Lecture 7

Lecture 7 - Lecture 7 Calculus is also a Game What is...

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Unformatted text preview: Lecture 7 Calculus is also a Game What is Calculus? It is short for “Calculus of Differentiation and Integration”. Thus, it’s the mathematical theory that studies differentiation and integration of function. Both of these are defined using limits: x x f x x f x f x ∆- ∆ + = ′ → ∆ ) ( ) ( lim ) ( , ) ( lim ) ( 1 x x k a f dx x f n k n b a ∆ ∆ + = ∑ ∫ = → n a b x where- = ∆ What is the Limit? Informally: iff f ( x ) approaches L as x approaches c. ( c may be ∞ ) Formally, is true iff f ( x ) is arbitrary close to L , whenever x is sufficiently close to c. For a precise (logical) definition, we make a distinction between c (and L ) being finite or infinite (i.e. ± ∞ ). L x f c x = → ) ( lim L x f c x = → ) ( lim Definition of Limits ( c and L finite) Definition: iff ( 2200ε > 0)( 5δ > 0)( 2200 x) ((0 < | x - c| < δ ) → (| f ( x ) - L| < ε )) Notes: This is the famous ε- δ definition of the limit....
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Lecture 7 - Lecture 7 Calculus is also a Game What is...

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