The final exam will be cumulative. However, 3/5 of the
final exam will be on Chapter

11, with 2/5 devoted to things covered in the rest of the course.

No calculators of any kind will be allowed.

**Formulas you will be given on the exam (if they are needed):**

• The transformation formulas x' = x cosθ + y sinθ and y' = −x sinθ
+ y cosθ , as well

as x = x' cosθ − y' sinθ and y = x' sinθ + y' cosθ .

• The formulas relating A' and C' to A, B, and C when a conic section is rotated
to

remove the xy -term.

**Formulas to remember:**

• The polar-rectangular transformation formulas:

• The most common polar graphs:

– r = c ( circle of radius c centered at the origin)

– θ = c (line of slope m = tan c through the origin)

– r = ±c cosθ (circle of radius c/2 passing through the origin)

– r = c cos nθ for an integer n (rose curves with 2n petals if n is even
and n petals

if n is odd; c is the length of each petal)

–
(lemniscate, i.e., infinity symbol ; c is the length of the petal)

– r = a + b cosθ (, i.e., a slug; if a = b, a cardioid, i.e., a
human heart)

– r = cθ or r = ce^{θ} or r = cθ^{2 }or any increasing function of θ (a
spiral expanding

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
> 1)

• Variations and techniques for getting the precise
graphs:

– If the equation only involves cosine, then replacing cosine with sine or
replacing

cosine with negative cosine results in the same shape rotated by some angle.
Thus

you only need to remember the basic forms for the polar curves.

– The effect of changing r = f(θ) to r = cf(θ) is to expand or contract the
entire

graph by a factor of c .

– 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
entire

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
in

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
remember.

• Concavity of a parametric equation:

where is computed as
above.

• Slope of the tangent to a polar equation r(θ): Using x = r(θ) cosθ and y =
r(θ)
sinθ

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):

Some subtleties:

– 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
of

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
and
, 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 c^{2} = a^{2} − b^{2}.

If the ellipse has b > a, then the foci are at (0, c) and (0,−c), where c^{2} = b^{2}
−a^{2}.

– Parabola : 4px = y^{2}. Focus at (p, 0), directrix at x = −p. If the parabola is
of the

form 4py = x^{2}, then the focus is at (0, p) and the directrix is y = −p.

– Hyperbola:. Foci at (c, 0) and (−c, 0),
where c^{2} = a^{2} + b^{2}. If the

hyperbola is instead , then the foci are at (0,
c) and (0,−c), where

c^{2} = a^{2} + b^{2} still. The asymptotes are in
either case.