P
p-series
A series of the form
or
comparison test and the limit comparison test.
Note: The harmonic series is a p-series with p =1.
Paired Data
Data that occurs in ordered pairs.
Pappus’s Theorem
Theorem of Pappus
, where p > 0. Often employed when using the
A method for finding the volume of a solid of revolution. The volume equals the product of the
area of the region being rotated times the distance traveled by the centroid of the region in one
rotation.
Volume by Parallel Cross Sections
The formula below gives the volume of a solid. A(x) is the formula for the area of parallel crosssections over the entire length of the solid.
Note: The disk method and the washer method are both derived from this formula.
Movie clips (with narration)
Volume by Parallel
Volume by Parallel
Cross-Sections:
Cross-Sections:
The Concept
Example
This can be a slow download.
Squares on an Ellipse
File size is 15.2M.
This can be a slow download.
File size is 16.4M.
Parallel Lines
Two distinct coplanar lines that do not intersect. Note: Parallel lines have the same slope.
Parallel Planes
Two distinct planes that do not intersect.
Parallel Postulate
The assumption that, given a point P and a line m not through P, there is exactly one line passing
through P that is parallel to m.
Parallelepiped
A polyhedron with six faces, all of which are parallelograms.
Parallelogram
A quadrilateral with two pairs of parallel sides.
Parameter (algebra)
The independent variable or variables in a set of parametric equations.
Parametric Derivative Formulas
The formulas for the first derivative
curve are given below.
and second derivative
of a parametrically defined
Parametric Equations
A system of equations with more than one dependent variable. Often parametric equations are
used to represent the position of a moving point.
Parametrize
To write in terms of parametric equations.
Example:
The line x + y = 2 can be parametrized as x = 1 + t, y = 1 – t.
Parent Functions
Toolkit Functions
A set of basic functions used as building blocks for more complicated functions. The list of
parent functions varies. A typical set of functions is listed below.
Parentheses
The symbols ( and ). Singular: parenthesis.
Partial Derivative
A derivative of a function that has more than one independent variable. Partial derivatives are
found by treating one independent variable as a variable and the rest as constants.
Partial Differential Equation
A differential equation that contains at least one partial derivative.
Partial Fractions
The process of writing any proper rational expression as a sum of proper rational expressions.
This method is use in integration as shown below.
Note: Improper rational expressions can also be rewritten using partial fractions. You must,
however, use polynomial long division first before finding a partial fractions representation.
Partial Sum of a Series
The sum of a finite number of terms of a series.
Partition of an Interval
A division of an interval into a finite number of sub-intervals. Specifically, the partition itself is
the set of endpoints of each of the sub-intervals.
Partition of a Positive Integer
Rewriting a positive integer as the sum of smaller positive integers.
Partition of a Set
A collection of disjoint subsets of a given set. The union of the subsets must equal the entire
original set.
For example, one possible partition of {1, 2, 3, 4, 5, 6} is {1, 3}, {2}, {4, 5, 6}.
Pascal's Triangle
The figure below, extended infinitely. A particular entry is found by adding the two numbers that
are above and on either side of the element. Note: The numbers which make up Pascal's triangle
are called binomial coefficients.
Pascal's Triangle
Note that the sum of any two
adjacent elements in a row
can be found between them
on the next row. Each row
begins and ends with 1.
etc.
Pentagon
A polygon with five sides.
Pentagon
Regular Pentagon
Per Annum
Per year. For example, 5% per annum means 5% per year.
Percentile
The pth percentile of a set of data is the number such that p% of the data is less than that number.
For example, a student whose SAT score is in the 78th percentile has scored higher than 78% of
the students taking the test.
Note: The median is the 50th percentile.
Perfect Number
A number n for which the sum of all the positive integer factors of n which are less than n add up
to n.
For example, 6 and 28 are perfect numbers. The number 6 has factors 1, 2, and 3, and 1 + 2 + 3 =
6. The number 28 has factors 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28.
Perfect Square
Any number that is the square of a rational number. For example, 0, 1, 4, 9, 16, 25, etc. are all
perfect squares. So are
and
.
Perimeter
The distance around the outside of a plane figure. For a polygon, the perimeter is the sum of the
lengths of the sides.
Period of a Periodic Function
The horizontal distance required for the graph of a periodic function to complete one cycle.
Formally, a function f is periodic if there exists a number p such that f(x + p) = f(x) for all x. The
smallest possible value of p is the period. The reciprocal of period is frequency.
Period of Periodic Motion
The time needed to complete a cycle. For example, a pendulum exhibits periodic motion. Its
period is the time it takes for the pendulum to swing from one side to the other and then back
again.
Note: Period is the reciprocal of frequency.
Periodic Function
A function which has a graph that repeats itself identically over and over as it is followed from
left to right. Formally, a function f is periodic if there exists a number p such that f(x + p) = f(x)
for all x.
Periodic Motion
Motion that repeats itself identically over and over, such as the swinging of a pendulum. If the
motion can be modeled using a sinusoid it is called simple harmonic motion.
Periodicity Identities
Trig identities showing the periodic behavior of the six trig functions.
Periodicity Identities, radians
sin (x + 2π) = sin x
csc (x + 2π) = csc x
Periodicity Identities, degrees
sin (x + 360°) = sin x
csc (x + 360°) = csc x
cos (x + 2π) = cos x sec (x + 2π) = sec x
cos (x + 360°) = cos x sec (x + 360°) = sec x
tan (x + π) = tan x
tan (x + 180°) = tan x cot (x + 180°) = cot x
cot (x + π) = cot x
Permutation
A selection of objects in which the order of the objects matters.
Example: The permutations of the letters in the set {a, b, c} are:
abc
bac
cab
acb
bca
cba
Permutation Formula
A formula for the number of possible permutations of k objects from a set of n. This is usually
written nPk .
Formula:
Example
How many ways can 4 students from a group of 15 be lined
:
up for a photograph?
Answer:
There are 15P4 possible permutations of 4 students from a
group of 15.
different lineups
Perpendicular Bisector
The line perpendicular to a segment passing through the segment's midpoint. Note: The
perpendicular bisectors of the sides of a triangle are concurrent, intersecting at the circumcenter.
Phase Shift
Horizontal shift for a periodic function.
For example, the function f(x) = 3sin (x – π) has a phase shift of π. That is, the graph of f(x) =
3sin x is shifted π units to the right.
Pi π
The ratio of the circumference of a circle to its diameter. Pi is written π and is a transcendental
number.
π ≈ 3.14159 26535 89793...
Plane Figure
A shape on a plane. Includes points, lines, polygons, polygon interiors, circles, disks, parabolas,
ellipses, hyperbolas, etc. Formally, a plane figure is any set of points on a plane.
Plane Geometry
The study of points, lines, polygons, shapes, and regions on a plane. Plane geometry typically
does not use Cartesian coordinates.
Platonic Solids
Regular Polyhedra
The solids which have faces that are all congruent regular polygons and which has dihedral
angles that are all congruent.
There are only five possible shapes for a regular polyhedron: regular tetrahedron, cube,
octahedron, dodecahedron, and icosahedron.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
If your browser is Java-enabled, try dragging, shift dragging, control dragging, and right-click dragging
both vertically and horizontally across these figures.
Point
The geometric figure formed at the intersection of two distinct lines.
Point of Division Formula
The formula for the coordinates of a point which part of the way from one point to another.
Note: The midpoint formula is a special case of the point of division formula in which t = ½.
Point of Symmetry
A special center point for certain kinds of symmetric figures or graphs. If a figure or graph can
be rotated 180° about a point P and end up looking identical to the original, then P is a point of
symmetry.
Example:
This is a graph of the curve
together with its point of symmetry (–2, 1).
The point of symmetry is marked in red.
Point-Slope Equation of a Line
y – y1 = m(x – x1), where m is the slope and (x1, y1) is a point on the line. Point-slope is the form
used most often when finding the equation of a line.
Movie Clips (with narration)
Point and Slope:
Two Points:
How to find the equation of a line
How to find the equation of a line
(4.13M)
(5.5M)
Polar Axis
The positive x-axis.
Polar-Rectangular Conversion Formulas
Rules for converting between polar coordinates and rectangular coordinates.
Polar Coordinates
A way to describe the location of a point on a plane. A point is given coordinates (r, θ). r is the
distance from the point to the origin. θ is the angle measured counterclockwise from the polar
axis to the segment connecting the point to the origin.
Note: With polar coordinates a given point has many possible representations. θ has many
possible values depending on which coterminal angle is chosen, and r can be positive or
negative.
Polar Curves
Curves commonly written with polar equations include cardioids, lemniscates, limaçons, rose
curves, and spirals.
Polar Derivative Formulas
The formula for the first derivative
of a polar curve is given below.
Polar Equation
An equation for a curve written in terms of the polar coordinates r and θ.
Polar Form of a Complex Number
The polar coordinates of a complex number on the complex plane. The polar form of a complex
number is written in any of the following forms: rcos θ + irsin θ, r(cos θ + isin θ), or rcis θ. In
any of these forms r is called the modulus or absolute value. θ is called the argument.
Polygon
A closed plane figure for which all sides are line segments. The name of a polygon describes the
number of sides. A polygon which has all sides mutually congruent and all angles mutually
congruent is called a regular polygon.
Number of
sides
Polygon name
3
triangle
4
quadrilateral
5
pentagon
6
hexagon
7
heptagon
8
octagon
9
nonagon
10
decagon
11
undecagon
12
dodecagon
n
n-gon
Polygon Interior
The points enclosed by a polygon.
Polyhedron
pentagon
regular undecagon
octagon
A solid with no curved surfaces or edges. All faces are polygons and all edges are line segments.
Polynomial
The sum or difference of terms which have variables raised to positive integer powers and which
have coefficients that may be real or complex.
The following are all polynomials: 5x3 – 2x2 + x – 13, x2y3 + xy, and (1 + i)a2 + ib2.
Note: Even though the prefix poly- means many, we use the word polynomial to refer to
polynomials with 1 term (monomials), 2 terms (binomials), 3 terms, (trinomials), etc.
Standard form for a polynomial in one variable:
anxn + an–1xn–1 + ··· + a2x2 + a1x + a0
Polynomial Facts
Facts about polynomials of the form p(x) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 are listed below.
Polynomial End Behavior:
1. If the degree n of a polynomial is even, then the arms of the graph are either both up or both
down.
2. If the degree n is odd, then one arm of the graph is up and one is down.
3. If the leading coefficient an is positive, the right arm of the graph is up.
4. If the leading coefficient an is negative, the right arm of the graph is down.
Extreme Values:
The graph of a polynomial of degree n has at most n – 1 extreme values.
Inflection Points:
The graph of a polynomial of degree n has at most n – 2 inflection points.
Remainder Theorem:
p(c) is the remainder when polynomial p(x) is divided by x – c.
Factor Theorem:
x – c is a factor of polynomial p(x) if and only if c is a zero of p(x).
Rational Root Theorem:
If a polynomial equation anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 = 0 has integer coefficients then it is
possible to make a complete list of all possible rational roots. This list consists of all possible
numbers of the form c/d, where c is any integer that divides evenly into the constant term a0 and
d is any integer that divides evenly into the leading term an.
Conjugate Pair Theorem:
If a polynomial has real coefficients then any complex zeros occur in complex conjugate pairs.
That is, if a + bi is a zero then so is a – bi, where a and b are real numbers.
Fundamental Theorem of Algebra:
A polynomial p(x) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 of degree at least 1 and with coefficients
that may be real or complex must have a factor of the form x – r, where r may be real or
complex.
Corollary of the Fundamental Theorem of Algebra:
A polynomial of degree n must have exactly n zeros, counting mulitplicity.
Population
The entire set of cases or individuals under consideration in a statistical analysis. For example, a
poll given to a sample of voters is designed to measure the preferences of the population of all
voters.
Positive Direction
A way of describing the scatterplot of positively associated data.
Positive Number
A real number greater than zero. Zero itself is not positive.
Positive Series
A series with terms that are all positive.
Positively Associated Data
A relationship in paired data in which the two sets of data tend to increase together or decrease
together. In a scatterplot, positively associated data tend to follow a pattern from the lower left to
the upper right. Positively associated data have a positive correlation coefficient.
Postulate
A statement accepted as true without proof. A postulate should be so simple and direct that it
seems to be unquestionably true.
Power
The result of raising a base to an exponent. For example, 8 is a power of 2 since 8 is 23.
Power Rule
The formula for finding the derivative of a power of a variable.
Power Series
A series which represents a function as a polynomial that goes on forever and has no highest
power of x.
Power Series Convergence
A theorem that states the three alternatives for the way a power series may converge.
Precision
The level of detail in a number or estimate. A precise number has many significant digits. Note:
An answer may be precise without being accurate.
Accuracy: 3.14 is a fairly accurate approximation of π (pi).
Precision: 3.199 is a more precise approximation, but it is less accurate.
Pre-Image of a Transformation
The original figure prior to a transformation. In the example below, the transformation is a
rotation and a dilation.
Prime Factorization
Writing an integer as a product of powers of prime numbers.
Prime Number
A positive integer which has only 1 and the number itself as factors. For example, 2, 3, 5, 7, 11,
13, etc. are all primes. By convention, the number 1 is not prime.
Principal
In finance, the original amount of money invested, deposited, or loaned.
Prism
A solid with parallel congruent bases which are both polygons. The bases must be oriented
identically. The lateral faces of a prism are all parallelograms or rectangles.
Probability
The likelihood of the occurrence of an event. The probability of event A is written P(A).
Probabilities are always numbers between 0 and 1, inclusive.
The four basic rules of probability:
1. For any event A, 0 ≤ P(A) ≤ 1.
2. P(impossible event) = 0.
Also written P(empty set) = 0 or P(
) = 0.
3. P(sure event) = 1.
Also written P(S) = 1, where S is the sample space.
4. P(not A) = 1 – P(A).
Also written P(complement of A) = 1 – P(A) or P(AC) = 1 –
P(A) or
.
If all outcomes of an experiment are equally likely, then
Product
The result of multiplying a set of numbers or expressions.
Product Rule
A formula for the derivative of the product of two functions.
Product to Sum Identities
Trig identities which show how to rewrite products of sines and/or cosines as sums.
Product to Sum Identities
Prolate Spheroid
A stretched sphere shaped like a watermelon. Formally, a prolate spheroid is a surface of
revolution obtained by revolving an ellipse about its major axis.
Proper Fraction
A fraction with a smaller numerator than denominator. For example,
is a proper fraction.
Proper Rational Expression
A rational expression in which the degree of the numerator polynomial is less than the degree of
the denominator polynomial.
Proper Subset
A subset which is not the same as the original set itself.
For example, {a, b} is a proper subset of {a, b, c}, but {a, b, c} is not a proper subset of {a, b, c}.
Proportional
Two variables are proportional if their ratio is constant. Proportionality is written using the ∝
symbol.
Example:
Suppose that a ∝ b. This means that a/b is constant, or that a = kb where k is a constant.
Pyramid
A polyhedron with a polygonal base and lateral faces that taper to an apex. A pyramid with a
triangular base is called a tetrahedron.
Solid view: right regular
Frame view: right regular
pyramid with a regular
pyramid with a regular
pentagon as base
pentagon as base
For any pyramid:
Lateral surface area for a right regular pyramid =
Total surface area for a right regular pyramid =
h = height of the pyramid
B = area of the base
P = perimeter of the base
s = slant height
All pyramids have the same volume formula. Here are some other types of pyramids:
Solid view: regular pyramid with a
Frame view: regular pyramid with a
regular pentagon as base
regular pentagon as base
Solid view: right pyramid with a square
Frame view: right pyramid with a
base
square base
Solid view: oblique pyramid with a
Frame view: oblique pyramid with a
square base
square base
Pythagorean Theorem
An equation relating the lengths of the sides of a right triangle. The sum of the squares of the
legs of a right triangle is equal to the square of the hypotenuse.
Pythagorean Triple
A set of three positive integers which satisfies the Pythagorean theorem a2 + b2 = c2.
Examples:
3, 4, 5
6, 8, 10
5, 12, 13
7, 24, 25
8, 15, 17
9, 40, 41
20, 21, 29
32 + 42 = 52
62 + 82 = 102
52 + 122 = 132
72 + 242 = 252
82 + 152 = 172
92 + 402 = 412
202 + 212 = 292