[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]