Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

While playing with a ball, Noor is curious about its movement. As he drops the ball to the ground, he asks himself, "What will its speed be when it reaches the middle of its path?" She drops the ball from a height 1 metre above the ground.

It Cover 50 centimetres to get to the midpoint. Noor knows that it took a second for the ball to reach midpoint B. With this information,

Can you find the velocity of the ball exactly when it is at point B?

As you might be thinking, Noor also thinks that the speed of the ball will be the distance traveled by the ball divided by the time it takes to get to that point.

So she comes up with the answer ‘50 centimetres per second’ or zero point five metres per second. 

But is this the speed of the ball when it’s at point ‘B’?

No, it's not. This answer would have been correct if the speed of the ball was constant throughout its motion. 

But we know that the speed of the ball increases as it falls. 

So, the answer Noor got is actually the AVERAGE speed of the ball as it reaches position ‘B’. But what we are interested in is the speed exactly at the INSTANT when the ball is at position ‘B’. 

That is called the instantaneous speed of the ball. Can you try finding the instantaneous speed? Let’s see what happens at the instant the ball is at position ‘B’. 

The distance traveled by the ball at this instant is zero and the time elapsed at this instant is zero. 

So we get the speed to be ‘zero divided by zero’ which is undefined. Doesn’t make any sense right! 

How do we then find the instantaneous speed of the ball? 

CALCULUS is the branch of mathematics that helps us answer this question. How? 

We will see that in the later section of this course. But wait… another thought puzzled Noor. As she drops the ball, she wonders, why the ball ever reaches the floor… This might seem to be a lame thought, but don’t forget that Noor’s smart. She thinks that mathematically, the ball should never touch the ground. 

So,what was her thought process? Let’s see! 

Suppose she drops the ball from a height one meter above the floor. Now, to get to the ground, the ball first has to cover half this distance to get to point 'B'.

Then the ball has to cover half the remaining distance, that's a quarter meter. Then the ball has to cover the next half, then the next half and so on. It means the number of steps the ball has to cover to reach the floor does not end. That is, there are an infinite number of steps the ball has to perform. To perform these steps, the ball takes an infinite amount of time. So according to this logic, Noor thinks the ball requires an INFINITE amount of time to reach the floor. 

So the ball should never hit the ground, right? Do you also think the same? Do you think Noor went wrong somewhere? Share your thoughts in the comments section Actually, Noor isn’t only the one who was puzzled by this. 

Many centuries ago, the same thought puzzled a Greek philosopher, Zeno of Elea . This is usually referred to as Zeno’s Dichotomy paradox. Although we know that when we drop the ball it hits the ground, this logical and mathematical conclusion tells us that it should never hit the ground.

Again, a satisfactory answer to Zeno's paradox is provided by Calculus. We saw two examples here that calculus can give us the answer to! But before looking at the central ideas of calculus, we will further explore what other real life problems calculus can help us with. If we are on a cliff next to the sea, it’s always tempting to randomly throw stones into the sea. 

It’s so much fun right! But have you ever wondered about the best possible way to throw a stone such that it covers the maximum distance? Knowing this was certainly important in the past to attack the enemy’s ship. Now,let’s get back to our question. If we throw a stone too high, we know that it will not cover the maximum distance. What if we throw the stone horizontally? Maybe not! 

By experience, we know instead of throwing the stone horizontally, if we throw it at an angle, it will cover greater distance. Of course the answer also depends on the speed with which you throw the stone. Let’s say, if you apply all your energy, you can throw it with a speed ‘V’. So if we throw the stone with a fixed speed ‘v’, at what angle should we throw it to cover the maximum possible distance? 

As the angle at which we throw the stone changes, the distance covered by it changes. And this is where calculus comes into play. To get the answer we need to know how the distance covered by the stone changes, as the angle we throw it at changes. And this is exactly the kind of problem that Calculus helps us with. 

Alright, so calculus helps us with analyzing things in motion. For instance, finding the instantaneous speed of an object, or finding the angle at which to throw the stone. But wait… 

Let me ask you a completely random question. Look at this trajectory of the stone. What do you think will be this area under the dashed curved path? 

We know how to find the area of a simple shape like the rectangle. 

It's area is equal to its length times its width. But how do we get this formula? Let’s say the length of the rectangle is ‘5’ centimetres and its width is ‘10’ centimetres. 

Then the area of the rectangle is fifty ‘square centimetres’. 

So,what does this mean? It means that if we take a square tile of length of one centimetre, that is a square tile of area one SQUARE centimetre, then fifty such tiles will cover this rectangle. 

Now let’s get back to our question. What will be the area under this curve? Should we cover this area also with square tiles? 

This will not work right! Look at the square tiles covering the curve. We have a problem here as they don’t fit perfectly. Then how can we figure out this area? You would have guessed by now that calculus helps us to find the answer. We know the area of simple shapes like rectangles, triangles, polygons and so on. 

Here are the formulas! This is easy because straight lines are involved. But the shapes that we encounter in our daily lives are not that simple, as curves are involved. That’s where calculus comes into the picture! So we have seen that other than finding the instantaneous speed of an object and the angle at which to throw an object to cover maximum distance, calculus also helps us to find the area of different shapes. 

In this course about Calculus, we will explore each of these examples in detail. But before moving on, let’s have a glimpse at the central idea around calculus. This idea was used by Greek mathematicians, to find the area of a shape, long before calculus was developed. 

Consider this Circle with radius ‘r’. How would you find its area? Consider these two triangles: one circumscribed around the circle, and the other inscribed inside it. 

We can say that the area of the circle will be between the areas of these two triangles. Now what if we used squares instead of triangles. We will get a better approximation of the area of the circle if instead of triangles, we use squares. 

We can further improve our results if we used Pentagons. 

Did you get the idea?

Can you tell me how we can improve the approximation Further? 

As we consider polygons with greater numbers of sides, we will get close to the circle. The area of the polygon inscribed in the circle and the area of the polygon circumscribing the circle get closer to each other. This was the method used by Greek mathematicians to find the area of a circle. It's called the method of exhaustion. This is the central idea of Calculus used to solve the problems we mentioned above. With this knowledge, do you think we can solve our problem of finding the instantaneous speed of an object?