Many people who are going to learn calculus for the first time may raise questions, why would one have to learn calculus? We learned arithmetic, the algebra learned, the geometry learned - can not we continue to work with them? Why should we learn a new mathematical named calculus?
The answer is very straightforward, in one word: If you do not know the calculus, then science can not be done very easily. We all know about speed, if anything goes, we say it has a speed. If we do not know calculus then we will not be able to find the value of the object.
| Time spent(s) | covered distance (m) |
|---|---|
| 1 | 5 |
| 2 | 20 |
| 3 | 45 |
| 4 | 80 |
Table 1
Let's ask a simple question to those who do not believe me. Suppose, A car has crossed the distance of 5m in the first second. And At the end of second second, the total distance has crossed 20m. In exactly the same way, at the end of the third, the total distance of the car is 45m and At the end of the fourth second, the total length of the car is 80m. you can clarify yourself after seeing the table 1.
Now let's ask you a simple question, How fast the car is running?
It is fairly certain that some of you are saying that since the total distance of 80 meters in 4 seconds has exceeded, the speed of the car is:
Some people say that this is not so easy, the velocity of the car is increasing gradually, so if I divide 80 meters with 4 seconds, it will be the average speed! That's not the instant real velocity. Those who are clever among you are quite understandable that the speed is growing slowly and that speed can not be evaluated! Those who are more clever, they can see the table and guess that a source can be found out for the distance.
With this formula, you can calculate the distance to T at any time, But there is no way to evaluate the speed of the car in 2.5 seconds or 3.9 seconds or 4.3 seconds. That is, any distance traveled can be evaluate at any time, but at any given moment its velocity can not be evaluate. After 2.5s, covered distance is.... S = 5(2.5)^2 m = 31.25m
But if you know calculus then you can easily say that the speed of the car is....
So the speed of the car in 2.5 seconds is v = 10 * 2.5 = 25m / s
The speed of the car could be determined for any time, not just in 2.5 seconds. It is possible to give thousands of such examples. Therefore, everyone who wants to learn science has to learn calculus.
HOW DID?
We might guess why passed out distance said as S = 5t ^ 2. But how did we determine the value of speed v 10t?, but how did we determine the value of speed v, that is 10t?
Actually it is not difficult at all
For better understanding of this, we first fixed a graph of passed distance and time.
Let's suppose, We want to figure out that what is velocity in t1 time by this graph.
We can see in the graph that the distance traveled in t1 is S1. Now suppose, after a short time of t1, traveled distance of t2 is s2. So we can say that the speed of the car is in the middle of t1 and t2:
But we do not want to make the average momentum between the two times, we want to get an instant momentum (i.e., t1).
This time we tell the time between t1 and t2 is Δ . So,
Then we can rewrite S1 and S2.
Since S = 5t ^ 2 , So......
So the average velocity between t1 and t2 time is:
Or
Now let us see the graph again. The time between t1 and t2 is Δ, so if t2 is closer to t1 then its value will be small. We have already said we have found the average velocity between t1 and t2. What would happen if I get t2 to bring it to T2? Of course, that speed will not be the average momentum. It will be the immediate velocity of t1!
That means, to get the immediate velocity of t1, t2 must be brought to t1.
That means Δ will be empty!
Now, if we put the value of Δ in the equation of v, we'll get:
It is true for any time even in a specific time. So, I can write in generally:
If you understand what has been done so far then you can be assumed that you have learned calculus. If you want, you can get out of the house by screaming Eureka Eureka like Archimedes - of course the clothes! Ha ha ha ha
Just look at one thing, we all know that nothing can be divided by 0. Here but we divided by Δ. When I divided, I did not say Δ = 0, after finishing everything i let you know Δ → 0, then there was no problem.
Thats all for today. Next day i will discuss about rate of change
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Thank you very much @ mdshahin786
For creating the contest, for helping many with this.
And also highly appreciative of that third prize.
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