Engineering Calculus Notes 278

Engineering Calculus Notes 278 - a = √ c about the origin...

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266 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Reconstructing Surfaces from Slices The level curves of a function f ( x,y ) can be thought of as a “topographical map” of the graph of f ( x,y ): a sketch of several level curves L ( f,c ), labeled with their corresponding c -values, allows us to formulate a rough idea of the shape of the graph: these are “slices” of the graph by horizontal planes at different heights. By studying the intersection of the graph with suitably chosen vertical planes, we can see how these horizontal pieces fit together to form the surface. Consider for example the function f ( x,y ) = x 2 + y 2 . We know that the horizontal slice at height c = a 2 > 0 is the circle x 2 + y 2 = a 2
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Unformatted text preview: a = √ c about the origin; in particular, L ( f,a 2 ) crosses the y-axis at the pair of points (0 , ± a ). To see how these circles ±t together to form the graph of f ( x,y ), we consider the intersection of the graph z = x 2 + y 2 with the yz-plane x = 0; the intersection is found by substituting the second equation in the ±rst to get z = y 2 and we see that the “pro±le” of our surface is a parabola, with vertex at the origin, opening up. (See ²igure 3.4 ) If instead we consider the function f ( x,y ) = 4 x 2 + y 2 , the horizontal slice at height c = a 2 > 0 is the ellipse x 2 ( a/ 2) 2 + y 2 a 2 = 1...
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