Engineering Calculus Notes 288

# Engineering Calculus Notes 288 - and hyperbolas the...

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276 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION of radius r = c 2 + 1 about the “origin”, provided c 2 > 1; for c = ± 1 we get a single point, and for | c | < 1 we get the empty set. In particular, our surface consists of two pieces, one for z 1 and another for z ≤ − 1. If we slice along the xz -plane y = 0 we get the hyperbola x 2 z 2 = 1 or z 2 x 2 = 1 which opens up and down; again, it is clear that the same thing happens along the yz -plane. Our surface consists of two “bowl”-like surfaces whose shadow on a vertical plane is a hyperbola. This is called a hyperboloid of two sheets (see Figure 3.10 ). The reader may have noticed that the equations we have considered are the three-variable analogues of the model equations for parabolas, ellipses
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Unformatted text preview: and hyperbolas, the quadratic curves; in fact, these are the basic models for equations given by quadratic polynomials in three coordinates, and are known collectively as the quadric surfaces . Exercises for § 3.4 Practice problems: 1. For each curve de±ned implicitly by the given equation, decide at each given point whether one can solve locally for (a) y = φ ( x ), (b) x = ψ ( y ), and ±nd the derivative of the function if it exists: (a) x 3 + 2 xy + y 3 = − 2, at (1 , − 1) and at (2 , − 6). (b) ( x − y ) e xy = 1, at (1 , 0) and at (0 , − 1). (c) x 2 y + x 3 y 2 = 0, at (1 , − 1) and at (0 , 1)...
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