This preview shows pages 1–3. 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: MATH 23b, SPRING 2003 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm Solutions (inclass portion) March 19, 2003 1. True or False T or F Every bounded infinite set in R n has an accumulation point. True. This is the BolzanoWeierstrass Theorem. T or F Let A R n . If f : A R is continuous and f attains its maximum on A , then A is compact. False. For example, f ( x ) = 1 x 2 is continuous on all of R , which is not compact, even though it does attain its maximum (of 1) at the point 0. T or F If f : R n R m is continuous and S R n is connected, then f ( S ) R m is connected. True. This is a theorem. T or F If f : R n R is differentiable at a R n , then all of its directional derivatives exist at a R n . True. This is a theorem. T or F If f : R n R is differentiable at a , and h is some direction vector, then D h f ( a ) = f ( a ) h . True. This is a theorem. T or F If f : R n R is differentiable at a R n , then [ D i D j f ]( a ) = [ D j D i f ]( a ), for all i and j . False. There are differentiable functions whose secondorder partial derivatives do not even exist. T or F On the set of points S = { ( x,y ) R 2  f ( x,y ) = 0 } , where f ( x,y ) = x 2 y 2 1, there is a neighborhood of the point (1 , 0) on which we may write y = h ( x ) with f ( x,h ( x )) = 0. False. First note that the Implicit Function Theorem does not apply because f y = 2 y and that this is zero at the point (1 , 0). Of course, this does not guarantee that y cannot be written as a function of x , but an examination of the graph of points where f ( x,y...
View
Full
Document
 Spring '08
 Davidson,M
 Linear Algebra, Algebra, Multivariable Calculus

Click to edit the document details