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Week 7
Outline
The Slope of a Parametric Curve
Arc Lengths for Parametric Curves
Surface Areas for Parametric Curves
Areas Bounded by Parametric Curves
The Slope of a Parametric Curve
Let
C
be the parametric curve
x
=
f
(
t
),
y
=
g
(
t
), where
f
0
(
t
) and
g
0
(
t
) are continuous
on an interval
I
. If
f
0
(
t
)
6
= 0 on
I
, then
d
y
d
x
=
g
0
(
t
)
f
0
(
t
)
.
Remark
We can rewrite the above equation in an easily remembered form:
d
y
d
x
=
d
y
d
t
d
x
d
t
,
if
d
x
d
t
6
= 0
It can be seen from this equation that the curve has a horizontal tangent when
d
y
d
t
= 0
(provided
d
x
d
t
6
= 0) and it has a vertical tangent when
d
x
d
t
= 0 (provided
d
y
d
t
6
= 0). This
information is useful for sketching parametric curves. It is also useful to consider
d
2
y
d
x
2
,
which can be found by
d
2
y
d
x
2
=
d
d
x
±
d
y
d
x
¶
=
d
d
t
(
d
y
d
x
)
d
x
d
t
.
Examples
1. A curve
C
is deﬁned by the parametric equations
x
=
t
2
,
y
=
t
3

3
t
.
(a) Show that
C
has two tangents at the point (3
,
0) and ﬁnd their equations.
(b) Find the points on
C
where the tangent is horizontal or vertical.
(c) Determine where the curve is concave upward or downward.
(d) Sketch the curve.
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This note was uploaded on 12/09/2010 for the course MATH 1120 taught by Professor Gross during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 GROSS
 Calculus, Arc Length, Slope

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