Engineering Calculus Notes 205

# Engineering Calculus Notes 205 - θ = π 2 or 3 π 2 Verify...

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2.4. REGULAR CURVES 193 13. Suppose f : R R is continuous on R . Show : (a) f is one-to-one if and only if it is strictly monotone. ( Hint: One direction is trivial. For the other direction, use the Intermediate Value Theorem: what does it mean to not be monotone?) (b) If f is locally one-to-one, then it is globally one-to-one. (c) Give an example of a function f ( x ) which is one-to-one on [ 1 , 1] but is not strictly monotone on [ 1 , 1]. 14. Recall the four-petal rose illustrated in Figure 2.17 on p. 149 , whose polar equation is r = sin 2 θ. This is the locus of the equation ( x 2 + y 2 ) 3 / 2 = 2 xy. The parametrization −→ p ( θ ) associated to the polar equation is b x = sin 2 θ cos θ = 2sin θ cos 2 θ y = sin 2 θ sin θ = 2sin 2 θ cos θ (a) Verify that as θ runs through the interval θ [0 , 2 π ], the origin is crossed four times, at θ = 0, θ = π 2 , θ = π , θ = 3 π 2 , and again at θ = 2 π , with a horizontal velocity when θ = 0 or π and a vertical one when
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Unformatted text preview: θ = π 2 or 3 π 2 . Verify also that the four “petals” are traversed in the order i , ii , iii , iv as indicated in Figure 2.17 . (b) Now consider the vector-valued function −→ q ( σ ) de±ned in pieces by b x = sin 2 σ cos σ = 2sin σ cos 2 σ y = sin 2 σ sin σ = 2sin 2 σ cos σ − π 2 ≤ t ≤ π 2 b x = − sin 2 σ cos σ = − 2sin σ cos 2 σ y = − sin 2 σ sin σ = − 2sin 2 σ cos σ π 2 ≤ t ≤ 3 π 2 . (2.25) Verify that this function is regular (the main point is di²erentiability and continuity of the derivative at the crossings of the origin). (c) Verify that the image of −→ q ( σ ) is also the four-leaf rose. In what order are the loops traced by −→ q ( σ ) as σ goes from 0 to 2 π ?...
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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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