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2.4. REGULAR CURVES
177
Now, consider the vectorvalued function
−→
q
(
θ
) = (cos
θ
)
−→
ı
+ (sin
θ
)
−→
,
0
< θ < π
giving another parametrization of the semicircle, in terms of polar
coordinates. This vectorvalued function is also onetoone and
C
1
, with
nonvanishing velocity vector at
θ
=
θ
0
−→
v
−→
q
(
θ
0
) =
V
q
′
(
θ
0
) = (
−
sin
θ
0
)
−→
ı
+ (cos
θ
0
)
−→
.
Comparing the two parametrizations, we see that each point of the
semicircle corresponds to a unique value of
θ
0
as well as a unique value of
x
0
; these are related by
x
0
= cos
θ
0
.
In other words, the function
−→
q
(
θ
) can be expressed as the composition of
−→
p
(
x
) with the changeofvariables function
x
= cos
θ.
Furthermore, substituting cos
θ
0
for
x
0
(and noting that for 0
< θ
0
< π
,
sin
θ
0
>
0, so
r
1
−
x
2
0
= sin
θ
0
) we see that at the point
P
=
−→
p
(
x
0
) =
p
x
0
,
R
1
−
x
2
0
P
=
−→
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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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