Why Calculus? - Lesson 1 | Don't Memorise - Free Educational videos for Students in K-12

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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