0126 - Math 31A 2010.01.26 MATH 31A DISCUSSION JED YANG 1...

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Unformatted text preview: Math 31A 2010.01.26 MATH 31A DISCUSSION JED YANG 1. Differentiation 1.1. Basics. Given a function f ( x ). The slope of the tangent line at x = c is f ′ ( c ). 1.1.1. Higher Derivatives. Recursively define higher derivatives: f ( n ) = ( f ( n − 1 )) ′ . 1.1.2. Chain Rule. If f and g are differentiable, then ( f ◦ g )( x ) = f ( g ( x )) is differ- entiable and ( f ( g ( x )) ′ = f ′ ( g ( x )) g ′ ( x ). 1.2. Exercise 3.6.51. Show that a nonzero polynomial function y = f ( x ) cannot satisfy the equation y ′′ = − y . Use this to prove that neither sin x nor cos x is a polynomial. Proof. Let y = a n x n + a n − 1 x n − 1 + . . . + a 1 x + a , with a n negationslash = 0. If the degree n < 2, then y ′′ ≡ 0 so y ′′ negationslash = − y . Otherwise, y ′ = na n x n − 1 +( n − 1) a n − 1 x n − 2 + . . . +2 a 2 x + a 1 , and y ′′ = ( n − 1) na n x n − 2 + . . . +2 a 2 . Since y ′′ lacks a monomial x n , we cannot have y ′′ = − y .....
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0126 - Math 31A 2010.01.26 MATH 31A DISCUSSION JED YANG 1...

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