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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 xaxis.
4. Show directly that the vectorvalued 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 onetoone on
each of the intervals (−∞, 0) and (0, ∞). →
5. Consider the vectorvalued 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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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.
 Fall '08
 ALL
 Calculus

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