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Engineering Calculus Notes 158

# Engineering Calculus Notes 158 - 146 CHAPTER 2 CURVES or in...

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146 CHAPTER 2. CURVES or, in terms of coordinates, braceleftBigg x = 1 + a 3 2 cos θ b 2 sin θ y = 2 a 2 cos θ + b 3 2 sin θ . The general relation between a plane curve, given as the locus of an equation, and its possible parametrizations will be clarified by means of the Implicit Function Theorem in Chapter 3 . Analyzing a Curve from a Parametrization The examples in the preceding section all went from a static expression of a curve as the locus of an equation to a dynamic description as the image of a vector-valued function. The converse process can be difficult, but given a function −→ p : R R 2 , we can try to “trace out” the path as the point moves. As an example, consider the function −→ p : R R 2 defined by x ( t ) = t 3 y ( t ) = t 2 with domain ( −∞ , ). We note that y ( t ) 0, with equality only for t = 0,
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