Engineering Calculus Notes 354

Engineering Calculus Notes 354 - 342 CHAPTER 3. REAL-VALUED...

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342 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION (b) If it is true (given α and β ) for a given pair ( x,y ), then it is also true for the pair ( t 1 x,t 1 y ) (verify this!), and so we can assume without loss of generality that xy = 1 (c) Prove the inequality in this case by minimizing f ( x,y ) = 1 α x α + 1 β y β over the hyperbola xy = 1 . 13. Here is a somewhat diFerent proof of Theorem 3.6.6 , based on an idea of Daniel Reem [ 44 ]. Suppose S R 3 is compact. (a) Show that for every integer k = 1 , 2 ,... there is a fnite subset S k S such that for every point x S there is at least one point in S k whose coordinates diFer from those of x by at most 10 k . In particular, for every x S there is a sequence of points { x k } k =1 such that x k S k for k = 1 ,... and x = lim x k . (b) Show that these sets can be picked to be nested:
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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