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[MUSIC] Two weeks ago, we started our 
journey together. 

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We've been looking at functions and 
limits. 

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We've mostly been asking what's the 
output value if the input is close to, 

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but not at, a particular point. 
That's one way to get information about a 

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function That's a limit. 
This week, in Week 3, we're going to look 

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at a different way to get some 
information about a function, a more 

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active perspective on functions. 
Instead of just asking about an output 

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value, I'm going to ask how changing the 
input affects the output, sort of a more 

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active way to get information about a 
function. 

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Changing one thing, affecting something 
else. 

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This sort of situation comes up 
throughout our lives. 

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There's some function whose input is 
money, and whose output is how much 

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coffee I'm drinking. 
Some change to the amount of money that 

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I'm making will affect how much coffee 
I'm drinking, and it keeps going. 

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Right? If I take one more sip. 
Of coffee, I'll be slightly more alert. 

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There's some function whose input is 
coffee, and whose output is alertness. 

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Some small change to my coffee, will 
effect how alert I am. 

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These kinds of relationships, one thing 
changing effecting something else, 

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prevade life, prevade science. 
We want to understand them 

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systematically, and that's what the 
Derivative offers us. 

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This idea is so improtant, that we're 
going to be spending 2 months on this 

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topic. 
We're going to be studying Derivatives; 

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numerically, algebraically, 
geometrically. 

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But our journey begins right now, in Week 
3, The Derivative. 

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[MUSIC]