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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 .....
View Full Document

## This note was uploaded on 09/19/2011 for the course MATH 31A taught by Professor Jonathanrogawski during the Winter '07 term at UCLA.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online