chaper 10 Parametric Equations and Polar Coordinates (10.3 Polar…
chaper 10 Parametric Equations and
Curves Defined by Parametric Equations
x = t2 – 2t y = t + 1
10.3 Polar Coordinates
A coordinate system represents a point in the plane by an
ordered pair of numbers called coordinates.
Here we describe a coordinate system introduced by Newton, called the polar coordinate system, which is more convenient for many purposes.
We choose a point in the plane that is called the pole (or origin) and is labeled O. Then we draw a ray (half-line)
starting at O called the polar axis.
Then the point P is represented by the ordered pair (r, θ )
and r, θ are called polar coordinates of P.
The graph of a polar equation r = f (θ ), or more generally
F (r, θ ) = 0, consists of all points P that have at least one
polar representation (r, θ ) whose coordinates satisfy the
The remainder of the curve is drawn in a similar fashion,
with the arrows and numbers indicating the order in which
the portions are traced out. The resulting curve has four
loops and is called a four-leaved rose.
Tangents to Polar Curves
10.4 Areas and Lengths in
Calculus with Parametric Curves
10.5 conic sectiors
a curve obtained as the intersection of the surface of a cone with a plane.
10.6 Conic Sections in Polar
let F be a fixed point and L be a fixed line in a plane. Let E be a fixed positive number. The set of all points P in the plane.