Lecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curve
Calculus with Parametric equations
Let C be a parametric curve described by the parametric equations
x = f (t), y = g (t). If the function f and g are differentiable and y is also a
differentiable function of x, the three derivatives
dy dx
,
dy dt
and
dx dt
are
related by
the Chain rule:
dy = dy dx dt dx dt
using this
we
can
obtain the formula
to
compute
dy dx
from
dx dt
and
dy dt
:
dy dx
=
dy dt dx
dt
if dx = 0 dt
Annette Pilkington
Lecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curve
Calculus with Parametric equations
Let C be a parametric curve described by the parametric equations
x = f (t), y = g (t). If the function f and g are differentiable and y is also a
differentiable function of x, the three derivatives
dy dx
,
dy dt
and
dx dt
are
related by
the Chain rule:
dy = dy dx dt dx dt
using this
we
can
obtain the formula
to
compute
dy dx
from
dx dt
and
dy dt
:
dy dx
=
dy dt dx
dt
if dx = 0 dt
The value of
dy dx
gives gives the slope of a tangent to the curve at any
given point. This sometimes helps us to draw the graph of the curve.
Annette Pilkington
Lecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curve
Calculus with Parametric equations
Let C be a parametric curve described by the parametric equations
x = f (t), y = g (t). If the function f and g are differentiable and y is also a
differentiable function of x, the three derivatives
dy dx
,
dy dt
and
dx dt
are
related by
the Chain rule:
dy = dy dx dt dx dt
using this
we
can
obtain the formula
to
compute
dy dx
from
dx dt
and
dy dt
:
dy dx
=
dy dt dx
dt
if dx = 0 dt
The value of
dy dx
gives gives the slope of a tangent to the curve at any
given point. This sometimes helps us to draw the graph of the curve.
The
curve
has
a
horizontal
tangent
when
dy dx
= 0,
and
has
a
vertical
tangent
when
dy dx
= .
Annette Pilkington
Lecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curve
Calculus with Parametric equations
Let C be a parametric curve described by the parametric equations
x = f (t), y = g (t). If the function f and g are differentiable and y is also a
differentiable function of x, the three derivatives
dy dx
,
dy dt
and
dx dt
are
related by
the Chain rule:
dy = dy dx dt dx dt
using this
we
can
obtain the formula
to
compute
dy dx
from
dx dt
and
dy dt
:
dy dx
=
dy dt dx
dt
if dx = 0 dt
The value of
dy dx
gives gives the slope of a tangent to the curve at any
given point. This sometimes helps us to draw the graph of the curve.
The
curve
has
a
horizontal
tangent
when
dy dx
= 0,
and
has
a
vertical
tangent
when
dy dx
= .
The second derivative
d2y dx 2
can also be obtained from
dy dx
and
dx dt
.
Indeed,
d2y dx 2
=
d ( dy ) = dx dx
d dt
(
dy dx
)
dx
if
dx = 0 dt
dt
Annette Pilkington
Lecture 35: Calculus with Parametric equations
Calculus with Parametric equations Example 2 Area under a curve Arc Length: Length of a curve
Example 1
Example 1 (a) Find an equation of the tangent to the curve x = t2 - 2t y = t3 - 3t when t = -2
Annette Pilkington
Lecture 35: Calculus with Parametric equations
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