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GLOSSARY
This page contains
definitions of terms which students may not have heard of or may forget
what they are. For the most part these are covered in other math
courses (such as MATH 1046 or MATH 1056) or should have been covered in
highschool.
The definitions included
here are not necessarily complete or written
in correct mathematical form. Instead they have been simplified
to give a general idea of the concept while minimizing the confusion
often associated with complete mathematical definitions.
Therefore the definitions should be used only to help understand a
concept not as a resource for mathematically correct definitions.
An excellent example of
this is the definition of a relation.
This definition is
incorrect to the extent that if you were to tell it to your professor
he/she would probably faint. However the actual definition of a
relation is far to complex to include in this glossary so the
definition provided is meant only to give the basics of a relation as
you may need in the MATH 1036/1037 course.
INDEX
- Antiderivative
- Antiderivative of a function f(x) is another
function derivative of which is equal to f(x). Thus, the process of
finding antiderivative is inverse to that of finding derivative.
Antidifferentiation is the process by which differentiation (taking the
derivative) is reversed and antiderivative is found. Antiderivative is
also called an indefinite integral.
- Arbitrary Number
- A number which could be any number it is
defined to be but for which no specific value is chosen. It is
often used in proofs since it can represent any number but does
actually have the value of any number so that the proof applies to more
than one
situation.
- Axiom
- A statement that is so self-evident no proof is
needed.
- Complex Numbers
- A number consisting of a two parts: a real part
and an imaginary part. Commonly expressed in the form a+bi where
a
and b are real numbers and i has the property that i2=-1.
Thus a is the real part and bi is the imaginary part. This allows
the number to be graphed in the complex plane (Argand plane) with
coordinates (a,b).
- Conditional Statement
- A logical statement of the form "if this then
that" meaning that "if this" happens "then that" will happen.
Commonly referred to as an if-then statement.
- Contrapositive
- A statement of the where hypothesis and
conclusion are reversed and negated. I.E. the conditional
statement "If this then that" becomes "If not that then not
this". The contrapositive is equivalent to the original statement.
- Converse
- A statement where the hypothesis and conclusion
are reversed. I.E. the conditional
statement "If this then that"
becomes "If that then this". The Converse is NOT equivalent to
the original statement, but is equivalent to the inverse.
- Counterexample
- A specific example used to disprove something.
For example if someone were to say "when you multiply two numbers the
answer is always even", a counterexample for this would be 1x3=3 since
it shows that two numbers can be multiplied can have an odd answer.
Most of the errors in the Common
Mistakes section are shown using
counterexamples.
- Domain
- The domain of a function is the set of all
values of a variable for which the function is defined. For
example, the domain of the function y=x2 is the whole real
line since y has a value for every real value of x whereas the domain
of the function y=(1 / x) x0 (the
whole
real line
except x=0) since y is undefined when x=0.
- Even Function
- A function with the property that
f(-x)=f(x). This means that when -x is substituted into f for x
the function stays the same. Functions of this type are symmetric about
the y-axis meaning that the graph of f(x) for x0 is a
reflection of the graph of f(x) for x0.
An example
of this is the
function f(x)=x2 since f(-x)=(-x)2=x2=f(x).
- Explicit Function
- A function where one variable is explicitly
(e.g. by means of formula or table of values) expressed in
terms of another. Examples are y=2x+2, q=3p2-2p-30,
and s=3sin t.
- Function
- A rule where each value from the domain set is
assigned to exactly one element in the range
set. A common test
for a function is the vertical line test
where on a graph if any
vertical line intersects the graph more than once it is not a graph of
a function.
- If and only if (IFF)
- An expression used to imply that a statement
holds in both directions and only in the described situations. This
means that if you have the
situation described on either side of the 'if and only if ' then you
will have the situation on the other side as well and if you do not
have one then you will not have the other. An if and only if
statement is also called a biconditional statement.
- Implicit Function
- A function where one variable is not expressed
explicitly in terms of another, but where it is still assumed that one
variable depends on another. Examples are y+x=2, pq2=2p-3q,
and s/t=2t
- Inverse
- A statement where the hypothesis and conclusion
are negated. I.E. the conditional
statement "If this then that"
becomes "If not this then not that". The inverse is NOT
equivalent to the original statement, but is equivalent to the converse.
- Horizontal Asymptote
- A horizontal line which the graph of a function
approaches as variable tends to positive or negative infinity. It
should be noted that the graph
can cross the horizontal asymptote as many times as it likes (as with
many oscillating functions). A
horizontal asymptote occurs when the limit of a function as the
variable approaches either
positive or negative infinity is a constant.
- Horizontal Line Test
- A technique used to test if a function is one-to-one
where if any horizontal line drawn on a graph intersects the graph of
the function more than once that function is not one-to-one.
- Lemma
- Often referred to as a 'mini theorem'. A Lemma
is
a fact which must be proved so it can be used in the proof of another
theorem.
- Math Amnesia
- The term given to describe the sudden loss of
mathematical knowledge. This commonly occurs at high stress times
such as tests and exams and is often blamed for poor scores received on
them. Math Amnesia can be overcome by stress reducing
techniques and the use of better preparation techniques. See Test/Exam Preparation
strategies in the Study Tips
tutorial for suggestions on how to prevent Math Amnesia.
-
- Mean Value Theorem
- A theorem which states that if a function f is
continuous on [a,b] and differentiable on (a,b) then there is a number
c in (a,b) that satisfies the property f '(c)=(f(b)-f(a)) /
(b-a). In words the theorem says that if the function is
continuous on the closed interval and differentiable on the open
interval then there is a point in the open interval such that the slope
of the tangent at that point is equal to the slope of the secant
joining the endpoints of the interval. I.E. there is a point
where the slope of the tangent equals the slope of the secant or the
instantaneous rate of change is equal to the average rate of change.
Geometrically, this means that there is a point on the graph where the
tangent line is parallel to the secant line. Note that there could be
more than one number in the interval with this
property.
- Odd Function
- A function with the property that
f(-x)=-f(x). This means that when -x is substituted into f for x
the sign of the function changes. Functions of this type are
symmetric about the origin: the graph of function for x0 appears as
the graph of function for x0
rotated 180 degrees
around the
origin. An example of this is the function f(x)=x3
since f(-x)=(-x)3=-(x3)=-f(x).
- One-to-One
- A function is one-to-one (also called
injective) if, and only if, it
has the property that if f(x1)=f(x2)
then x1=x2. In other words it never takes
on
the same value twice. A quick test for this property is the horizontal line
test where if a horizontal line intersects the graph of
a function more than once the function is not one-to-one.
-
- Oscillating Function
- A function whose graph continuously switches
between increasing and decreasing causing the graph to have a series of
local maxima and minima resembling waves in water or a vibrating string.
- Periodic Function
- A function f is said to be periodic if
f(x+p)=f(x) for all x in the domain
of f where p is the period
(smallest positive number for which this property holds). In
other words the graph of function repeats itself indefinitely. An
example of this is f(x)=sin(x) where the period is 2 since sin(x+2)=sin(x)
for all x.
- Proof by Contradiction
- A type of proof whereby the opposite of what is
being proved is assumed to be true and a sequence of statements is
obtained until a contradiction
(an impossible scenario) is reached thus indicating that the assumption
is false, so the original statement must be true.
- Proof by Induction
- A type of proof where the following series of
steps is followed. The statement is proved true for one specific
integer value (usually 0, 1 or 2). This is called the base of
induction. The statement is then assumed to be true for some arbitrary
value (which is a positive integer). This is called the inductive
assumption. A proof is then formulated to prove the statement true for
the arbitrary value +1 making use of the assumption that the statement
is true for the arbitrary value. This is called inductive step. It is
then concluded that since
the statement is true for an arbitrary value +1 given that it is true
for an arbitrary value it should be true for any positive integer.
Therefore
the entire statement is proved. This type of proof can only be
used for number sets who increase and decrease by integer values I.E.
the set of whole numbers, natural numbers, positive integers,
non-negative integers, negative integers, etc. For more information,
see the Induction
tutorial.
- Range
- The range of a function is the set of values
the function takes as the variable goes through all the values of the domain.
For example the range of the function y=x3 is
the whole real line since all the values in the real line have
corresponding values in the domain that the function takes them to
whereas the range of y=x2 is y0 (any value greater than or equal to
0) since the function does not take any
values below y=0 for any value in the domain.
- Relation
- A relation (or, more precisely, a binary
relation), is a condition on two numbers that is either satisfied for a
given pair of numbers, or is not satisfied. For example, ">" is a
relation. A relation between two real numbers can be viewed as a subset
of a coordinate plane. For example, relation y > x distinguishes the
region strictly above the line y = x.
- Rolle's Theorem
- A theorem which states that if a function f is
continuous on [a,b], is differentiable on (a,b), and f(a)=f(b) then
there will be a point c in (a,b) with the property that f '(c)=0.
In words the theorem says that if the function is continuous on a
closed interval, differentiable on the open interval, and the value of
the function at the endpoints of the interval are equal, then there is
a
point in the open interval such that the slope of the tangent line at
that point is 0. I.E. There is a local maximum or minimum on the
interval or the function is constant. Geometrically, Rolle's Theorem
states that there is a point on the graph where the tangent line is
horizontal. Note that there could be
more than one number in the interval with this property.
- Slant Asymptote
- An oblique line (neither horizontal or
vertical) which the graph of a function approaches as the variable goes
to positive or negative infinity. Often occurs in rational functions
when the degree of the
numerator is one
higher than the degree of the denominator.
- Vertical Asymptote
- A vertical line which the graph of the of a
function approaches but never reaches. Occurs when the limit of a
function as it approaches a number (from the left, right, or at the
number) equals either positive or negative infinity.
- Vertical Line Test
- A technique used to determine whether a curve
in coordinate plane
is a function or not where if a vertical line crosses the graph more
than once that relation is not a function.
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