Engineering Calculus Notes 280

Engineering Calculus Notes 280 - 268 CHAPTER 3 REAL-VALUED...

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Unformatted text preview: 268 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION is a different parabola z = 4x2 . One might say that the “shadow” of the graph on the xz -plane is a narrower parabola than the shadow on the yz -plane. (See Figure 3.35.) This surface is called an elliptic paraboloid. A more interesting example is given by the function f (x, y ) = x2 − y 2 . The horizontal slice at height c = 0 is a hyperbola which opens along the x-axis if c > 0 and along the y -axis if c < 0; the level set L(f, 0) is the pair of diagonal lines y = ±x which are the common asymptotes of each of these hyperbolas. (See Figure 3.6.) To see how these fit together to form the graph, we again slice along the coordinate planes. The intersection of the graph z = x2 − y 2 with the xz -plane y=0 is a parabola opening up : these points are the “vertices” of the hyperbolas L(f, c) for positive c. The intersection with the yz -plane x=0 is a parabola opening down, going through the vertices of the hyperbolas L(f, c) for negative c. Fitting these pictures together, we obtain a surface shaped like a saddle (imagine the horse’s head facing parallel to the x-axis, and the rider’s legs parallel to the yz -plane). It is often called the saddle surface, but its official name is the hyperbolic paraboloid. (See Figure 3.7.) ...
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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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