Calculus+Glossary

Calculus Glossary
In words, the chain rule says the derivative of a composite function is the derivative of the outside function, done to the inside function, times the derivative of the inside function. gives the signed area between //f (x //) and the //x //-axis from //x // =; //a // to //x // =; //b //, with area above the //x //-axis counting positive and area below the //x //-axis counting negative. or in words the limit of the slopes of the secant lines through the point (//x, f(x //)) and a second point on the graph of //f(x //) as that second point approaches the first. The derivative can be interpreted as the slope of a line tangent to the function, the instantaneous velocity of the function, or the instantaneous rate of change of the function.
 * antiderivative: ** A function //<span style="font-family: 'Verdana','sans-serif';">F(x //) is called an antiderivative of a function //<span style="font-family: 'Verdana','sans-serif';">f(x //) if //<span style="font-family: 'Verdana','sans-serif';">F'(x //) =; //<span style="font-family: 'Verdana','sans-serif';">f(x //) for all //<span style="font-family: 'Verdana','sans-serif';">x // in the domain of //<span style="font-family: 'Verdana','sans-serif';">f //. In words, this means that an antiderivative of //<span style="font-family: 'Verdana','sans-serif';">f // is a function which has //<span style="font-family: 'Verdana','sans-serif';">f // for its derivative.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">chain rule: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The chain rule tells how to find the derivative of composite functions. In symbols, the chain rule says
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">change of variables: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A term sometimes used for the technique of integration by substitution.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">concave downward: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A function is concave downward on an interval if //<span style="font-family: 'Verdana','sans-serif';">f"(x //) is negative for every point on that interval.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">concave upward: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A function is concave upward on an interval if //<span style="font-family: 'Verdana','sans-serif';">f"(x //) is positive for every point on that interval.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">continuous: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A function //<span style="font-family: 'Verdana','sans-serif';">f(x //) is continuous at a point //<span style="font-family: 'Verdana','sans-serif';">x // =; //<span style="font-family: 'Verdana','sans-serif';">c // when //<span style="font-family: 'Verdana','sans-serif';">f(c //) exists, exists, and . In words, this means the curve could be drawn without lifting the pencil. To say that a function is continuous on some interval means that it is continuous at each point in that interval.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">critical point: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A critical point of a function is a point (//<span style="font-family: 'Verdana','sans-serif';">x, f(x //)) with //<span style="font-family: 'Verdana','sans-serif';">x // in the domain of the function and either //<span style="font-family: 'Verdana','sans-serif';">f'(x //) =; 0 or //<span style="font-family: 'Verdana','sans-serif';">f'(x //) undefined. Critical points are among the candidates to be maximum or minimum values of a function.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">cylindrical shell method: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A procedure for finding the volume of a solid of revolution by treating it as a collection of nested thin rings.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">definite integral: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The definite integral of //<span style="font-family: 'Verdana','sans-serif';">f(x //) between //<span style="font-family: 'Verdana','sans-serif';">x // =; //<span style="font-family: 'Verdana','sans-serif';">a // and //<span style="font-family: 'Verdana','sans-serif';">x // =; //<span style="font-family: 'Verdana','sans-serif';">b //, denoted
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">derivative: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The derivative of a function //<span style="font-family: 'Verdana','sans-serif';">f (x //) is a function that gives the slope of //<span style="font-family: 'Verdana','sans-serif';">f (x //) at each value of //<span style="font-family: 'Verdana','sans-serif';">x //. The derivative is most often denoted . The mathematical definition of the derivative is
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">differentiable: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A function is said to be differentiable at a point when the function's derivative exists at that point. A function will fail to be differentiable at places where the function is not continuous or where the function has corners.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">disk method: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with circular cross sections.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">Extreme Value Theorem: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A theorem stating that a function which is continuous on a closed interval [//<span style="font-family: 'Verdana','sans-serif';">a, b //] must have a maximum and a minimum value on [//<span style="font-family: 'Verdana','sans-serif';">a, b //].
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">First Derivative Test for Local Extrema: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A method used to determine whether a critical point of a function is a local maximum or local minimum. If a continuous function changes from increasing (first derivative positive) to decreasing (first derivative negative) at a point, then that point is a local maximum. If a function changes from decreasing (first derivative negative) to increasing (first derivative positive) at a point, then that point is a local minimum.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">general antiderivative: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> If //<span style="font-family: 'Verdana','sans-serif';">F(x //) is an antiderivative of a function //<span style="font-family: 'Verdana','sans-serif';">f(x //), then //<span style="font-family: 'Verdana','sans-serif';">F(x //) + //<span style="font-family: 'Verdana','sans-serif';">C // is called the general antiderivative of //<span style="font-family: 'Verdana','sans-serif';">f(x //).
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">general form: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The general form (sometimes also called standard form) for the equation of a line is //<span style="font-family: 'Verdana','sans-serif';">ax // + //<span style="font-family: 'Verdana','sans-serif';">by // =; //<span style="font-family: 'Verdana','sans-serif';">c //, where //<span style="font-family: 'Verdana','sans-serif';">a // and //<span style="font-family: 'Verdana','sans-serif';">b // are not both zero.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">higher order derivatives: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The second derivative, third derivative, and so forth for some function.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">implicit differentiation: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A procedure for finding the derivative of a function which has not been given explicitly in the form "//<span style="font-family: 'Verdana','sans-serif';">f(x //) =;".
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">indefinite integral: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The indefinite integral of //<span style="font-family: 'Verdana','sans-serif';">f(x //) is another term for the general antiderivative of //<span style="font-family: 'Verdana','sans-serif';">f(x //). The indefinite integral of //<span style="font-family: 'Verdana','sans-serif';">f (x //) is represented in symbols as
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">instantaneous rate of change: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> One way of interpreting the derivative of a function is to understand it as the instantaneous rate of change of that function, the limit of the average rates of change between a fixed point and other points on the curve that get closer and closer to the fixed point.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">instantaneous velocity: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> One way of interpreting the derivative of a function //<span style="font-family: 'Verdana','sans-serif';">s(t //) is to understand it as the velocity at a given moment //<span style="font-family: 'Verdana','sans-serif';">t // of an object whose position is given by the function //<span style="font-family: 'Verdana','sans-serif';">s(t //).
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">integration by parts: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> One of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">intercept form: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The intercept form for the equation of a line is //<span style="font-family: 'Verdana','sans-serif';">x/a // + //<span style="font-family: 'Verdana','sans-serif';">y/b // =; 1, where the line has its //<span style="font-family: 'Verdana','sans-serif';">x //-intercept (the place where the line crosses the //<span style="font-family: 'Verdana','sans-serif';">x //-axis) at the point (//<span style="font-family: 'Verdana','sans-serif';">a //,0) and its //<span style="font-family: 'Verdana','sans-serif';">y //-intercept (the place where the line crosses the //<span style="font-family: 'Verdana','sans-serif';">y //-axis) at the point (0,//<span style="font-family: 'Verdana','sans-serif';">b //).
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">limit: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A function //<span style="font-family: 'Verdana','sans-serif';">f(x //) has the value //<span style="font-family: 'Verdana','sans-serif';">L // for its limit as //<span style="font-family: 'Verdana','sans-serif';">x // approaches //<span style="font-family: 'Verdana','sans-serif';">c // if as the value of //<span style="font-family: 'Verdana','sans-serif';">x // gets closer and closer to //<span style="font-family: 'Verdana','sans-serif';">c //, the value of //<span style="font-family: 'Verdana','sans-serif';">f(x //) gets closer and closer to //<span style="font-family: 'Verdana','sans-serif';">L //.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">Mean Value Theorem: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> If a function //<span style="font-family: 'Verdana','sans-serif';">f(x //) is continuous on a closed interval [//<span style="font-family: 'Verdana','sans-serif';">a //,//<span style="font-family: 'Verdana','sans-serif';">b //] and differentiable on the open interval (//<span style="font-family: 'Verdana','sans-serif';">a //,//<span style="font-family: 'Verdana','sans-serif';">b //), then there exists some //<span style="font-family: 'Verdana','sans-serif';">c // in the interval [//<span style="font-family: 'Verdana','sans-serif';">a //,//<span style="font-family: 'Verdana','sans-serif';">b //] for which
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">normal line: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The normal line to a curve at a point is the line perpendicular to the tangent line at that point.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">point of inflection: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A point is called a point of inflection of a function if the function changes from concave upward to concave downward, or vice versa, at that point.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">point-slope form: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The point-slope form for the equation of a line is //<span style="font-family: 'Verdana','sans-serif';">y // – //<span style="font-family: 'Verdana','sans-serif';">y //1 =; //<span style="font-family: 'Verdana','sans-serif';">m(x // – //<span style="font-family: 'Verdana','sans-serif';">x //1), where //<span style="font-family: 'Verdana','sans-serif';">m // stands for the slope of the line and (//<span style="font-family: 'Verdana','sans-serif';">x //1,//<span style="font-family: 'Verdana','sans-serif';">y //1) is a point on the line.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">Riemann sum: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A Riemann sum is a sum of several terms, each of the form //<span style="font-family: 'Verdana','sans-serif';">f //(//<span style="font-family: 'Verdana','sans-serif';">xi //)Δ//<span style="font-family: 'Verdana','sans-serif';">x //, each representing the area below a function //<span style="font-family: 'Verdana','sans-serif';">f //(//<span style="font-family: 'Verdana','sans-serif';">x //) on some interval if //<span style="font-family: 'Verdana','sans-serif';">f //(//<span style="font-family: 'Verdana','sans-serif';">x //) is positive or the negative of that area if //<span style="font-family: 'Verdana','sans-serif';">f //(//<span style="font-family: 'Verdana','sans-serif';">x //) is negative. The definite integral is mathematically defined to be the limit of such a Riemann sum as the number of terms approaches infinity.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">Second Derivative Test for Local Extrema: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A method used to determine whether a critical point of a function is a local maximum or local minimum. If //<span style="font-family: 'Verdana','sans-serif';">f'(x //) =; 0 and the second derivative is positive at this point, then the point is a local minimum. If //<span style="font-family: 'Verdana','sans-serif';">f'(x //) =; 0 and the second derivative is negative at this point, then the point is a local maximum.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">slope of the tangent line: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> One way of interpreting the derivative of a function is to understand it as the slope of a line tangent to the function.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">slope-intercept form: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The slope-intercept form for the equation of a line is //<span style="font-family: 'Verdana','sans-serif';">y // =; //<span style="font-family: 'Verdana','sans-serif';">mx // + //<span style="font-family: 'Verdana','sans-serif';">b //, where //<span style="font-family: 'Verdana','sans-serif';">m // stands for the slope of the line and the line has its //<span style="font-family: 'Verdana','sans-serif';">y //-intercept (the place where the line crosses the //<span style="font-family: 'Verdana','sans-serif';">y //-axis) at the point (0,//<span style="font-family: 'Verdana','sans-serif';">b //).
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">standard form: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The standard form (sometimes also called general form) for the equation of a line is //<span style="font-family: 'Verdana','sans-serif';">ax // + //<span style="font-family: 'Verdana','sans-serif';">by // =; //<span style="font-family: 'Verdana','sans-serif';">c //, where //<span style="font-family: 'Verdana','sans-serif';">a // and //<span style="font-family: 'Verdana','sans-serif';">b // are not both zero.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">substitution: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> Integration by substitution is one of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">tangent line: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> The tangent line to a function is a straight line that just touches the function at a particular point and has the same slope as the function at that point.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">trigonometric substitution: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A technique of integration where a substitution involving a trigonometric function is used to integrate a function involving a radical.
 * <span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;">washer method: **<span style="color: black; font-family: 'Verdana','sans-serif'; font-size: 12px;"> A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with cross sections shaped like washers.