Engineering Calculus Notes 264

Engineering Calculus Notes 264 - t = 0 and g ( t ) = f x (...

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252 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 5. A moving point starts at location (1 , 2) and moves with a fxed speed; in which oF the Following directions is the sum oF its distances From ( 1 , 0) and (1 , 0) increasing the Fastest? −→ v 1 is parallel to −→ ı −→ v 2 is parallel to −→ −→ v 3 is parallel to −→ ı + −→ −→ v 4 is parallel to −→ −→ ı . In what direction (among all possible directions) will this sum increase the Fastest? Theory problems: 6. ±ill in the Following details in the prooF oF Theorem 3.3.4 : (a) Show that iF f ( x,y ) is di²erentiable at ( x,y ) and g ( t ) is defned by g ( t ) = f ( x + t x,y + y ) then g is di²erentiable at
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Unformatted text preview: t = 0 and g ( t ) = f x ( x + t x,y + y ) x. (b) Show that we can write f ( x,y + y ) f ( x,y ) = f y ( x,y + 2 ) y where | 2 | | y | . 7. (a) Use Proposition 3.3.7 to prove Equation ( 3.11 ). (b) Use Defnition 3.3.3 to prove Equation ( 3.11 ) directly. 8. Show that iF f ( x,y ) and g ( x,y ) are both dierentiable real-valued Functions oF two variables, then so is their product h ( x,y ) = f ( x,y ) g ( x,y ) and the Following Leibniz Formula holds: h = f g + g f....
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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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