Transcript Parametric Function

Parametric Functions
© Annie Patton
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Aim of Lesson
To establish what is a
set of Parametric
Functions and how to
differentiate them.
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What is a Parametric Function?
• If x = f (t) and y = g (t) , such that x and y
are both expressed in terms of a third
variable say t.
• Then t is called the Parameter.
• Note t could be replaced by any other
variable for example .
• Example x=3t and y=t2
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To plot a set of parametric functions
x=3t and y=t2
t
x=3t
y=t2
-3
-9
9
-2
-6
4
-1
-3
1
10
y
5
0
0
0
1
3
1
2
6
4
3
9
9
Equation 1: x=2t, y=t²
x
–10
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–5
5
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10
t2. Find
If x = 3t and y =
10
dx
3
dt
dy
dx
y
dy
 2t
dt
5
Equation 1: x=2t, y=t²
dy
dy dt

dx
dt dx
x
–6
Because
dy
1
 2t.
dx
3
–4
–2
2
4
6
dx
dt 1
 3, then 
dt
dx 3
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t2
If x 
1 t2
t3
dy
y
.Find
.
2
1 t
dx
Because of the
Quotient Rule
dx (1  t )(2t )  (t )(2t )

dt
(1  t 2 )2
2
2
dy (1  t 2 )(3t 2 )  (t 3 )(2t )

dt
(1  t 2 ) 2
dy dy dt (1  t 2 )(3t 2 )  (t 3 )(2t )
(1  t 2 )2


.
2 2
dx dt dx
(1  t )
(1  t 2 )(2t )  (t 2 )(2t )
dy 3t 2  3t 4  2t 4
3t 2  t 4
3t  t 3



3
3
dx
2t  2t  2t
2t
2
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dy x+y
x=e cos t and y=e sint. Show that
=
.
dx x-y
t
t
Leaving Certificate Higher No 7(b) (ii) Paper 1 2007
Start clicking when you want to see the answer.
y=et sint
dy t
=e cost+sint(et )
dt
x=et cost
dx t
=e (-sint)+cost(et )
dt
t
t
dy dy dt e cost+e sint x+y
=
= t
=
t
dx dt dx e cost-e sint x-y
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The parametric equations of the curve are:

x=cost+tsin t y=sin t-tcost where 0<t< .
2
dy
Find
and write your answer in its simplest form.
dx
Leaving Certificate Higher No 7(b) (i) Paper 1 2003
Start clicking when you want to see the answer.
dx
  sin t  t cos t  sin t  t cos t
dt
dy
 cos t  t ( sin t )  cos t  t sin t
dt
dy dy dt t sin t


 tan t
dx dt dx t cos t
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t 2
t2
dy
1
x
and y 
. If
 .Find k
t 1
t 1
dx
k
Start clicking when you want to see the answer.
dt
(t  1) 2

dx
3
dx (t  1)(1) (t  2)(1) t  1  t  2
3



dt
(t  1)2
(t  1)2
(t  1)2
dy (t  1)(1) (t  2)(1) t  1  t  2
1



2
2
dt
(t  1)
(t  1)
(t  1)2
dy dy dt
1 (t  1)2 1


.

2
dx dt dx (t  1)
3
3
dy
1 1
So if
= - = .Then k= -3
dx
3 k
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The parametric equations of a curve are:
x=3cosθ-cos 3θ
y=3sinθ-sin 3θ,where0<θ<
(i) Find
π
.
2
dy
dx
and
. (ii ) Hence show that
dθ
dθ
Leaving Certificate Higher No 7(b) Paper 1 2006
dy
1

dx
tan 3 
Start clicking when you want to see the answer.
dx
2
=-3 sinθ +3cos θsinθ
d
dy dy dθ 3cosθ-3sin 2θcosθ
=
=
dx dθ dx -3sinθ+3cos 2sinθ
dy
= 3cosθ - 3sin 2 cosθ
d
dy
cosθ(1-sin 2θ)
cosθ(cos 2θ)
=
=
2
dx -sinθ(1-cos θ) -sinθ(sin 2θ)
dy
cos3 θ
-1
==
3
dx
sin θ
tan 3θ
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The parametric equations of a curve are:
x=8+lnt 2
y=ln(2+t 2 ), where t>0.
dy
Find
in terms t and calculate its value at t=
dx
Leaving Certificate Higher No 7(b) (1) Paper 1 2005
dx
1
2
=0+ 2 .2t=
dt
t
t
2.
Start clicking when you want to
see the answer.
dy
1
=
.2t
2
dt 2+t
2
dy dy dt
2t t
t
=
=
. 
2
dx dt dx 2+t 2 2  t 2
dy
( 2)2
2
1
At t  2



2
dx 2  ( 2)
22 2
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The parametric equation of a curve are:
x=2θ - sin2θ
y=1-cos2θ ,where 0<θ<π. Find
Show that the tangent to the curve at θ=
dy
.
dx
π
2π
is perpendicular to the tangent at θ=
.
6
3
Leaving Certificate Higher No 7(b) Paper 1 2004
Start clicking when you want to see the answer.
dx
=2-2cos 2θ
dθ
dy
= -(-2sin2θ) = 2sin2θ
dθ
dy dy d 2sin 2
sin 2



dx d dx 2  2cos2 1  cos2
Continued
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The parametric equation of a curve are:
x=2θ-sin2θ
y=1-cos2θ ,where 0<θ<π. Find
Show that the tangent to the curve at θ=
Leaving Certificate Higher No 7(b) Paper 1 2004
2
At  =
3
dy
sin 2

dx 1  cos 2
4
sin
2 dy
3 
At  

3 dx 1  cos 4
3
3

dy
2   3  1

dx 1  ( 1 )
3
3
2
dy
.
dx
π
2π
is perpendicular to the tangent at θ=
.
6
3
Start clicking when you want to see the answer.
At  =

6
dy
sin 2

dx
1  cos 2
At  
 dy
6 dx
sin


3
1  cos


3
3
dy
3
 2 
1
dx
1
1
2
3 -1
.
=-1 perpendicular
1
3
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Butterfly Function
Click to see an interesting parametric Function
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dy
Homework. Calculate
in each case.
dx
1. x=e t y=e-t
2. x=t 2 +2 y=t 3
3. x=1+3 sin θ y=sin 2θ
 et 
 1  et 
4. x=ln 
y  ln  t 
t 
1 e 
 e 
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Parametric Functions
If x = f (t) and y = g (t) , such that x and y
are both expressed in terms of a third
variable say t. Then it is a parametric
Function.
10
y
The End
5
Equation 1: x=2t, y=t²
x
–6
–4
–2
2
4
© Annie
Patton
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