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[MUSIC] So, there's all these different ways of taking two functions and

producing new functions. You could add, subtract, multiply, divide

two functions, and could take two functions and compose them, meaning that

the output for the one becomes the input for the other.

In light to this, I'd encourage you just to pick up your pen and just write down

some extraordinarily complicated functions,

alright? The function that you write down has

probably never been written down in the history of humankind.

I mean, there's just so many different choices that you could make when you are

combining all the algebraic operations. And that's part of what makes Calculus so

amazing, right?

There's just a huge variety of functions out there.

But not, not every function really has its source in just combinations of

algebraic symbols, right?

A lot of the functions that we want to study are really functions that are

somehow coming from the real world. So, I want to see some real world

examples of, of functions right now. So, here's one unit conversion from

Celsius to Fahrenheit. These are two different temperature

scales. So, the function would be f of x,

it's 9 * x / 5 + 32. So, this is just a linear function.

It's a number times x plus a number. let's take a look what's f of zero.

And that would be 9 * 0 / 5 + 32. Well, that's zero plus 32, that just 32

and, of course, zero degrees Celsius is the same thing as 32 degrees Fahrenheit.

Here's another example. What's f of, say, 37?

Well, that's nine * 37 / 5 + 32. 9 * 37 is 333 / 5 + 32.

333 / 5 is 66.6, so 66.6 + 32 is 98.6. And indeed, 37 degrees Celsius is the

same thing as 98.6 degrees Fahrenheit. So, this function takes in something in

Celsius and spits out something in Fahrenheit.

Unit conversion is an example of a function, but hardly the coolest example.

This is a much cooler example from the real world.

What is this thing? Well, this thing here is a

microcontroller. So, a very small computer and it's

attached to a couple light emitting diodes, LEDs.

With a name like light emitting diode, you might think that they light up, and

they could. But in this circuit, I'm using the light

emitting diodes in reverse. I'm using them as light sensors.

This one happens to be a red one. This one happens to be a green one.

So, what this circuit does is let me detect how much red and green light is

falling on these sensors. At the other end is a USB cable and it

plugs into my computer so I can record the results.

The data that I gathered from the real world using the microcontroller.

It's really two different functions. A function for the red LED and a function

for the green LED. Along the x-axis, I've plotted the number

of seconds that have elapsed since June 5th, 2012 at 6:0303 p.m.

And on the y-axis, I'm plotting the number of clock cycles it took to

discharge the LED. So, what is the red function do?

It's input is a number a number of seconds that have elapsed since this

particular moment in time. It's output is how many clock cycles it

takes at that particular moment in time to discharge the red LED.

Now, this thing was sitting in my windowsill, right?

And the sun was rising. And as the sun rises, there's more light

shining on the sensors which means fewer clock cycles are necessary to discharge

the LED. And you can see that in this graph,

right? The red function is decreasing as the sun

rises. There's tons more examples of functions

coming from the real world. Here's one, human population.

It's a function. The input is a year, the output is the

number of people alive during that year. If you want to see this function just

take a look at Wikipedia and their article on population growth. There's a

graph of that function, along the x-axis is years and along the y-axis is human

population. And as long as researching the Internet,

here's another example of a real world function.

It's a function I'll call f of n, and it'll be defined by the rule f of n

equals the number of Google hits when we search for the number n.

Let's try it out. Let's figure out some values of this

function like f of 188. So I plug 188 into Google, and I find

that there's about 1.08 billion hits. So, the function at 188 is about a

billion, alright, the input is 188 and the output of this function is the number

of Google hits. let's try about 4 * 188, that's 752. And

if I search for that, there's 308 million hits,

alright? So, f, the function, at 752 is about 300

million. if we're persistent, we can plug inn lots

of numbers, and make a really nice-looking chart like this.

Now, you do this for hundreds of numbers, right?

You type them into Google, you see how many Google hits you get.

And you can plot them, right? It's a function so you can the graph of

the function. Along the x-axis is the number that I

typed into Google. On the y-axis is the millions of Google hits that I get.

And when you look at the graph of this function, it's not random.

There's real structure here, right? The function is decreasing, right?

Larger values get smaller outputs because, you know, there is fewer

webpages that talk about really large numbers than about popular small numbers.

But even more dramatically, when you plot this on this special log, log graph

paper, the graph looks like it's sitting near a straight line.

I mean, that's, that's really amazing when you think about it.

I mean, this is some sort of pattern that's just hidden in the number of pages

that talk about numbers. I mean why?

Where is this coming from, right? It's a function from the real world.

Input is a number. The output is a number.

We want to understand that function. Calculus is part of the tool kit for

analyzing problems like this. So, we've seen what functions do.

They take their input and they transform in into some output.

And we've even sort of got this mental image now,

this metaphor of a machine. A conveyor belt that's transforming the

input into the output. We've seen how to build a lot of new

functions using algebra. or say, composing two functions.

And we've thought about some real world examples.

[MUSIC] Now, we're going to be thinking more about functions for the rest of the

term. But if you've got questions right now, I

encourage you to contact me as soon as possible.

And I encourage you to get started on the homework right away.

Good luck. [MUSIC]