CCalculus

The branch of mathematics dealing with limits, derivatives, definite integrals, indefinite integrals, and power series.

Common problems from calculus include finding the slope of a curve, finding extrema, finding the instantaneous rate of change of a function, finding the area under a curve, and finding volumes by parallel cross-sections.

 

Cardinal Numbers

The numbers 1, 2, 3, . . . as well as some types of infinity. Cardinal numbers are used to describe the number of elements in either finite or infinite sets.

 

Cardinality of a Set

The number of elements in a set., whether the set is finite or infinite. Note: Not all infinite sets have the same cardinality.

Cardioid

A curve that is somewhat heart shaped. A cardioid can be drawn by tracing the path of a point on a circle as the circle rolls around a fixed circle of the same radius. The equation is usually written in polar coordinates.

Note: A cardioid is a special case of the limaçon family of curves.

 

Cardioid:   r = a ± a cos θ (horizontal) or  r = a ± a sin θ (vertical)

Mathwords for ICP Program

r = 2 + 2 cos θ r = 2 + 2 sin θ

 

Cartesian CoordinatesRectangular Coordinates

(x, y) or (x, y, z) coordinates.

 

Cartesian FormRectangular Form

A function (or relation) written using (x, y) or (x, y, z) coordinates.

 

Coordinate PlaneCartesian Plane

The plane formed by a horizontal axis and a vertical axis, often labeled the x-axis and y-axis, respectively.

Catenary

The curve naturally formed by a slack rope or wire hanging between two fixed points. A catenary is NOT a parabola, even though it looks like one. Note: The graph of the hyperbolic cosine function is a catenary.

 

 

Mathwords for ICP Program

 

Cavalieri’s Principle

A method, with formula given below, of finding the volume of any solid for which cross-sections by parallel planes have equal areas. This includes, but is not limited to, cylinders and prisms.

 

Formula: Volume = Bh, where B is the area of a cross-section and h is the height of the solid.

Ceiling FunctionLeast Integer Function

A step function of x which is the least integer greater than or equal to x. The ceiling function of x

is usually written . Sometimes this function is written with reversed floor function brackets

, and other times it is written with reversed boldface brackets ]x[ or reversed plain brackets ]x[.

Examples: and .

 

Mathwords for ICP Program

 

Center of Mass Formula

The coordinates of the center of mass of a plane figure are given by the formulas below. The formulas only apply for figures of uniform (constant) density.

 

 

Center of Rotation

In a rotation, the point that does not move. The rest of the plane rotates around this one fixed point.

 

Mathwords for ICP Program

Centers of a Triangle

The main centers of a triangle are listed in the table below along with selected properties.

 

Central Angle

An angle in a circle with vertex at the circle's center.

 

Centroid

For a triangle, this is the point at which the three medians intersect. In general, the centroid is the center of mass of a figure of uniform (constant) density.

 

Mathwords for ICP Program

Centroid of a Triangle(Click on the image or here to launch an interactive java applet)

Centroid Formula

The coordinates of the centroid of a triangle are found by averaging the x- and y-coordinates of the vertices. This method will also find the centroid (center of mass) of any set of points on the x-y plane.

 

Ceva's Theorem

A theorem relating the way three concurrent cevians of a triangle divide the triangle's three sides.

Cevian

Mathwords for ICP Program

A line segment, ray, or line that extends from a vertex of a triangle to the opposite side (which may be extended). Medians, altitudes, and angle bisectors are all examples of cevians.

 

Chain Rule

A method for finding the derivative of a composition of functions. The formula is

. Another form of the chain rule is .

 

 

Change of Base Formula

A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e.

 

Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.

Change of base formula:

 

Mathwords for ICP Program

Example 1:

 

Example 2:    (note that

)

 

Example 3:

Verify a SolutionCheck a Solution

The process of making sure a solution is correct by making sure it satisfies any and all equations and/or inequalities in a problem.

 

1Example: Verify that x = 3 is a solution of the equation x2 – 5x + 6 = 0.

To do this, substitute x = 3 into the equation.

32 – 5·3 + 6 = 0?  9 – 15 + 6 = 0?                0 = 0 confirmed

Chord

A line segment on the interior of a circle. A chord has both endpoints on the circle.

 

Mathwords for ICP Program

Circle

The locus of all points that are a fixed distance from a given point.

 

Pythagorean IdentitiesCircle Identities

Trig identities relating sine with cosine, tangent with secant, and cotangent with cosecant. Derived from the Pythagorean theorem.

 

Pythagorean Identities

sin2 θ + cos2 θ = 1

tan2 θ + 1 = sec2 θ

cot2 θ + 1 = csc2 θ

 

Circle Trig Definitions

A set of definitions of the six trig functions: sine, cosine, tangent, cosecant, secant, and cotangent.

Mathwords for ICP Program

 

 

Circular Cone

A cone which has a circle as base.

 

Circular Cylinder

A cylinder with a circles as bases.

 

Trig FunctionsCircular Functions

The six functions sine, cosine, tangent, cosecant, secant, and cotangent. These functions can be defined several different ways. These include SOHCAHTOA definitions, circle definitions (below), and unit circle definitions.

 

Mathwords for ICP Program

Circumcenter

The center of a circumcircle. For any circumscribable polygon, the circumcenter is found at the point of intersection of the perpendicular bisectors of the sides.

 

CircumcircleCircumscribed Circle

A circle that passes through all vertices of a plane figure and contains the entire figure in its interior.

Note: All triangles have circumcircles and so do all regular polygons. Most other polygons do not.

 

 

Circumference

Mathwords for ICP Program

A complete circular arc. Circumference also means the distance around the the outside of a circle.

 

Circumscribable

Describes a plane figure that has a circumcircle.

 

Circumscribable

Describes a plane figure that has a circumcircle.

 

Cis

A complex-valued function made from sine and cosine with definition cis θ = cos θ + isin θ.

Note: cis θ is the same as eiθ.

Clockwise

The direction that the hands of a clock move.

Closed Interval

An interval that contains its endpoints.

 

Mathwords for ICP Program

Coefficient

The number multiplied times a product of variables or powers of variables in a term. For example, 123 is the coefficient in the term 123x3y.

Coefficient Matrix

The matrix formed by the coefficients in a linear system of equations.

 

Cofactor

The determinant obtained by deleting the row and column of a given element of a matrix or determinant. The cofactor is preceded by a + or – sign depending whether the element is in a + or – position.

 

Cofactor MatrixMatrix of Cofactors

A matrix with elements that are the cofactors, term-by-term, of a given square matrix.

 

Mathwords for ICP Program

Cofunction Identities

Trig identities showing the relationship between sine and cosine, tangent and cotangent, and secant and cosecant. The value of a trig function of an angle equals the value of the cofunction of the complement of the angle.

 

Cofunction Identities, radians   Cofunction Identities, degrees

  sin (90° – x) = cos x

cos (90° – x) = sin x

 tan (90° – x) = cot x cot (90° – x) = tan x

 sec (90° – x) = csc x csc (90° – x) = sec x

Coincident

Identical, one superimposed on the other. That is, two or more geometric figures that share all points. For example, two coincident lines would look like one line since one is on top of the other.

Collinear

Mathwords for ICP Program

Lying on the same line.

 

Column of a Matrix

A vertical set of numbers in a matrix.

Combination

A selection of objects from a collection. Order is irrelevant.

 

Example: A poker hand is a combination of 5 cards from a 52 card deck. This is a combination since the order of the 5 cards does not matter.

Combination Formula

A formula for the number of possible combinations of r objects from a set of n objects. This is written in any of the ways shown below.

All forms are read aloud "n choose r."

 

Mathwords for ICP Program

Formula:

  Note: , where nPr is the formula for permutations of n

objects taken r at a time.

Example: How many different committees of 4 students can be chosen from a group of 15?

Answer:There are possible combinations of 4 students from a set of 15.

 

There are 1365 different committees.

 

Combinatorics

The mathematics of counting, especially counting how many elements are in very large sets.

Common Logarithm

The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 102 = 100.

 

Common Ratio

For a geometric sequence or geometric series, the common ratio is the ratio of a term to the previous term. This ratio is usually indicated by the variable r.

 

Mathwords for ICP Program

Example: The geometric series 3, 6, 12, 24, 48, . . . has common ratio r = 2.

Commutative Operation

Any operation ⊕ for which a⊕b = b⊕a for all values of a and b. Addition and multiplication are both commutative. Subtraction, division, and composition of functions are not. For example, 5 + 6 = 6 + 5 but 5 – 6 ≠ 6 – 5.

More: Commutativity isn't just a property of an operation alone. It's actually a property of an operation over a particular set. For example, when we say addition is commutative over the set of real numbers, we mean that a + b = b + a for all real numbers a and b. Subtraction is not commutative over real numbers since we can't say that a – b = b – a for all real numbers a and b. Even though a – b = b – a whenever a and b are the same, that still doesn't make subtraction commutative over the set of all real numbers.

Further examples: In this more formal sense, it is correct to say that matrix multiplication is not commutative for square matrices. Even though AB = BA for some square matrices A and B, commutativity does not hold for all square matrices. It is also correct to say composition is not commutative for functions, even though one-to-one functions commute with their inverses.

Comparison Test

A convergence test which compares the series under consideration to a known series. Essentially, the test determines whether a series is "better" than a "good" series or "worse" than a "bad" series. The "good" or "bad" series is often a p-series.

 

If ∑ an , ∑ cn , and ∑ dn are all positive series, where ∑ cn converges and ∑ dn diverges, then:

1. If an ≤ cn for all n ≥ N for some fixed N, then ∑ an converges.

2. If an ≥ dn for all n ≥ N for some fixed N, then ∑ an diverges.

Compatible Matrices

Mathwords for ICP Program

Two matrices with dimensions arranged so that they may be multiplied. The number of columns of the first matrix must equal the number of rows of the second.

 

 

Complement of an Angle

The complement of an acute angle A is the angle 90° – A. For example, the complement of 20° is 70°.

Complement of an Event

The opposite of an event. That is, the set of all outcomes of an experiment that are not included in an event. The complement of event A is written AC and is often read aloud as "not A".

 

Complement of a Set

The elements not contained in a given set. The complement of set A is indicated by AC.

 

Mathwords for ICP Program

 

Complementary Angles

Two acute angles that add up to 90°. For example, 40° and 50° are complementary. In the diagram below, angles 1 and 2 are complementary.

 

Complex Conjugate

The complex conjugate of a + bi  is a – bi, and similarly the complex conjugate of a – bi  is a + bi. This consists of changing the sign of the imaginary part of a complex number. The real part is left unchanged.

Complex conjugates are indicated using a horizontal line over the number or variable. For example, .

Note: Complex conjugates are similar to, but not the same as, conjugates.

Expression

Complex

Conjugate

5 – 2i

4i + 1–5i12

5 + 2i

–4i + 15i12

 

Mathwords for ICP Program

Compound FractionComplex Fraction

A fraction which has, as part of its numerator and/or denominator, at least one other fraction.

Examples:1. 

   

 2.  

 

Complex Number Formulas

Algebra rules and formulas for complex numbers are listed below.

 

Mathwords for ICP Program

 

Complex Numbers

Numbers like 3 – 2i or that can be written as the sum or difference of a real number and an imaginary number. Complex numbers are indicated by the symbol .

Note: All real numbers and all pure imaginary numbers are complex. Sometimes, however, mathematicians use the phrase complex numbers to refer strictly to numbers which have both nonzero real parts and nonzero imaginary parts.

 

 

 Composite

Built from more than one thing.

Composite Number

A positive integer that has factors other than just 1 and the number itself. For example, 4, 6, 8, 9, 10, 12, etc. are all composite numbers. The number 1 is not composite.

Composition

Mathwords for ICP Program

Combining two functions by substituting one function's formula in place of each x in the other function's formula. The composition of functions f and g is written f ° g, and is read aloud "f composed with g." The formula for f ° g is written (f ° g)(x). This is read aloud "f composed with g of x."

Note: Composition is not commutative. That is, (f ° g)(x) is usually different from (g ° f)(x). The example below illustrates this.

 

Example:    f(x) = 3x2 + 12x – 1  and  g(x) = 4x + 1

   1(f ° g)(x) = 3(4x + 1)2 + 12(4x + 1) – 1

   = 3(16x2 + 8x + 1) + 48x + 12 – 1= 48x2 + 72x + 14

   1(g ° f)(x) = 4(3x2 + 12x – 1) + 1

   = 12x2 + 48x – 4 + 1

= 12x2 + 48x – 3

 

Compound FractionComplex Fraction

A fraction which has, as part of its numerator and/or denominator, at least one other fraction.

Examples:1. 

   

 2.  

 

Compound Inequality

Two or more inequalities taken together. Often this refers to a connected chain of inequalities, such as 3 < x < 5.

Mathwords for ICP Program

Formally, a compound inequality is a conjunction of two or more inequalities.

 

Compound Interest

A method of computing interest in which interest is computed from the up-to-date balance. That is, interest is earned on the interest and not just on original balance.

 

Continuously Compounded Interest

Interest that is, hypothetically, computed and added to the balance of an account every instant. This is not actually possible, but continuous compounding is well-defined nevertheless as the upper bound of "regular" compound interest. The formula, given below, is sometimes called the shampoo formula (Pert®).

Note: This same formula can be used for exponential growth and exponential decay.

 

CompressionContraction

Mathwords for ICP Program

A transformation in which a figure grows smaller. Compressions may be with respect to a point (compression of a geometric figure) or with respect to the axis of a graph (compression 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. Compression (or contraction) refers to transformations in which the figure becomes smaller. Dilation properly refers only to transformations in which the figure grows larger. Unfortunately the English language has no word that refers collectively to both stretching and shrinking.

 

Compression of a Geometric FigureContraction of a Geometric Figure

A transformation in which all distances are shortened by a common factor. This is done by contracting all points toward some fixed point P.

Note: The common factor is less than 1 for a contraction. When the common factor is greater than 1 the transformation is called a dilation.

 

 

 

ShrinkCompression of a Graph

A transformation in which all distances on the coordinate plane are shortened by multiplying either all x-coordinates (horizontal compression) or all y-coordinates (vertical compression) of a graph by a common factor less than 1.

Note: When the common factor is greater than 1 the transformation is called a dilation or a stretch.

 

Mathwords for ICP Program

 

 

Compute

To figure out or evaluate. For example, "compute 2 + 3" means to figure out that the answer is 5.

 

ConcaveNon-Convex

A shape or solid which has an indentation or "cave". Formally, a geometric figure is concave if there is at least one line segment connecting interior points which passes outside of the figure.

 

 

 

Concave Down

A graph or part of a graph which looks like an upside-down bowl or part of an upside-down bowl.

 

Mathwords for ICP Program

 

Concave Up

A graph or part of a graph which looks like a right-side up bowl or part of an right-side up bowl.

 

 Concentric

Similar geometric figures that share a common center.

 

 

Conclusion

The part of a conditional statement after then.

For example, the conclusion of "If a line is horizontal then the line has slope 0" is "the line has slope 0".

Mathwords for ICP Program

Concurrent

Lines or curves that all intersect at a single point.

 

 

Conditional

An "If . . . then . . ." statement. For example, "If it is raining, then the grass is wet."

Conditional Convergence

Describes a series that converges but does not converge absolutely. That is, a convergent series that will become a divergent series if all negative terms are made positive.

 

Conditional Equation

An equation that is true for some value(s) of the variable(s) and not true for others.

 

Example: The equation 2x – 5 = 9 is conditional because it is only true for x = 7. Other

values of x do not satisfy the equation.

Conditional Probability

Mathwords for ICP Program

A probability that is computed based on the assumption that some event has already occurred. The probability of event B given that event A has occurred is written P(B|A).

 

 

Cone

A three dimensional figure with a single base tapering to an apex. The base can be any simple closed curve. Often the word cone refers to a right circular cone.

 

                                                                                               Double Cone

 

Cone Angle

Mathwords for ICP Program

The angle remaining in a sheet of paper after a sector has been cut out so that the paper can be rolled into a right circular cone.

 

 

Congruence Tests for Triangles

SSS, SAS, ASA, AAS, and HL. These tests describe combinations of congruent sides and/or angles that are used to determine if two triangles are congruent.

 

Conic Sections

The family of curves including circles, ellipses, parabolas, and hyperbolas. All of these geometric figures may be obtained by the intersection a double cone with a plane, hence the name conic section. All conic sections have equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

 

 

 Conjecture

An educated guess.

Conjugates

Mathwords for ICP Program

The result of writing sum of two terms as a difference or vice-versa. Note: Conjugates are similar to, but not the same as, complex conjugates.

 

Expression Conjugate

x + 2 x – 2

 

Conjugate Pair Theorem

An assertion about the complex zeros of any polynomial which has real numbers as coefficients.

Theorem: If a polynomial

p(x) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0

has real coefficients, then any complex zeros occur in conjugate pairs. That is, if a + bi is a zero then so is a – bi and vice-versa.

Example: 2 – 3i is a zero of p(x) = x3 – 3x2 + 9x + 13 as shown here:

p(2 – 3i) = (2 – 3i)3 – 3(2 – 3i)2 + 9(2 – 3i) + 13              = (–46 – 9i) – 3(–5 – 12i) + (18 – 27i) + 13              = –46 – 9i + 15 + 36i + 18 – 27i + 13              = 0.

By the conjugate pair theorem, 2 + 3i is also a zero of p(x).

p(2 + 3i) = (2 + 3i)3 – 3(2 + 3i)2 + 9(2 + 3i) + 13              = (–46 + 9i) – 3(–5 + 12i) + (18 + 27i) + 13              = –46 + 9i + 15 – 36i + 18 + 27i + 13              = 0.

Mathwords for ICP Program

Conjunction

A statement which connects two words or statements using the word and. For example, "peas and carrots" is a conjunction.

 

Consecutive Interior Angles

In the figure below, angles 4 and 6 are consecutive interior angles. So are angles 3 and 5. Consecutive interior angles are supplementary. Formally, consecutive interior angles may be defined as two interior angles lying on the same side of the transversal cutting across two parallel lines.

 

Parallel lines cut

by a transversal

Consistent System of Equations

A system of equations that has at least one solution.

 

Constant

As a noun, a term or expression with no variables. Also, a term or expression for which any variables cancel out. For example, –42 is a constant. So is 3x + 5 – 3x, which simplifies to just 5.

As an adjective, constant means the same as fixed. That is, not changing or moving.

 

Constant Term

The term in a simplified algebraic expression or equation which contains no variable(s). If there is no such term, the constant term is 0.

Mathwords for ICP Program

 

Example:   –5 is the constant term in p(x) = 2x3 – 4x2 + 9x – 5

Sigma NotationContinued Sum

A notation using the Greek letter sigma (Σ) that allows a long sum to be written compactly.

 

Continuous

Describes a connected set of numbers, such as an interval. For example, the set of real numbers is continuous. The set of integers is not continuous; it is discrete.

 

Continuous Function

A function with a connected graph.

Continuously Differentiable Function

A function which has a derivative that is itself a continuous function.

Mathwords for ICP Program

Contrapositive

Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining."

Note: As in the example, the contrapositive of any true proposition is also true.

 

Converge

To approach a finite limit. There are convergent limits, convergent series, convergent sequences, and convergent improper integrals.

 

Convergence Tests

Limit test for divergenceIntegral testComparison testLimit comparison testAlternating series testRatio testRoot test

Convergent Sequence

A sequence with a limit that is a real number. For example, the sequence 2.1, 2.01, 2.001, 2.0001, . . . has limit 2, so the sequence converges to 2. On the other hand, the sequence 1, 2, 3, 4, 5, 6, . . . has a limit of infinity (∞). This is not a real number, so the sequence does not converge. It is a divergent sequence.

 

Convergent Series

An infinite series for which the sequence of partial sums converges. For example, the sequence of partial sums of the series 0.9 + 0.09 + 0.009 + 0.0009 + ··· is 0.9, 0.99, 0.999, 0.9999, .... This sequence converges to 1, so the series 0.9 + 0.09 + 0.009 + 0.0009 + ··· is convergent.

Mathwords for ICP Program

Converse

Switching the hypothesis and conclusion of a conditional statement. For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining."

Note: As in the example, a proposition may be true but have a false converse.

 

Convex

A geometric figure with no indentations. Formally, a geometric figure is convex if every line segment connecting interior points is entirely contained within the figure's interior.

 

 

CoordinatesOrdered PairOrdered Triplen-tuple

On the coordinate plane, the pair of numbers giving the location of a point (ordered pair). In three-dimensional coordinates, the triple of numbers giving the location of a point (ordered triple). In n -dimensional space , a sequence of n numbers written in parentheses.

 

Ordered pair: Two numbers written in the form (x, y).

Ordered triple: Three numbers written in the form (x, y, z).

n-tuple: n numbers written in the form (x1, x2, x3, . . . , xn).

 

 

Coplanar

Mathwords for ICP Program

Lying in the same plane. For example, any set of three points in space are coplanar.

 

Corollary

A special case of a more general theorem which is worth noting separately. For example, the Pythagorean theorem is a corollary of the law of cosines.

 

Correlation

The degree to which two variables are associated. For example, height and weight have a moderately strong positive correlation.

Correlation Coefficient

A number that is a measure of the strength and direction of the correlation between two variables. Correlation coefficients are expressed using the variable r, where r is between 1 and –1, inclusive. The closer r is to 1 or –1, the less scattered the points are and the stronger the relationship. Only data points with a scatterplot which is a perfectly straight line can have r = –1 or r = 1. When r < 0 the data have a negative association, and when r > 0 the data have a positive association.

Corresponding

Two features that are situated the same way in different objects.

 

cosinecos

The trig function cosine, which is written cos θ. For acute angles, cos θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = cos x is a periodic function with period 2π.

Mathwords for ICP Program

 

cosecantcsccosec

The trig function cosecant, written csc θ. csc θ equals . For acute angles, csc θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = csc x is a periodic function with period 2π.

 

cotangentcotctg

The trig function cotangent, written cot θ. cot θ equals or . For acute angles, cot θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = cot x is a periodic function with period π.

 

Mathwords for ICP Program

 

Coterminal Angles

Angles which, drawn in standard position, share a terminal side. For example, 60°, -300°, and 780° are all coterminal.

 

 

Countable

Describes the cardinality of a countably infinite set. Aleph null (0א) is the symbol for this.

Countably Infinite

Describes a set which contains the same number of elements as the set of natural numbers. Formally, a countably infinite set can have its elements put into one-to-one correspondence with the set of natural numbers.

Note: The symbol aleph null (0א) stands for the cardinality of a countably infinite set.

Mathwords for ICP Program

Counterexample

An example which disproves a proposition. For example, the prime number 2 is a counterexample to the statement "All prime numbers are odd."

Natural NumbersCounting Numbers

The numbers used for counting. That is, the numbers 1, 2, 3, 4, etc.

 

 

 

CPCFC

"Corresponding parts of congruent figures are congruent." A theorem stating that if two figures are congruent, then so are all corresponding parts.

 

CPCTC

"Corresponding parts of congruent triangles are congruent." A theorem stating that if two triangles are congruent, then so are all corresponding parts.

Mathwords for ICP Program

Cramer’s Rule

A method for solving a linear system of equations using determinants. Cramer’s rule may only be used when the system is square and the coefficient matrix is invertible.

 

 

 Critical NumberCritical Value

The x-value of a critical point.

 

Critical Point

A point (x, y) on the graph of a function at which the derivative is either 0 or undefined. A critical point will often be a minimum or maximum, but it may be neither.

Note: Finding critical points is an important step in the process of curve sketching.

 

Cross Product

A way of multiplying two vectors, written u × v, in which the product is another vector. The cross product of two vectors results in a vector which is orthogonal to both the vectors being multiplied. The magnitude of the cross product of two vectors is found by the formula |u × v| = |u| |v| sin θ, where θ is the smaller angle between the vectors.

Mathwords for ICP Program

Note: Cross products are not commutative. That is, u × v ≠ v × u. The vectors u × v and v × u have the same magnitude but point in opposite directions.

 

 

CubeRegular Hexahedron

A regular polyhedron for which all faces are squares.

Note: It is one of the five platonic solids.

Cube

a = length of an edge

Volume = a3

Surface Area = 6a2

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Java-enabled.

 

Cube Root

A number that must be multiplied times itself three times to equal a given number. The cube root of x is written or .

For example, since .

Cubic Polynomial

A polynomial of degree 3. For example, x3 - 1, 4a3 - 100a2 + a - 6, and m2n + mn2 are all cubic polynomials.

Mathwords for ICP Program

Curly d

The symbol ∂ used in the notation for partial derivatives.

Curve

A word used to indicate any path, whether actually curved or straight, closed or open. A curve can be on a plane or in three-dimensional space (or n-dimensional space, for that matter). Lines, circles, arcs, parabolas, polygons, and helixes are all types of curves.

Note: Typically curves are thought of as the set of all geometric figures that can be parametrized using a single parameter. This is not in fact accurate, but it is a useful way to conceptualize curves. The exceptions to this rule require some cleverness, or at least some exposure to space-filling curves.

Cusp

A sharp point on a curve. Note: Cusps are points at which functions and relations are not differentiable.

 

 

Curve Sketching

The process of using the first derivative and second derivative to graph a function or relation. As a result the coordinates of all discontinuities, extrema, and inflection points can be accurately plotted.

Cycloid

Mathwords for ICP Program

The path traced by a point on a wheel as the wheel rolls, without slipping, along a flat surface. The standard parametrization is x = a(t – sin t), y = a(1 – cos t), where a is the radius of the wheel.

Note: Cycloids are periodic functions.

 

 

 

Cylinder

A three-dimensional geometric figure with parallel congruent bases. The bases can be shaped like any closed plane figure (not necessarily a circle) and must be oriented identically.

Note: The word cylinder often refers to a right circular cylinder.

 

 

 

Cylindrical Shell MethodShell Method

A technique for finding the volume of a solid of revolution.

 

Mathwords for ICP Program

 

 

Mathwords for ICP Program