# Review Sheet for Mathematics 2300 Final Exam

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 = ceθ or r = cθ2 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

* If replacing (r, θ) with (r, π − θ) gives the same equation, the graph is symmetric

* 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 c2 = a2 − b2.
If the ellipse has b > a, then the foci are at (0, c) and (0,−c), where c2 = b2 −a2.

Parabola : 4px = y2. Focus at (p, 0), directrix at x = −p. If the parabola is of the
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 either case.

 Prev Next

Start solving your Algebra Problems in next 5 minutes!

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of June 22nd you will pay only \$39.99 instead of our regular price of \$74.99 -- this is \$35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1\$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebra Helper can solve problems in all the following areas:

• simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
• factoring and expanding expressions
• finding LCM and GCF
• (simplifying, rationalizing complex denominators...)
• solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
• solving a system of two and three linear equations (including Cramer's rule)
• graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
• graphing general functions
• operations with functions (composition, inverse, range, domain...)
• simplifying logarithms
• basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
• arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!