Engineering Calculus Notes 267

Engineering Calculus Notes 267 - the locus of an equation...

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3.4. LEVEL CURVES 255 for c = a 2 > 0, L ( f,c ) is the hyperbola x 2 a 2 y 2 a 2 = 1 which “opens” left and right, and for c = a 2 < 0 we have x 2 a 2 y 2 a 2 = 1 which “opens” up and down. For c = 0 we have the common asymptotes of all these hyperbolas. (a) f ( x,y ) = x 2 + y 2 (b) f ( x,y ) = x 2 4 + y 2 (c) f ( x,y ) = x 2 y 2 Figure 3.1: Level Curves We would like to establish criteria for when a level set of a function f ( x,y ) will be a regular curve. This requires in particular that the curve have a well-de±ned tangent line. We have often found the slope of the tangent to
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Unformatted text preview: the locus of an equation via implicit dierentiation: for example to nd the slope of the tangent to the ellipse (Figure 3.2 ) x 2 + 4 y 2 = 8 (3.12) at the point (2 , 1), we think of y as a function of x and dierentiate both sides to obtain 2 x + 8 y dy dx = 0; (3.13) then substituting x = 2 and y = 1 yields 4 8 dy dx = 0...
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