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Engineering Calculus Notes 264

Engineering Calculus Notes 264 - t = 0 and g ′ t = ∂f...

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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 + δ 2 ) △ y where | δ 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 di²erentiable 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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