Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
DDe Moivre’s Theorem
A formula useful for finding powers and roots of complex numbers.
Decagon
A polygon with ten sides.
Decagon Regular Decagon
Mathwords for ICP Program
Deciles
The 10th and 90th percentiles of a set of data.
Decreasing Function
A function with a graph that moves downward as it is followed from left to right. For example, any line with a negative slope is decreasing.
Note: If a function is differentiable, then it is decreasing at all points where its derivative is negative.
Definite Integral
An integral which is evaluated over an interval. A definite integral is written . Definite integrals are used to find the area between the graph of a function and the x -axis . There are many other applications.
Formally, a definite integral is the limit of a Riemann sum as the norm of the partition
approaches zero. That is, .
Integral Rules
For the following, a, b, c, and C are constants; for definite integrals, these represent real number constants. The rules only apply when the integrals exist.
Indefinite integrals (These rules all apply to definite integrals as well)
Mathwords for ICP Program
1.
2.
3.
4.
5. Integration by parts:
Definite integrals
1.
2.
3. If f(u) ≤ g(u) for all a ≤ u ≤ b, then
4. If f(u) ≤ M for all a ≤ u ≤ b, then
5. If m ≤ f(u) for all a ≤ u ≤ b, then
6. If a ≤ b, then
Mathwords for ICP Program
Degenerate
An example of a definition that stretches the definition to an absurd degree.
A degenerate triangle is the "triangle" formed by three collinear points. It doesn’t look like a triangle, it looks like a line segment.
A parabola may be thought of as a degenerate ellipse with one vertex at an infinitely distant point.
Degenerate examples can be used to test the general applicability of formulas or concepts. Many of the formulas developed for triangles (such as area formulas) apply to degenerate triangles as well.
Degenerate Conic Sections
Plane figures that can be obtained by the intersection of a double cone with a plane passing through the apex. These include a point, a line, and intersecting lines. Like other conic sections, all degenerate conic sections have equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
Degree of a Polynomial
The highest degree of any term in the polynomial.
Mathwords for ICP Program
Degree of a Term
For a term with one variable, the degree is the variable's exponent. With more than one variable, the degree is the sum of the exponents of the variables.
Del Operator
The symbol , which stands for the "vector" or .
Dependent Variable
A variable that depends on one or more other variables. For equations such as y = 3x – 2, the dependent variable is y. The value of y depends on the value chosen for x. Usually the dependent variable is isolated on one side of an equation. Formally, a dependent variable is a variable in an expression, equation, or function that has its value determined by the choice of value(s) of other variable(s).
Mathwords for ICP Program
Derivative
A function which gives the slope of a curve; that is, the slope of the line tangent to a function. The derivative of a function f at a point x is commonly written f '(x). For example, if f(x) = x3 then f '(x) = 3x2. The slope of the tangent line when x = 5 is f '(x) = 3·52 = 75.
Derivative of a Power Series
The derivative of a function defined by a power series can be found by differentiating the series term-by-term.
Mathwords for ICP Program
Derivative Rules
A list of common derivative rules is given below.
Descartes' Rule of Signs
A method for determining the maximum number of positive zeros for a polynomial. This maximum is the number of sign changes in the polynomial when written as shown below.
Mathwords for ICP Program
Determinant
A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors.
Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.
Diagonal Matrix
A square matrix which has zeros everywhere other than the main diagonal. Entries on the main diagonal may be any number, including 0.
Determinant
A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors.
Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.
Mathwords for ICP Program
Diagonal of a Polygon
A line segment connecting non-adjacent vertices of a polygon. Note: An n -gon has diagonals.
Diameter of a Circle or Sphere
A line segment between two points on the circle or sphere which passes through the center. The word diameter is also also refers to the length of this line segment.
Diametrically Opposed
Mathwords for ICP Program
Two points directly opposite each other on a circle or sphere. More formally, two points are diametrically opposed if they are on opposite ends of a diameter.
Difference
The result of subtracting two numbers or expressions. For example, the difference between 7 and 12 is 12 – 7, which equals 5.
Sum/Difference Identities
Trig identities which show how to find the sine, cosine, or tangent of the sum or difference of two given angles.
Sum/Difference Identities
Difference Quotient
Mathwords for ICP Program
For a function f, the formula . This formula computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition the derivative.
Differentiable
A curve that is smooth and contains no discontinuities or cusps. Formally, a curve is differentiable at all values of the domain variable(s) for which the derivative exists.
Differential
An tiny or infinitesimal change in the value of a variable. Differentials are commonly written in the form dx or dy.
Differential Equation
An equation showing a relationship between a function and its derivative(s). For example,
is a differential equation with solutions y = Ce–x.
Differentiation
The process of finding a derivative.
Digit
Mathwords for ICP Program
Any of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 used to write numbers. For example, the digits in the number 361 are 3, 6, and 1.
Dihedral Angle
An angle formed by intersecting planes.
Dilation
A transformation in which a figure grows larger. Dilations may be with respect to a point (dilation of a geometric figure) or with respect to the axis of a graph (dilation of a graph).
Note: Some high school textbooks erroneously use the word dilation to refer to all transformations in which the figure changes size, whether the figure becomes larger or smaller. Unfortunately the English language has no word that refers collectively to both stretching and shrinking.
Pronunciation: Dilation (die-LAY-shun) has only three syllables, not four.
Dilation of a Geometric Figure
A transformation in which all distances are lengthened by a common factor. This is done by stretching all points away from some fixed point P.
StretchDilation of a Graph
Mathwords for ICP Program
A transformation in which all distances on the coordinate plane are lengthened by multiplying either all x-coordinates (horizontal dilation) or all y-coordinates (vertical dilation) by a common factor greater than 1.
Note: When the common factor is less than 1 the transformation is called a compression.
Dimensions
On the most basic level, this term refers to the measurements describing the size of an object. For example, length and width are the dimensions of a rectangle.
In a more advanced sense, the number of dimensions a set, region, object, or space possesses indicates how many mutually perpendicular directions of movement are possible. For example, a line is one dimensional, a plane is two dimensional, the space in which we live is three dimensional, and a point is zero dimensional.
Dimensions of a Matrix
The number of rows and columns of a matrix, written in the form rows×columns. The matrix below has 2 rows and 3 columns, so its dimensions are 2×3. This is read aloud, "two by three."
Note: One way to remember that Rows come first and Columns come second is by thinking of RC Cola®.
Mathwords for ICP Program
Direct ProportionDirect VariationDirectly Proportional
A relationship between two variables in which one is a constant multiple of the other. In particular, when one variable changes the other changes in proportion to the first.
If b is directly proportional to a, the equation is of the form b = ka (where k is a constant).
Equation: y = 4x
Variable y is directly proportional to x.
Doubling x causes y to double. Tripling x causes y to triple.
x y
1 4
2 8
3 12
Directrices of an Ellipse
Two parallel lines on the outside of an ellipse perpendicular to the major axis. Directrices can be used to define an ellipse. Formally, an ellipse is the locus of points such that the ratio of the distance to the nearer focus to the distance to the nearer directrix equals a constant that is less than one. This constant is the eccentricity.
Mathwords for ICP Program
Directrices of a Hyperbola
Two parallel lines which are perpendicular to the major axis of a hyperbola. The directrices are between the two parts of a hyperbola and can be used to define it as follows: A hyperbola is the locus of points such that the ratio of the distance to the nearer focus to the distance to the nearer directrix equals a constant that is greater than one. This constant is the eccentricity.
Directrix of a Parabola
A line perpendicular to the axis of symmetry used in the definition of a parabola. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.
Mathwords for ICP Program
Example:
This is a graph of the parabola with all its major features labeled: axis of symmetry, focus, vertex, and directrix.
Discontinuity
A point at which the graph of a relation or function is not connected. Discontinuities can be classified as either removable or essential. There are several kinds of essential discontinuities, one of which is the step discontinuity.
Discontinuous Function
A function with a graph that is not connected.
Mathwords for ICP Program
Discrete
A set with elements that are disconnected. The set of integers is discrete. The set of real numbers is not discrete; it is continuous.
Formally, a set of numbers is discrete if each number in the set is contained in a neighborhood that contains no other elements of the set.
Discriminant of a Quadratic
The number D = b2 – 4ac determined from the coefficients of the equation ax2 + bx + c = 0. The discriminant reveals what type of roots the equation has.
Note: b2 – 4ac comes from the quadratic formula.
Mutually ExclusiveDisjoint Events
Mathwords for ICP Program
Events that have no outcomes in common.
Disjoint SetsNon-Overlapping Sets
Two or more sets which have no elements in common. For example, the sets A = {a,b,c} and B = {d,e,f} are disjoint.
Disjunction
A statement which connects two other statements using the word or.
For example, "A polygon with four sides can be called a quadrilateral or a quadrangle" contains the disjunction "quadrilateral or quadrangle".
Disk
The union of a circle and its interior.
Disk Method
A technique for finding the volume of a solid of revolution. This method is a specific case of volume by parallel cross-sections.
Mathwords for ICP Program
Distance Formula
The formula is the distance between points (x1, y1) and (x2, y2).
Distance from a Point to a Line
The length of the shortest segment from a given point to a given line. A formula is given below.
Distinct
Different. Not identical.
DistributeExpand
Mathwords for ICP Program
To multiply out the parts of an expression. Distributing is the opposite of factoring.
Example 1: 3x(x + 8) = 3x·x + 3x·8 = 3x2 + 24x
Example 2: (x + 2)(x + 5) = x(x + 5) + 2(x + 5) = x·x + x·5 + 2·x + 2·5 = x2 + 7x + 10
Example 3: (x – 3)4 = (x – 3)(x – 3)(x – 3)(x – 3) = (x·x – x·3 – 3·x + 3·3)(x·x – x·3 – 3·x + 3·3) = (x2 – 6x + 9)(x2 – 6x + 9) = x2(x2 – 6x + 9) – 6x(x2 – 6x + 9) + 9(x2 – 6x + 9) = x2·x2 – x2·6x + x2·9 – 6x·x2 + 6x·6x – 6x·9 + 9·x2 – 9·6x + 9·9 = x4 – 6x3 + 9x2 – 6x3 + 36x2 – 54x + 9x2 – 54x + 81 = x4 – 12x3 + 54x2 – 108x + 81
Distributing Rules
Algebra rules for distributing expressions. See factoring rules as well.
A. Multiplication
1. addition: a(b + c) = ab + ac and (b + c)a = ba + ca
2. subtraction: a(b – c) = ab – ac and (b – c)a = ba – ca
3. FOIL: (a + b)(c + d) = ac + ad + bc + bd
Careful!!
a(bc) ≠ ab·ac
B. Division
Mathwords for ICP Program
1. addition:
2. subtraction:
Careful!!
C. Exponents (a ≥ 0, b ≥ 0)
1. multiplication: (ab)x = axbx
2. division: (b ≠ 0)
Careful!!
(a + b)x ≠ ax + bx
(a – b)x ≠ ax – bx
D. Roots (x ≥ 0, y ≥ 0)
1. multiplication:
2. division: ( y ≠ 0)
Careful!!
Mathwords for ICP Program
E. Logarithms (x > 0, y > 0, a > 0, a ≠ 1)
1. multiplication: loga (xy) = loga x + loga y
2. division:
3. powers: loga (xp) = p loga x
Careful!!
loga (x + y) ≠ loga x + loga y
loga (x – y) ≠ loga x – loga y
F. Trig
1.
2.
3.
Diverge
To fail to approach a finite limit. There are divergent limits, divergent series, divergent sequences, and divergent improper integrals.
Divergent Sequence
A sequence that does not converge. For example, the sequence 1, 2, 3, 4, 5, 6, 7, ... diverges since its limit is infinity (∞). The limit of a convergent sequence must be a real number.
Divergent Series
Mathwords for ICP Program
A series that does not converge. For example, the series 1 + 2 + 3 + 4 + 5 + ··· diverges. Its sequence of partial sums 1, 1 + 2, 1 + 2 + 3 , 1 + 2 + 3 + 4 , 1 + 2 + 3 + 4 + 5, ... diverges.
Dodecagon
A polygon with 12 sides.
Dodecagon Regular Dodecagon
Domain
The set of values of the independent variable(s) for which a function or relation is defined. Typically, this is the set of x-values that give rise to real y-values.
Note: Usually domain means domain of definition, but sometimes domain refers to a restricted domain.
Mathwords for ICP Program
DodecahedronRegular Dodecahedron
A polyhedron with 12 faces. A regular dodecahedron has faces that are all regular pentagons.
Note: It is one of the five platonic solids.
Regular Dodecahedron
a = length of an edge
Volume =
Surface Area =
Rotate me if your browser is
Java-enabled.
Domain of DefinitionNatural Domain
Alternate terms for domain used to make it clear that the domain being referred to is not a restricted domain.
Dot Product
In two dimensions, (ai + bj)•(ci + dj) = ac + bd. In three dimensions, (ai + bj + ck)•(di + ej + fk) = ad + be + cf. In either case, u • v = |u| |v| cos θ, where θ is the angle between the vectors.
Double Cone
A geometric figure made up of two right circular cones placed apex to apex as shown below. Typically a double cone is considered to extend infinitely far in both directions, especially when working with conic sections and degenerate conic sections.
Note: The graph of the equation z2 = x2 + y2 is a standard way to represent a double cone. That is the equation for the image below.
Mathwords for ICP Program
Double Cone
Double Angle IdentitiesDouble Number Identities
Trig identities that show how to find the sine, cosine, or tangent of twice a given angle.
Double Angle Identities
Doubling Time
For a substance growing exponentially, the time it takes for the amount of the substance to double.
Mathwords for ICP Program
Double Root
A root of a polynomial equation with multiplicity 2. Also refers to a zero of a polynomial function with multiplicity 2.
Mathwords for ICP Program