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Engineering Calculus Notes 273

# Engineering Calculus Notes 273 - 261 3.4 LEVEL CURVES then...

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3.4. LEVEL CURVES 261 then g a ( y ) = ∂f ∂y ( a,y ) > 0 so g a ( y ) is strictly increasing on [ y δ 2 ,y + δ 2 ]. In particular, when a = x 0 , g x 0 ( y 0 δ 2 ) < 0 <g x 0 ( y 0 + δ 2 ) and we can pick δ 1 > 0 ( δ 1 δ ) so that g a ( y 0 δ 2 ) < 0 <g a ( y 0 + δ 2 ) for each a [ x 0 δ 1 ,x 0 + δ 1 ]. The Intermediate Value Theorem insures that for each such a there is at least one y [ y 0 δ 2 ,y 0 + δ 2 ] for which g a ( y ) = f ( a,y ) = 0, and the fact that g a ( y ) is strictly increasing insures that there is precisely one . Writing x in place of a , we see that the definition φ ( x ) = y ⇐⇒ f ( a,y ) = 0 and | y y 0 | 2 gives a well-defined function φ ( x ) on [ x 0 δ 1 ,x 0 + δ 1 ] satisfying Equation ( 3.17 ). Second we show that this function satisfies Equation ( 3.18 ). We fix ( x,y ) = ( x,φ ( x )) in our rectangle and consider another point
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