تدريب Shadowing: ALL OF MATH explained in 14 minutes - تعلم التحدث بالإنجليزية عبر الفيديو

Hi, you're on a rocket ship, flying to the edge of the universe.

Wait, how big is the universe?

It's 1 plus 2 times 99 times e plus 100 to the power of- Stop!

What the heck is going on here?

The universe is infinite.

Infinity can consist of numbers which are symbols that represent a value.

We can add these values, we can subtract them, and we can also do some strange things like this.

But, we'll get to that later.

Besides numbers, there's also these things representing instructions, relationships, operations, and concepts called signs and symbols.

Some of these signs are inverses, meaning that putting them together just make them want to undo each other, or, as the nerds call it, cancel each other.

This little fixed number right here is known as a constant, and then there's also these unknown mysterious letters called variables.

One of them by themself don't really tell us that much, but put them together, and now we have an equation.

Like, you know, this one for example.

The goal of every equation is to get solved.

We do it by rearranging it in a way that isolates the variables from the constants.

Think of it like this.

If we would like to solve this equation, we would use the inverse operation to cancel out the constant on this side.

By doing that, we've isolated the variable and thus gotten its value.

Now, there are also different types of equations.

There's the quadratic equation, which always has one variable raised by no more than two.

And then there's also the polynomial equation, which can have an unlimited amount of numbers their variables can be raised by.

Meaning, they can look like this sometimes.

These two are basically more hardcore versions of the linear equation, which can't be raised by anything more than one.

A key thing to remember is that the variables in a quadratic

and polynomial equation can be equal to multiple values at the same time.

To determine how many possible solutions an equation has, we look at the largest number of variables in the equation

is raised by x cubed minus 3x equals zero for example will have these three solutions.

Okay, now that we know about these equations, we need to, you know, actually be able to solve them.

These four formulas plus some extra methods pretty much explain step by step how.

Looking at a quadratic equation, we will see that it's basically the same as the first formula.

If we then plug the number in the formula and then simplify, we will then find out that the value x1 equals negative 1 and x2 equals negative 5.

This is pretty neat, but that's too complicated for us.

The second, third, and fourth formula tells us that we can rewrite these equations by factoring them into smaller parts.

We can then put each of them equal to zero and then solve them separately.

There is a twist here though.

Not every equation will be this simple.

They can also look like this, which are close to impossible to factorize, meaning if you're not smart enough to come up with your own formula, you're gonna have to use this one instead.

Now, when it comes to the polynomial equation, things can get.. weird.

Except if you're four steps ahead and see that we actually can factorize the equation, and again solve each of them separately.

Oh yeah, did I mention that there's also equations that have a constant, or as they say, a base raised by a variable?

They're called exponential equations, and tell us how a starting value grows or declines exponentially at a certain rate.

The base of the formula tells us the rate of growth or decline, the constant tells us the initial value, and the exponent tells us for how long it grows.

All right, enough about equations.

Let's get ourselves out of the atmosphere.

If you've ever been to space, you'd know that the Earth looks like this.

Oh, sorry, wrong picture.

You would know that the Earth looks like this.

That, my friend, is a shape, aka a sphere.

There are two-dimensional shapes.

Fuse some of them together, and now you got a three-dimensional shape.

Now combine some of these and you have yourself this thing.

You see, everything around you is just a bunch of shapes combined to make objects.

Now, let's say you wanted to know the size of this thing's surface, or as most people call it, the area.

First stretch the surfaces out, which gives us a rectangle.

Then measure the base and height, and apply this formula.

Area is measured in square units, and every shape uses the base formula B times H.

Actually, that was kind of a lie.

The circle uses its radius squared times pi, but you get the idea.

Now, if we instead wanted to know how much we could stuff inside of an object, we would need something called volume.

Calculating how much candy you could fit inside of your cylindrical shaped bucket on Halloween could be pretty useful, so in that case, you would use this formula.

Similar to area, every shape pretty much uses the same base formula B, which is the surface area of the base times the height, with a few adjustments for each shape.

Everyone except for, you guessed it right, the sphere again, which now times pi by the radius squared.

Hey, wanna see something cool?

Said a man named Pythagoras almost 2,500 years ago as he formalized the Pythagorean theorem.

It tells us that this side, called a hypotenuse and a right-angled triangle, equals the sum of the squares of the other two sides.

But that's freaking whack, said this guy who 300 years later gave us a way

to calculate the crazy relationship a triangle's angle has with its sides, aka trigonometry.

If we look at a triangle, we will see this angle.

In trigonometry, it's called theta.

These sides have a particular ratio to one another based on the value of the angle theta and are labeled hypotenuse, opposite, and adjacent.

There are three functions that pretty much sum up all of trigonometry.

Actually, there's one more thing too but we'll get to that later.

The sin function tells us the ratio between the opposite and the hypotenuse.

The cos function tells us the ratio between adjacent and the hypotenuse.

And the tan function tells us the ratio between opposite and the adjacent.

Imagine a triangle with these values.

By using the sin function and then calculating it as an equation, we can then find the value for x, which equals 6.88.

The other key thing to remember is this little circle divided into 4 quadrants with 12 points called a unit circle.

The points in the circle are defined by various measurements like, for example, degrees.

But the most common measurement, however, is radians, which basically is the radius of the circle stretched out over its side, creating an arc.

6.283 radians make up a whole circle, which is equal to the circumference of the circle, 2 pi radius.

But, since we only use radians as the measurement and not the radius, we can rename 2 pi radius to 2 pi radians.

Yeah, I know, they literally sound the same.

Deal with it.

Congratulations!

Now that we know the circle in total is 2 pi radians, we can also say that the points in our circle become various fractions of pi over radians.

Now, here is when things get kind of freaky.

If we draw a line from the center of the circle, this line according to trigonometry will have the unit 1.

One what, you may ask?

Just one.

Yeah.

You see, since we're dealing with ratios between lengths, they'll cancel each other out, therefore we just use 1.

If we know that this length is one unit long, we can also figure out that the end of each axis have these coordinates.

Now let's place this angle theta in the circle and draw one line alongside the x-axis, and then another one from the point where the angle intercepts the circle.

Wait a minute, that's the same triangle we drew earlier!

This side, aka the ipotenuse, will have the unit 1.

If we look at the functions sin, cos, and tan, we can then plug our values into one of them and then simplify it.

Do you see what's going on here?

We're marking out the coordinates for the angle theta.

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In simple terms, we're figuring out the lengths of the triangle's sides.

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Knowing this, we can then say that the coordinates for the angle theta are cos theta, sin theta.

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Now, to most normal people, this tells us about as much as this kid trying to explain his dream.

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So, we're gonna need some actual numbers.

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The three key numbers to remember are square 3 over 2, square 2 over 2, and square 1 over 2, which is simplified to 1 over 2.

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Remember those dots in the circle representing the fractions of 2 pi radians?

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Turns out, if we draw a right triangle with a 30 degree angle to pi over 6 radians, then draw a mirror image triangle below it, we can then see that the length of the original line,

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aka the wide coordinate, aka sin theta equals 1 over 2.

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You see, there's a pattern here.

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If we instead draw a triangle with a 45 degree angle to pi over 4 radians, we will see indeed that sin theta equals square 2 over 2.

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Doing the same with a triangle with the angle 60 degrees

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to pi over 3 radians will give us the conclusion that sin theta equals to square 3 over 2.

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Long line big number, medium line medium number, short line small number, and the same pattern applies to cos theta.

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Fun fact!

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We can actually draw out the function of cos and sin in a graph.

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It shows us the variation of cos and sin with respect to the angle theta.

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You can think of cos theta as the x-axis.

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When theta increases, so does the cos theta line, until theta goes over the second quadrant, which sits on the negative side of cos theta, making the line slope down.

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Sin theta?

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Pretty much does the same thing, just in reverse.

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All that talk about graphs and functions, I think it's about time we talk about calculus.

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Using a function, we can understand the change in motion in whatever your imagination can come up with.

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This is f of x, and this is an equation.

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Put them together, and then they become a function.

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A function tells us that for a certain input x, we get a certain output y.

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To join the special club of functions, their inputs and outputs need to either have a one-to-one relationship or a many-to-one relationship.

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If we then plug all the input values

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and output values into a graph and then connect all the values with a line, we will then get a longer line.

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This line can tell us multiple things.

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Like, for example, this kid's heart rate when he realizes someone threw his diamond sword into lava.

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Functions come in multiple variants.

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Hey look, those are the equations we looked at earlier.

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Depending on the equation, the function fx equals, you will get a certain type of line.

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A linear function gives us a straight line where the constant

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tells us how much the output y increases in relation to the input x.

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These lines can also get very bendy, and can look like this, or even this.

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One thing many people confuse is that while these three lines might look similar, they are vastly different.

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The function of a quadratic equation will always have one turning point where it either turns up or down, whereas the function of a polynomial equation will have multiple turning points.

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Fun fact!

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The more variables you have in a function, the more of these points you'll have.

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Not so fun fact, this means a line can look like this.

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Okay, but this graph, doesn't it also have a turning point right there?

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No, that is an exponential function which, as you know, grows or declines exponentially, meaning it won't at any point turn around.

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Speaking of points, limits, describe the behavior of a function as it approaches a specific point.

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If you're not confused by that definition, you're probably like, four steps ahead.

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But, for the rest of us, think of it as a point on a graph that, when the line gets closer and closer to it, it can start acting weird.

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For example, why don't you try to evaluate this function?

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You can't because it equals zero, which means we can't find out what the line does at x equals two.

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What we do instead is that we find out what it does as it approaches two, aka the limit.

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In this case, we can then plug in a number that is just over two, which then gives us 4.1, and to prove we're right, we use a limit which gives us the number 4.

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Now, when we have a function that equals zero, it becomes discontinuous.

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Continuous functions are functions that can be drawn without lifting the pencil of the paper, aka, they won't have any holes or gaps that break its flow.

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Discontinuous functions, which have the three states, whole, jump, and infinite, are the total opposite.

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They couldn't think of anything worse than to stay in flow.

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If you thought this was the limit when it comes to the usage of limits, you were wrong.

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They were also used to point out the holy grail of calculus, or at least the second holy grail, derivatives.

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Derivatives are used to calculate the rate of change in a function at a certain point.

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If we zoom in on a function, we can see that it's basically just a straight line.

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Now let's draw a line right here.

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This is called a tangent, and is the derivative of the original function.

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Now, I'm sorry to disappoint you, but getting the derivative isn't just zooming in on one line and drawing another.

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Let's say you have the function f of x equals 5x minus 4.

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You would then have to plug everything into this formula by first replacing x with x plus h, and then divide everything with h, simplify, to then get the derivative 5.

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Nah, screw that, said this guy who basically saved every future calculus student from getting an f by formalizing the power rule.

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It's basically an idiot's guide to derivatives, and tells us that the derivative of x raised to some power is equal to

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that number times x raised to the power of that number minus 1.

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For example, x cubed when derived becomes 3x squared, and just x when derived becomes 1, and a constant derived is just completely cancelled out.

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Now, the crazy thing here is

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that you can actually derive the derivative of a function to get a function for the derivative, and then you can derive

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that function to get a function of the derivative of a function of the derivative of a function of a function.

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Yeah.

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Okay, that's cool and all, but what if we have an exponential function where a constant is raised by a variable?

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Well, then you would derive using the exponential rule.

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You see, there is also these things called the differentiation rules, which gives us the formulas for how to derive much more complex functions.

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Like, you know, this one.

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Okay, now let's say you would like to know the area of the surface under the function.

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This is where we use the second holy grail of calculus.

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Integrals.

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By slicing up the area under the line into an infinite amount of slices, you can then calculate the area of each slice, add them up, and then get the whole area under the function.

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Now, I know what you're thinking.

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Infinite rectangles.

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Isn't that like impossible?

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Technically, yeah.

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But not if we use an integral, which gives us a really specific estimate of the area under the function as we approach an infinite number of squares.

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This is the formula which is used by first deciding the intervals we want to calculate the area of, and then we do something that literally can't stand derivatives so much,

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they've named themselves antiderivatives, which are basically derivatives in reverse.

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3x squared, for example, is x cubed.

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6x antiderived is 3x squared.

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Eh, yeah, there's actually a problem here, said some mathematicians.

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Since the derivative of a constant equals zero, we wouldn't know if the anti-derivative has a constant.

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Now, here is when an absolute genius came along and said, what if we make the constants subtract each other out by adding this part of the integral formula, and then, boom, we get the area.

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Hey, wanna play a game?

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If you were on a train station and this weird guy came up to you with two pieces of paper, what would be the chance of you being able to flip this little piece of paper on your first try?

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Well, assuming that the game isn't completely rigged, you would firstly have to calculate the amount of preferred outcomes, then divide it by the total number of possible outcomes, and then you get the number 0.5.

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This means that there's a chance of 50%.

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Now, there are two other key rules that you probably should remember.

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The first one tells us that the probability of any event must fall somewhere between 0 and 1,

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with 0 being something totally impossible and 1 being something 100% certain to happen.

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The second rule tells us that if you were to add up all the probabilities of a certain event, you would get the number 1, and the last one tells us

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that the probability of something happening plus the probability of it not happening is also equal to 1.

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Anyways, if you've been listening really well, you should at this point be able to solve this mathematical problem.

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Now, rumor goes that first person to comment the right answer gets a shoutout in the next video. So, subscribe.

ما هي تقنية التظليل الصوتي؟

التظليل الصوتي (Shadowing) تقنية تعلم لغة مدعومة علمياً، طُورت أصلاً لتدريب المترجمين الفوريين المحترفين. الطريقة بسيطة لكنها قوية: تستمع لصوت إنجليزي أصلي وتكرره فوراً بصوت عالٍ — كظل يتبع المتحدث بتأخير 1-2 ثانية. تُظهر الأبحاث تحسناً كبيراً في دقة النطق والتنغيم والإيقاع وربط الأصوات والاستماع والطلاقة.

تدريب Shadowing: ALL OF MATH explained in 14 minutes - تعلم التحدث بالإنجليزية عبر الفيديو

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