[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.
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