It can be turned into finding the original function, then substituting it into the boundary, and finally y=12=1."

"Whether the area surrounded by the boundary between y=2x and x and y is reflected in the triangle area formula, the bottom is 1, the height is 2, 1×2÷2=1, which is also equal to 1."

"If you substitute another number, x=2, the answer to the original function is 4, and the area surrounded by y=2x is, 2×4÷2=4, which is also equal to 4."

"The following and so on, the answer is exactly the same."

"Even the area of ​​the trapezoid is actually the same."

Li Zong used a very coincidental example to illustrate that after a given boundary, the area of ​​the figure can indeed be calculated through the formula of the original function. The calculated area is completely consistent, which just confirms Li Zong's hypothesis above.

Although this is just an isolated case, this method is enough to refresh the two of them.

The area of ​​a triangle can be calculated like this, who could have thought of this!

Then Li Zong said: "In fact, there is a more stringent proof process, which is just for you to understand. I will use this as an example."

“Suppose that is right!”

"So, did we write an equation about the equation of a circle before, does it also have xy? And we calculated the boundary at that time. If I remember correctly, the coordinates of point b are one quarter."

"If we can also know the original function of the equation of that circle, can we directly substitute one quarter of it? Of course, the starting point is 0, so we don't have to calculate, but we need to calculate the area of ​​that small area s(abd)."

After hearing this, the two of them felt that Li Zong was a genius!

This made Li Zong think of it!

but……

Next, after Li Zong wrote down the equation of the circle, how to find the original function, it stumped everyone.

"How do you need to find the original function for this formula?"

"Just now, we were blind cats and we encountered a dead mouse. We just calculated that it was 2x through differentials. So what is the difference of the original function next to be equal to (x-x2), and then the root number is opened."

Zhang Gongchuo and the others were immediately stunned.

Even after reading this formula, I seem to have forgotten all the differentials in front of it. This is what is called. After reading it, you feel that you understand it yourself, but in fact, you don’t understand anything. (Picture)

"This is indeed a very complicated formula, and although we know the differential process from beginning to end, it is impossible for us to push forward from behind."

"Especially, this kind of subtraction and even squared method."

"What should I do?"

"Let's simplify it."

“That’s the result.”

"Then let's ignore the half of the x in the first place. Let's look at the half of the (1-x) in the latter. Does the formula of f(m) look very similar to what we mentioned before?"

"Then can we expand this formula according to the formula of f(m)?"

"Finally get it."
Chapter completed!