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Engineering Calculus Notes 201

# Engineering Calculus Notes 201 - 189 2.4 REGULAR CURVES →...

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Unformatted text preview: 189 2.4. REGULAR CURVES → − → (a) Show that if f (θ0 ) = 0, we have − (θ0 ) = 0 . (Hint: Calculate v → − (θ ) and the speed |− (θ )|.) → v v → (b) Show that if f (θ ) = 0 but f ′ (θ ) = 0, we still have − (θ ) = 0. v 0 0 0 (c) Show that in the second case (i.e., when the curve goes through the origin) the velocity makes angle θ0 with the positive x-axis. 4. Show directly that the vector-valued function giving the Spiral of Archimedes → − (θ ) = (θ cos θ, θ sin θ ), p θ ≥ 0. is regular. (Hint: what is its speed?) Show that it is one-to-one on each of the intervals (−∞, 0) and (0, ∞). → 5. Consider the vector-valued function − (t) given by Equation (2.24), p tracing out the Folium of Descartes. → (a) Show that − (t) has image in the locus C of the equation p 3 + y 3 = 3xy . x → (b) Show that − (t) is regular. p → − (t) is onto, by establishing (c) Show that p i. ii. iii. iv. v. limt→−∞ x(t) = −∞ limt→−∞ y (t) = ∞ limt→∞ x(t) = ∞ limt→∞ y (t) = −∞ → − (0) = 3 , 3 p 22 → (Why does this prove that − (t) is onto?) p (d) Verify that the velocity is horizontal when t = −1 and vertical when t = 1. 6. Consider the curve C given by the polar equation r = 2 cos θ − 1 known as the Lima¸on of Pascal (see Figure 2.27). c (a) Find a regular parametrization of C . (b) Verify that this curve is the locus of the equation (x2 − 2x + y 2 )2 = x2 + y 2 . ...
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