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

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

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

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

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

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some extraordinarily complicated 
functions, 

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alright? 
The function that you write down has 

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probably never been written down in the 
history of humankind. 

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

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

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amazing, 
right? 

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

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

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algebraic symbols, 
right? 

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

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

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

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Celsius to Fahrenheit. 
These are two different temperature 

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

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

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

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

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

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

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

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

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

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Celsius and spits out something in 
Fahrenheit. 

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

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This is a much cooler example from the 
real world. 

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

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

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attached to a couple light emitting 
diodes, LEDs. 

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

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

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

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

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

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

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plugs into my computer so I can record 
the results. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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right? 
The red function is decreasing as the sun 

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rises. 
There's tons more examples of functions 

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

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

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

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

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

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population. 
And as long as researching the Internet, 

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here's another example of a real world 
function. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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that talk about numbers. 
I mean why? 

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

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Input is a number. 
The output is a number. 

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

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

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

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

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

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

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functions using algebra. 
or say, composing two functions. 

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And we've thought about some real world 
examples. 

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

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

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encourage you to contact me as soon as 
possible. 

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

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Good luck. 
[MUSIC]