– r = a + b cosθ (, i.e., a slug; if a = b, a cardioid, i.e., a
– r = cθ or r = ceθ or r = cθ2 or any increasing function of θ (a
around the origin)
(a conic section with a focus at the origin and opening along the
positive x-axis; an ellipse if e < 1, a parabola if e = 1, and a hyperbola if e
• Variations and techniques for getting the precise
– If the equation only involves cosine, then replacing cosine with sine or
cosine with negative cosine results in the same shape rotated by some angle.
you only need to remember the basic forms for the polar curves.
– To plot the standard curves, you only need to know a few special (r, θ) points.
For a circle, lemniscate,
, spiral, or conic section, you can get the
graph just by knowing r at the cardinal angles θ = 0,
. For a rose curve,
it is sufficient just to know which θ gives the maxima, minima, and zeroes of r .
– If you have a polar curve you don’t recognize, you can figure out its graph by
plotting (r, θ) pairs for those values of θ that maximize r, minimize r, or result
r = 0. This will always give the essential features of the graph.
– Symmetry can help.
If replacing (r, θ) with (r,−θ ) gives the same equation, the graph is symmetric
about the x-axis.
If replacing (r, θ) with (r, π − θ) gives the same equation, the graph is symmetric
about the y-axis.
If replacing (r,θ ) with either (−r, θ) or (r, θ+π ) gives the same equation, the
graph is symmetric about the origin.
• Slope of the tangent to a parametric equation:
The cancelation of the dt terms makes this formula easy to
• Concavity of a parametric equation:
where is computed as
• Slope of the tangent to a polar equation r(θ): Using x = r(θ) cosθ and y =
and the product rule , you can derive
It’s better to derive this formula as needed than to try
to memorize it. NEVER make
the mistake of saying the slope of a polar equation is
. A slope must always be
expressed in rectangular coordinates.
• Arc length of a polar curve r(θ):
• Area enclosed by a polar curve r(θ):
– If r ≥ 0 always, then you can find the area by integrating from 0 to 2π .
– If r < 0 for some values of θ, make sure you understand the graph before you
try to integrate. Usually you can use symmetry (for example, when finding area
in a rose, it’s best to find the area in one petal, then multiply by the number
petals ). Good bounds to use are consecutive values of θ which make r = 0; for
example, to find the area in cos 3θ , integrate from
to to get
the area of one
petal, then multiply by three to get the whole area.
– To find the area enclosed by two curves
, first understand both
graphs. (This is crucial! If you don’t know what the region looks like, you’ll
probably get the wrong answer.) Find the angles of intersection by setting
, then compute
• Conic section formulas in rectangular coordinates
(oriented along x-axis):
– Ellipse:, with a > b. Foci at (c, 0) and
(−c, 0), where c2 = a2 − b2.
If the ellipse has b > a, then the foci are at (0, c) and (0,−c), where c2 = b2
– Parabola : 4px = y2. Focus at (p, 0), directrix at x = −p. If the parabola is
form 4py = x2, then the focus is at (0, p) and the directrix is y = −p.
– Hyperbola:. Foci at (c, 0) and (−c, 0),
where c2 = a2 + b2. If the hyperbola is instead , then the foci are at (0,
c) and (0,−c), where
c2 = a2 + b2 still. The asymptotes are in
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