Math 242 Review — Chapter 10 Highlights
For
slope
or
concavity
of parametric curves, use the following:
dy
dx
=
dy
dt
dx
dt
d
2
y
dx
2
=
d
dx
dy
dx
=
d
dt
(
dy
dx
)
dx
dt
The latter equation comes from
d
dx
(
u
) =
du
dt
dx
dt
.
For
converting
between rectangular and polar coordinates, remember
x
=
r
cos
θ
y
=
r
sin
θ
r
2
=
x
2
+
y
2
Be aware of how to correctly interpret polar coordinates when
r
is
negative
.
If you define a
polar
curve with an equation
r
=
f
(
θ
), then you can regard
it as a
parametric
curve
x
=
f
(
θ
) cos
θ
,
y
=
f
(
θ
) sin
θ
.
For example, if
r
=
e
θ
, then
x
=
e
θ
cos
θ
and
y
=
e
θ
sin
θ
. You can then say
dx
dθ
=
e
θ
cos
θ

e
θ
sin
θ
and
dy
dθ
=
e
θ
sin
θ
+
e
θ
cos
θ
, and so on.
Area
in polar coordinates is given by
Z
b
a
1
2
r
2
dθ
. You find
a
and
b
by drawing
a picture and/or considering where
r
= 0.
ELLIPSE:
Two special points, “focuses”
F
1
and
F
2
(distance to
F
1
) + (distance to
F
2
) = constant
PARABOLA:
One special point, “focus”
F
; one special line, “directrix”
L
(distance to
F
) = (distance to
L
)
HYPERBOLA:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 wang
 Slope, Sin, Cos, dx, Conic section, θ

Click to edit the document details