[MUSIC] Two weeks ago, we started our 
journey together. 
We've been looking at functions and 
limits. 
We've mostly been asking what's the 
output value if the input is close to, 
but not at, a particular point. 
That's one way to get information about a 
function That's a limit. 
This week, in Week 3, we're going to look 
at a different way to get some 
information about a function, a more 
active perspective on functions. 
Instead of just asking about an output 
value, I'm going to ask how changing the 
input affects the output, sort of a more 
active way to get information about a 
function. 
Changing one thing, affecting something 
else. 
This sort of situation comes up 
throughout our lives. 
There's some function whose input is 
money, and whose output is how much 
coffee I'm drinking. 
Some change to the amount of money that 
I'm making will affect how much coffee 
I'm drinking, and it keeps going. 
Right? If I take one more sip. 
Of coffee, I'll be slightly more alert. 
There's some function whose input is 
coffee, and whose output is alertness. 
Some small change to my coffee, will 
effect how alert I am. 
These kinds of relationships, one thing 
changing effecting something else, 
prevade life, prevade science. 
We want to understand them 
systematically, and that's what the 
Derivative offers us. 
This idea is so improtant, that we're 
going to be spending 2 months on this 
topic. 
We're going to be studying Derivatives; 
numerically, algebraically, 
geometrically. 
But our journey begins right now, in Week 
3, The Derivative. 
[MUSIC]