Why Calculus? - Lesson 1 | Don't Memorise - By JenniferESL
Transcript
00:03 | while playing with the ball . Nora gets curious about | |
00:07 | its motion as she drops the ball on the floor | |
00:10 | . She asks herself what will be its speed as | |
00:14 | it reaches midway in its path , She drops the | |
00:18 | ball from a height one m above the ground . | |
00:21 | It covers 50 cm to reach the midpoint . Nora | |
00:25 | knows that it took one second for the ball to | |
00:28 | reach the mid point B . With this information . | |
00:32 | Can't you find the speed of the ball exactly when | |
00:34 | it's at point B . Like you'd probably be thinking | |
00:40 | ? Nora also thinks the speed of the ball will | |
00:42 | be the distance traveled by , it divided by the | |
00:45 | time taken to reach that point . So she comes | |
00:49 | up with the answer 50 cm/s or 0.5 m/s . | |
00:56 | But is this the speed of the ball when it's | |
00:58 | at point B . No , it's not . This | |
01:01 | answer would have been correct if the speed of the | |
01:04 | ball was constant throughout its motion . But we note | |
01:08 | that the speed of the ball increases as it falls | |
01:11 | . So the answer Nora got is actually the average | |
01:15 | speed of the ball as it reaches the position be | |
01:18 | . But what we are interested in is the speed | |
01:21 | exactly at the instant when the ball is at position | |
01:24 | being that is called the instantaneous speed of the ball | |
01:29 | . Can you try finding the instantaneous speed ? Let's | |
01:34 | see what happens at the instant . The ball is | |
01:37 | at position . Be the distance traveled by the ball | |
01:41 | at this instant is zero at the time elapsed at | |
01:44 | this instant is zero . So we get the speed | |
01:48 | to be zero divided by zero , which is undefined | |
01:52 | . Doesn't make any sense . Right ? So how | |
01:55 | do we then find the instantaneous speed of the ball | |
01:59 | ? Calculus is the branch of mathematics that helps us | |
02:02 | answer this question . How we will see that in | |
02:06 | the latest section of this course . But wait another | |
02:10 | thought , puzzled Nora as she drops the ball . | |
02:14 | She wonders why the ball ever reaches the floor . | |
02:18 | This might seem to be a lame thought , but | |
02:20 | don't forget that Nora smart , she thinks that mathematically | |
02:25 | the ball should never touch the ground . So what | |
02:28 | was her thought process ? Let's see , suppose she | |
02:32 | drops the ball from a height one m above the | |
02:35 | floor now to reach the floor first the ball has | |
02:38 | to cover half this distance to reach point B . | |
02:42 | Then the ball has to cover half of the remaining | |
02:44 | distance , that is 1/4 of a meter . Then | |
02:49 | the ball has to cover the next half , then | |
02:51 | the next half and so on . It means the | |
02:55 | number of steps the ball has to cover to reach | |
02:58 | the floor does not end . That is there are | |
03:01 | infinite number of steps the ball has to perform and | |
03:05 | to perform these steps , the ball takes an infinite | |
03:08 | amount of time . So according to this logic Nora | |
03:11 | thinks the ball requires an infinite amount of time to | |
03:15 | reach the floor . Therefore the ball should never reach | |
03:18 | the floor right . Do you also think the same | |
03:22 | ? Do you think Nora went wrong somewhere ? Share | |
03:27 | your thoughts in the comments section ? Yeah , actually | |
03:32 | , Nora isn't the only one who was puzzled by | |
03:34 | this many centuries ago . The same thought puzzled a | |
03:38 | greek philosopher Zeno of Elia . This is usually referred | |
03:43 | to as Zinos dichotomy paradox . Even though we know | |
03:47 | that when we dropped the ball it reaches the floor | |
03:50 | . This logical and mathematical conclusion tells us that it | |
03:54 | should never reach the floor again . A satisfactory answer | |
03:59 | to the Zeno . S paradox is provided by calculus | |
04:04 | . We saw two examples here that calculus can give | |
04:07 | us the answer to . But before looking at the | |
04:10 | central ideas of calculus we will further explore what other | |
04:15 | real life problems calculus can help us with . If | |
04:22 | we're on a cliff next to the sea it's always | |
04:25 | tempting to randomly throw stones into the sea . It's | |
04:28 | so much fun . Right ? But have you ever | |
04:31 | wondered about the best possible way to throw a stone | |
04:34 | ? Such that it covers the maximum distance ? Knowing | |
04:38 | this was certainly important in the past to attack the | |
04:42 | enemy ship . Now let's get back to our question | |
04:46 | . If we throw a stone too high we know | |
04:49 | it will not cover maximum distance . What to feed | |
04:52 | through the stone horizontally . Mhm . Maybe not by | |
04:57 | experience . We know instead of throwing the stone horizontally | |
05:01 | if we throw it at an angle it will cover | |
05:03 | greater distance . Of course the answer also depends on | |
05:08 | the speed with which you throw the stone . Let's | |
05:11 | say if you apply all your energy , you can | |
05:14 | throw it with a speed V . So if we | |
05:17 | throw the stone with a speed , we at what | |
05:19 | angle should we throw it to cover ? Maximum possible | |
05:22 | distance ? As the angle at which we throw the | |
05:26 | stone changes . The distance covered by it changes . | |
05:30 | And this is where calculus comes into play . To | |
05:33 | get the answer , we need to know how the | |
05:36 | distance covered by the stone changes as the angle we | |
05:40 | throw it at changes . And this is exactly the | |
05:43 | kind of problem that calculus helps us with . All | |
05:47 | right , so calculus helps us with analyzing things in | |
05:50 | motion . For instance , finding the instantaneous speed of | |
05:54 | an object or finding the angle at which to throw | |
05:57 | the stone . But wait , let me ask you | |
06:00 | a completely random question . Look at this trajectory of | |
06:04 | the stone . What do you think will be this | |
06:07 | area ? under the dashed curved path . We know | |
06:12 | how to find the area of a simple shape . | |
06:14 | Like the rectangle . Its area is equal to its | |
06:18 | length time , its width . But how do we | |
06:20 | get this formula ? Let's say the length of the | |
06:24 | rectangle is five cm and its width is 10 cm | |
06:29 | . Then the area of the rectangle is 50 square | |
06:32 | cm . So what does this mean ? It means | |
06:37 | that if we take a square tile of length of | |
06:39 | one centimeter , that is a square tile of area | |
06:42 | one square centimeter , Then 50 such tiles will cover | |
06:46 | this rectangle . Now let's get back to our question | |
06:51 | . What will the area be under this curve ? | |
06:54 | Should we cover this area also with square tiles . | |
06:57 | This will not work right . Look at the square | |
07:00 | tiles covering the curve . We have a problem here | |
07:04 | , as they don't fit perfectly . Then how can | |
07:07 | we figure out this area you would have guessed by | |
07:11 | now that calculus helps us to find the answer . | |
07:15 | We know the area of simple shapes like rectangles , | |
07:18 | triangles , polygons , and so on Here at the | |
07:22 | formulas . This is easy because straight lines are involved | |
07:27 | , but the shapes that we encounter in our daily | |
07:30 | lives are not that simple as curves are involved . | |
07:34 | That's where calculus comes into the picture . So we've | |
07:38 | seen that other than finding the instantaneous speed of an | |
07:42 | object and the angle at which to throw an object | |
07:45 | to cover maximum distance , calculus also helps us to | |
07:49 | find the area of different shapes . In this course | |
07:52 | about calculus , we will explore each of these examples | |
07:56 | in detail , but before moving on , let's have | |
07:59 | a glimpse at the central idea around calculus . This | |
08:03 | idea was used by greek mathematicians to find the area | |
08:07 | of a shape . Long before calculus was developed , | |
08:11 | consider this circle with radius R . How would you | |
08:15 | find its area ? Considered these two triangles . One | |
08:21 | circumscribed around the circle and the other inscribed inside it | |
08:26 | . We can say that the area of the circle | |
08:28 | will be between the areas of these two triangles . | |
08:32 | Now , what if he used squares instead of triangles | |
08:36 | ? We will get a better approximation of the area | |
08:39 | of circle . If instead of triangles , we used | |
08:42 | squares , we can further improve our results If we | |
08:46 | used pentagon's , Did you get the idea ? Can | |
08:51 | you tell me how we can improve the approximation further | |
08:55 | ? As we consider polygons with greater number of sites | |
08:59 | , we will get close to the circle , the | |
09:02 | area of the polygon inscribed in the circle and the | |
09:06 | area of the polygon , circumscribing the circle , get | |
09:09 | closer to each other . This was the method used | |
09:12 | by greek mathematicians to find the area of the circle | |
09:17 | . It's called the method of exhaustion . This is | |
09:20 | the central idea of calculus used to solve the problems | |
09:23 | we mentioned above with this knowledge . Do you think | |
09:28 | we can solve our problem of finding the instantaneous speed | |
09:31 | of an object ? Think about the ways in which | |
09:35 | you can approach the problem and share your thoughts in | |
09:38 | the comments section . Mhm . In the next part | |
09:42 | we will see how to find the instantaneous speed of | |
09:45 | an object , and the idea applied to calculate the | |
09:48 | area of a shape . We will also discover that | |
09:52 | these two ideas are related to each other . See | |
09:56 | you in the next part . |
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Why Calculus? - Lesson 1 | Don't Memorise is a free educational video by JenniferESL.
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