parametrics
TRANSCRIPT
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8/19/2019 Parametrics
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ParametricsLauren Henry Noah Stephens-Davidowitz
In Wan Ahn
Parametrics are functions given with euations for !oth " and y in terms of a
third va#ue $often time%
y&'$t%
"&($t%
)o graph a parametric euation* convert to $"*y% coordinates+
e"+ y&,t y&,t "&t . y&,$"-.%
y&," / ,
)o ta0e the derivative of a parametric* the form is as fo##ows+
e"+ e" .+ e" 1+
)o ta0e the second derivative* the form is as fo##ows+
1
1
$ %dyd
dx dx
dxdt
d y
dx=
dydt
dxdt
dy
dx=
1,
sin
2
cos
2
cos
dy
dt
dxdt
y t
x t
t
t
dy t
dx t
==
=
=
=
1
1
.
1
.
1
#im
1
t
t
dy
dt t
t dxdt
dy
dx e
y t
x e
e
==
=
=
=
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e"+
e" .+
1 1
1
3
33
3
33
2cos 2 sin
1 1cos
$ % 2
$ % 2
$ % cos
$ % sin
2 sec 2 tan seccos
t t t
d y t
dx
f t t
f t
g t t
g t t
t t t t t
+
=
=
=
= −
= = +
e" 1+
1
. 1. .
11 t
dy t
dx te t e
− −
= =
Product 4u#e+
1
1
1
1
..1
1.1
1
1
. 1 1 1.1
1
. 1 1 1.1
1
1 5.1
$ %
3$ %
$ %
3$ % 1
1
1
.
$ %1
t
t
t t
t dxdt
t t d y
t dx
d y
t dx
f x t
f x t
g x e
g x e
t e t e
e
t e t e
e
t t e
−
−
−
−
− − − −
− − − −
=
= −
=
= −
− −
=
− −=
= −+
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Arc Length+ )he arc #ength of a parametric euation can !e e"pressed as+
1 1$ % $ %b
dy dxdt dt
a
+∫
6"+ 7ind the arc #ength from .*1 of
"&,ty&t1 1t / ,
y&$ , x %1 1, x -,
dxdt
&,t
dy
dt &1t
1
1 1
.
, $1 %t +∫
arc length = 4.263