### Calculus Encyclopedia If you would like to prepare for school subjects or simply increase your general knowledge, then enjoy our calculus encyclopedia. We tried to focus only on very important terms and definitions. We also kept our terminology very brief so that you absorb the concept more quickly and easily.

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 ABSOLUTE MAXIMUM: The highest point on a graph, especially over a specified domain. It is the greatest value of f(x) over a defined interval of x, provided y=f(x). ABSOLUTE MINIMUM: The lowest point on a graph, especially over a specified domain. It is the least value of f(x) over a defined interval of x, provided y=f(x). ANTIDERIVATIVE: Given a function with a derivative, the antiderivative of that derivative function returns the original function. 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. AREA UNDER A CURVE: If we have limits of integration, it is most simply the definite integral of the function defined between those limits of integration. ASYMPTOTE: A line (or curve) that a function approaches without actually reaching the line as the domain either grows unbounded or approaches a limit. AXIOM: A statement that is so self-evident no proof is needed. BETA: Beta is the second letter of the Greek alphabet. BOUNDARY: Some functions are bounded, some are not. Some regions are bounded, some are not. To be bounded means to have a limit; its extent only goes so far, and then it stops or ends. BOUNDED FUNCTION: A bounded function approaches or reaches a limit. If a function goes toward infinity it is generally considered unbounded. CALCULUS: Calculus is the branch of mathematics concerned with the rates of changes between variables (derivatives) as well as areas under curves that represent functions (integrals). CHAIN RULE: The Chain Rule is a basic rule in calculus to find the derivative of a composite function. 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. CONTINUOUS: A function is considered Continuous if its graph has no gaps, no holes, no steps, and no cusps or discontinuities. CONTINUOUS FUNCTION: When the graph of a function has no holes, no gaps, no steps, or no discontinuities, then it is considered Continuous. It may have cusps. CONTINUOUSLY DIFFERENTIABLE: When a function is Continuously Differentiable it is both continuous and smooth. 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. CONVERGENCE: To approach a limit is to experience Convergence. Mathematical series experience convergence when the sum of their expanded terms reaches a boundary or limit. CONVERGENT SERIES: A series is said to be Convergent when its sum approaches a limit. 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. CRITICAL NUMBER: While there are many ways to define Critical Numbers, depending on the circumstances, most generally we're interested in places where a function generates either an extreme value or a discontinuity. CURVE: Beware that mathematicians consider straight lines to be Curves!. CUSP: When the graph of a function comes to a sharp point, we say that point on the graph is a Cusp. CYLINDRICAL SHELL: A method for volumetric calculations especially for rotated bodies around an axis. The small thickness of the shell is typically the differential "dx" (or dy or whatever differential represents the incremental thickness of the cylinder). DEFINITE INTEGRAL: An integral evaluated between limits of integration is termed a Definite Integral. DELTA: Delta is the fourth letter of the Greek alphabet. Upper-case Delta looks like a triangle and is used to mean "the change in...". DERIVATIVE: A first Derivative is the slope of the line tangent to a function. A Derivative provides an instantaneous rate of change between variables. DIFFERENTIABLE: If a function is smooth and continuous it is differentiable. DIFFERENTIAL EQUATION: A Differential Equation employs derivatives and algebra to solve for variables that represent functions. DISCONTINUITY: When a function is literally not continuous because of a gap, a step, a hole, or any kind of "break" it is considered discontinuous. 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. DOT PRODUCT: A product of vector multiplication, the Dot Product is a scalar, which means it has magnitude only and not an associated direction. The Dot Product does not result in another vector. 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. EXTREME VALUE THEOREM: On any continuous function graphed on a closed interval from a domain, we are guaranteed to have a maximum and a minimum value if the range of the function is not constant. FINITE: The common meaning of Finite and its meaning to mathematicians are not quite the same. In everyday language, Finite means countable within a reasonable time. To math people, Finite means not infinite; it means, simply, having a bound. FIRST DERIVATIVE: The First Derivative of a typical function, say, y = f(x), is the slope of the line tangent to a point on the graph of the original function f(x). FIRST ORDER DIFFERENTIAL EQUATION: This type of equation includes first derivatives and employs algebra to treat those derivative functions as variables. 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. 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. 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=2tinverse. INDEFINITE INTEGRAL: An integral with no limits of integration, an Indefinite Integral, can be thought of an an antiderivative. INDETERMINATE: Often a resultant fraction like 0/0 is an Indeterminate form that requires more analysis to determine its true nature, depending on the functions involved. INFINITE: In common language, not countable in any practical manner. In math, having no bounds or boundary. INFINITE SERIES: Any series of terms whose progression has an unlimited (limitless) number of terms is an Infinite Series. INFINITESIMAL: Infinitely small is Infinitesimal, so tiny that it occupies no space. While in human terms anything really small (a molecule) is Infinitesimal, in math the term means approaching zero in size. INFINITY: That without bound; limitless. INFLECTION: On the graph of a function, a point of Inflection is where the curve begins to "bend the other way.". INSTANTANEOUS RATE OF CHANGE: The value of the first derivative of a standard function of the form y = f(x). INSTANTANEOUS VELOCITY: The reading at any instant on a speedometer gives an Instantaneous Velocity. To be precise, the speedometer gives an instant snapshot of speed (only) with no direction; physical velocity has both magnitude and direction, as a vector. INTEGRAL: A specific function in calculus. Or, simply related to integers. Integral might also mean "important" in common language. INTEGRAND: The function that undergoes integration is the Integrand. INTEGRATION: A process, or function, in calculus to sum an infinite number of infinitesimal increments. INTERMEDIATE VALUE THEOREM: The IVT basically says that between two different values is an intermediate value somewhere between the extremes. INVERSE FUNCTION: For most functions in Cartesian coordinates, the inverse function is the mirror image around the x=y line. INVERSE, MATRIX: When two matrices multiply to produce the identity matrix, each is said to be the Inverse Matrix of the other. LAMBDA: Lambda is the eleventh letter of the Greek alphabet and is used for wavelength in physics. 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. LIMIT: Some functions have a Limit, a bound beyond which they may not realize. LOWER BOUND: As the name suggests, some functions are limited on the low side. MAGNITUDE, VECTORS: The Magnitude of a vector is the length of the vector. We may apply a Pythagorean relation to the perpendicular components of the vector to find the length. 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: Essentially, between any two extremes is an average value. MODULUS: Most typically it is the length of a vector. MOMENT: Moment takes on many meanings in statistics and physics. MOMENT OF INERTIA: Each shape or body has an associated Moment of Inertia related to mass distribution and the choice of the axis around which the body is rotated. MULTIVARIABLE: Having more than one variable. Also multivariate. MULTIVARIATE: Having more than one variable. Also multivariable. NEWTON'S METHOD: An iterative method for finding roots of polynomials. NORMALIZE: We might Normalize data by culling errors. Or we might Normalize a vector by assigning a unit vector in its direction. OCTANT: As we have four quadrants in the rectangular plane, we have eight Octants in rectangular space. In three dimensions the three axes divide space into eight sections, each termed an Octant. ODD FUNCTION: An Odd Function adheres to this property: f(-x) = -f(x). The standard sine function is an odd function. 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. ORDER, MATRIX: The Order of a Matrix is its size, expressed as "rows by columns.". ORDERED TRIPLE: Three coordinates are required to label a point in space, typically (x, y, z). ORDINARY DIFFERENTIAL EQUATION: A Differential Equation with no partial derivatives is considered an Ordinary Differential Equation. 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. PARTIAL DERIVATIVE: The derivative with respect to a single variable is a Partial Derivative. PARTIAL DIFFERENTIAL EQUATION: A Differential Equation with a Partial derivative. PARTIAL FRACTION: A Fraction built from the decomposition of other terms. PARTIAL SUM: A Partial Sum occurs when we sum only a finite number of terms from a larger or infinite series of terms. 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. POWER RULE: A simple device in calculus to determine the derivative of a monomial. PRODUCT RULE: An algorithm within the calculus to find the derivative of the Product of two functions. PROJECTILE MOTION: Projectile Motion is a parabolic arc caused by gravity. 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. PSI: The 23rd letter (next-to-last) of the Greek alphabet. PURE IMAGINARY NUMBER: Given a complex number of the form a + bi, when a = 0 we say that the number is a Pure (or purely) Imaginary Number. 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. RIEMANN SUM: Effectively the definite integral in calculus. ROLLE'S THEOREM: A principle from first-semester calculus that asserts a first derivative of zero exists on a smooth, continuous, differentiable function between constant range values. SCALAR PRODUCT: A Product of vector multiplication, such as a dot product, that results in a value that is Scalar with size (magnitude) but no associated direction. SECOND DERIVATIVE: A Derivative taken of a first Derivative is termed a Second Derivative. SECOND-ORDER DIFFERENTIAL EQUATION: An ordinary Differential Equation in which the highest derivative is a second derivative is called a Second-Order Differential Equation.