[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
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
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.