midinb03sol

midinb03sol - MATH 23b, SPRING 2003 THEORETICAL LINEAR...

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Unformatted text preview: MATH 23b, SPRING 2003 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm Solutions (in-class 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 Bolzano-Weierstrass 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 second-order 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...
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midinb03sol - MATH 23b, SPRING 2003 THEORETICAL LINEAR...

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