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
Unformatted text preview: MINIQUIZ 10/2/08  REVISED SOLUTIONS Name: Topics covered: • § 4.1  Maximum and minimum values • § 4.2  The mean value theorem TrueFalse: (3 pts. each) 1. If f is differentiable at x = a , then f is continuous at x = a . TRUE. The following proof, which you do NOT need to know, is taken from pg. 128 of Stewart 6e. Assume f is differentiable at x = a . Then f ( a ) = lim x → a f ( x ) f ( a ) x a exists (and is finite). To show that f is continuous at x = a , we need to show that lim x → a f ( x ) = f ( a ), or equivalently, lim x → a ( f ( x ) f ( a )) = 0. Trivially, we have f ( x ) f ( a ) = f ( x ) f ( a ) x a ( x a ) and so lim x → a [ f ( x ) f ( a )] = lim x → a f ( x ) f ( a ) x a ( x a ) = lim x → a f ( x ) f ( a ) x a · lim x → a ( x a ) = f ( a ) · = . 2. If f is continuous at x = a , then f is differentiable at x = a . FALSE. A counterexample is f ( x ) =  x  . We claim that this function is continuous at x = 0. For this we need to show that lim x → f ( x ) = f (0), or in our case that lim x →  x  = 0. We compute the left and righthand limits and show they are equal:...
View
Full
Document
This note was uploaded on 03/02/2010 for the course AST 00000 taught by Professor Edwardl.robinson during the Fall '10 term at University of Texas.
 Fall '10
 EdwardL.Robinson

Click to edit the document details