Pratica di Shadowing: ALL OF MATH explained in 14 minutes - Impara a parlare inglese con i video

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.

100

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

101

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

102

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

103

So, we're gonna need some actual numbers.

104

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.

105

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

106

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,

107

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

108

You see, there's a pattern here.

109

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.

110

Doing the same with a triangle with the angle 60 degrees

111

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

112

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

113

Fun fact!

114

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

115

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

116

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

117

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.

118

Sin theta?

119

Pretty much does the same thing, just in reverse.

120

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

121

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

122

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

123

Put them together, and then they become a function.

124

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

125

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.

126

If we then plug all the input values

127

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

128

This line can tell us multiple things.

129

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

130

Functions come in multiple variants.

131

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

132

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

133

A linear function gives us a straight line where the constant

134

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

135

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

136

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

137

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.

138

Fun fact!

139

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

140

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

141

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

142

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

143

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

144

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

145

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.

146

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

147

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

148

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

149

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.

150

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

151

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.

152

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

153

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

154

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

155

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

156

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

157

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

158

Now let's draw a line right here.

159

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

160

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

161

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

162

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.

163

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

164

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

165

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

166

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.

167

Now, the crazy thing here is

168

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

169

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

170

Yeah.

171

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

172

Well, then you would derive using the exponential rule.

173

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

174

Like, you know, this one.

175

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

176

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

177

Integrals.

178

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.

179

Now, I know what you're thinking.

180

Infinite rectangles.

181

Isn't that like impossible?

182

Technically, yeah.

183

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.

184

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,

185

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

186

3x squared, for example, is x cubed.

187

6x antiderived is 3x squared.

188

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

189

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

190

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.

191

Hey, wanna play a game?

192

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?

193

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.

194

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

195

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

196

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

197

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

198

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

199

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

200

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

201

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

Cos'è la tecnica dello Shadowing?

Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.

Pratica di Shadowing: ALL OF MATH explained in 14 minutes - Impara a parlare inglese con i video

Contesto e Sfondo

Top 5 Frasi per la Comunicazione Quotidiana

Guida Passo-Passo per il Shadowing

Cos'è la tecnica dello Shadowing?