| Term | Definition |
|---|---|
| average rate of change | The change in the value of a function divided by the change in the input over an interval [a, b], calculated as (f(b) - f(a))/(b - a). |
| dynamic change | Change that occurs over time or as variables vary, which calculus uses limits to understand and model. |
| instantaneous rate of change | The rate at which a function is changing at a specific point, represented by the derivative at that point. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| rate of change at an instant | The instantaneous rate of change of a function at a specific point, interpreted through the limiting behavior of average rates of change over intervals containing that point. |
| Term | Definition |
|---|---|
| continuity | A property of a function at a point where the function is defined, the limit exists, and the limit equals the function value at that point. |
| jump discontinuity | A type of discontinuity where the left-hand and right-hand limits of a function exist but are not equal, causing the function to jump from one value to another. |
| removable discontinuity | A discontinuity that can be eliminated by defining or redefining the function value at that point to equal the limit. |
| vertical asymptote | A vertical line where a function approaches positive or negative infinity, resulting in a discontinuity that cannot be removed. |
| Term | Definition |
|---|---|
| continuity | A property of a function at a point where the function is defined, the limit exists, and the limit equals the function value at that point. |
| function | A mathematical relationship that assigns exactly one output value to each input value of an independent variable. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| Term | Definition |
|---|---|
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| domain | The set of all input values (x-values) for which a function is defined. |
| exponential function | A function of the form f(x) = a^x, where a is a positive constant not equal to 1. |
| interval | A connected set of real numbers, typically expressed as a range between two endpoints. |
| logarithmic function | A function of the form f(x) = log_a(x), the inverse of an exponential function. |
| polynomial function | A function composed of terms with non-negative integer exponents and real coefficients. |
| power function | A function of the form f(x) = x^n, where n is a real constant. |
| rational function | A function expressed as the ratio of two polynomial functions. |
| trigonometric function | Functions such as sine, cosine, and tangent that relate angles to ratios of sides in a right triangle. |
| Term | Definition |
|---|---|
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| discontinuity | A point where a function is not continuous, often due to a break, jump, or hole in the graph. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| piecewise-defined function | A function defined by different expressions over different intervals or regions of its domain. |
| removable discontinuity | A discontinuity that can be eliminated by defining or redefining the function value at that point to equal the limit. |
| Term | Definition |
|---|---|
| asymptotic behavior | The behavior of a function as it approaches a line (asymptote) but never reaches it, often described using limits. |
| infinite limit | Limits that describe the behavior of a function as it approaches infinity or negative infinity, or as the function values grow without bound. |
| unbounded behavior | The behavior of a function whose values grow without limit, either positively or negatively, as the input approaches a particular value or infinity. |
| Term | Definition |
|---|---|
| end behavior | The behavior of a function as the input values approach positive or negative infinity. |
| limits at infinity | The value that a function approaches as the input variable increases or decreases without bound. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| closed interval | An interval that includes both of its endpoints, denoted as [a, b]. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| Intermediate Value Theorem | A theorem stating that if a function is continuous on a closed interval [a, b] and d is a value between f(a) and f(b), then there exists at least one number c in the interval where f(c) = d. |
| Term | Definition |
|---|---|
| analytic notation | The symbolic mathematical representation of a limit, typically written as lim(x→a) f(x) = L. |
| approaches | In the context of limits, the behavior of a function's output as the input gets arbitrarily close to a specific value. |
| function | A mathematical relationship that assigns exactly one output value to each input value of an independent variable. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| limit notation | The symbolic representation of a limit, written as lim[x→c] f(x) = R, indicating that f(x) approaches R as x approaches c. |
| Term | Definition |
|---|---|
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| one-sided limit | A limit that considers the function's behavior as the input approaches a value from only one direction (either from the left or from the right). |
| oscillating | A function behavior where the output repeatedly fluctuates between values without settling on a single limit as the input approaches a particular value. |
| unbounded | A function behavior where the output grows without bound (approaches positive or negative infinity) as the input approaches a particular value. |
| Term | Definition |
|---|---|
| composite function | A function formed by combining two functions where the output of one function becomes the input of another. |
| differences | The result of subtracting one function from another. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| limit theorems | Algebraic rules that allow limits of combined functions to be determined from the limits of individual functions. |
| one-sided limit | A limit that considers the function's behavior as the input approaches a value from only one direction (either from the left or from the right). |
| products | The result of multiplying two or more functions together. |
| quotient | The result of dividing one function by another. |
| sums | The result of adding two or more functions together. |
| Term | Definition |
|---|---|
| conjugate | An expression formed by changing the sign between two terms, such as the conjugate of (a + b) being (a - b). |
| equivalent expressions | Different algebraic forms of the same function that have the same value. |
| factoring | The process of breaking down an expression into its multiplicative components. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| rational function | A function expressed as the ratio of two polynomial functions. |
| squeeze theorem | A method for determining the limit of a function by showing that the function is bounded between two other functions that have the same limit at a point. |
| trigonometric function | Functions such as sine, cosine, and tangent that relate angles to ratios of sides in a right triangle. |
| Term | Definition |
|---|---|
| equivalent expressions | Different algebraic forms of the same function that have the same value. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| squeeze theorem | A method for determining the limit of a function by showing that the function is bounded between two other functions that have the same limit at a point. |
| Term | Definition |
|---|---|
| cosecant | A trigonometric function defined as the reciprocal of sine. |
| cotangent | A trigonometric function defined as the ratio of cosine to sine. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| derivative rules | Formulas and procedures used to calculate derivatives, such as the product rule and quotient rule. |
| differentiable function | Functions that have a derivative at every point in their domain, meaning they are smooth and continuous without sharp corners or breaks. |
| identities | Equations that are true for all values of the variables, used to rewrite trigonometric expressions. |
| products | The result of multiplying two or more functions together. |
| quotient | The result of dividing one function by another. |
| secant | A trigonometric function defined as the reciprocal of cosine. |
| tangent | A trigonometric function defined as the ratio of sine to cosine. |
| Term | Definition |
|---|---|
| average rate of change | The change in the value of a function divided by the change in the input over an interval [a, b], calculated as (f(b) - f(a))/(b - a). |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| difference quotient | The expression [f(x+h) - f(x)]/h used to calculate the average rate of change and find the derivative as a limit. |
| instantaneous rate of change | The rate at which a function is changing at a specific point, represented by the derivative at that point. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| difference quotient | The expression [f(x+h) - f(x)]/h used to calculate the average rate of change and find the derivative as a limit. |
| dy/dx | Leibniz notation for the derivative of y with respect to x. |
| f'(x) | Lagrange notation for the derivative of function f at x. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| slope | The steepness or rate of change of a line, calculated as the change in y-values divided by the change in x-values. |
| tangent line | A line that touches a curve at a single point and has a slope equal to the derivative of the function at that point. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| estimate | To find an approximate value of a derivative using available information such as tables, graphs, or numerical methods. |
| Term | Definition |
|---|---|
| continuity | A property of a function at a point where the function is defined, the limit exists, and the limit equals the function value at that point. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| difference quotient | The expression [f(x+h) - f(x)]/h used to calculate the average rate of change and find the derivative as a limit. |
| differentiability | A property of a function at a point where the derivative exists; a function is differentiable at a point if the limit of the difference quotient exists at that point. |
| domain | The set of all input values (x-values) for which a function is defined. |
| left hand limit | The value that a function approaches as the input approaches a point from values less than that point. |
| right hand limit | The value that a function approaches as the input approaches a point from values greater than that point. |
| slope | The steepness or rate of change of a line, calculated as the change in y-values divided by the change in x-values. |
| tangent line | A line that touches a curve at a single point and has a slope equal to the derivative of the function at that point. |
| Term | Definition |
|---|---|
| definition of the derivative | The formal mathematical definition using limits: f'(x) = lim(h→0) [f(x+h) - f(x)]/h, which defines the derivative as the instantaneous rate of change. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| power rule | A derivative rule stating that the derivative of x^n is n·x^(n-1), where n is a constant. |
| Term | Definition |
|---|---|
| constant multiple rule | A derivative rule stating that the derivative of a constant times a function equals the constant times the derivative of the function. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| difference rule | A derivative rule stating that the derivative of a difference of functions equals the difference of their individual derivatives. |
| polynomial function | A function composed of terms with non-negative integer exponents and real coefficients. |
| power rule | A derivative rule stating that the derivative of x^n is n·x^(n-1), where n is a constant. |
| sum rule | A derivative rule stating that the derivative of a sum of functions equals the sum of their individual derivatives. |
| Term | Definition |
|---|---|
| cosine | A trigonometric function, denoted as cos x, for which the derivative is -sin x. |
| definition of the derivative | The formal mathematical definition using limits: f'(x) = lim(h→0) [f(x+h) - f(x)]/h, which defines the derivative as the instantaneous rate of change. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| exponential function | A function of the form f(x) = a^x, where a is a positive constant not equal to 1. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| logarithmic function | A function of the form f(x) = log_a(x), the inverse of an exponential function. |
| sine | A trigonometric function, denoted as sin x, for which the derivative is cos x. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| differentiable function | Functions that have a derivative at every point in their domain, meaning they are smooth and continuous without sharp corners or breaks. |
| product rule | A differentiation rule that states the derivative of a product of two functions equals the first function times the derivative of the second plus the second function times the derivative of the first. |
| quotient | The result of dividing one function by another. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| differentiable function | Functions that have a derivative at every point in their domain, meaning they are smooth and continuous without sharp corners or breaks. |
| products | The result of multiplying two or more functions together. |
| quotient | The result of dividing one function by another. |
| quotient rule | A differentiation rule used to find the derivative of a quotient of two differentiable functions. |
| Term | Definition |
|---|---|
| chain rule | A differentiation rule that provides a method for finding the derivative of a composite function by multiplying the derivative of the outer function by the derivative of the inner function. |
| composite function | A function formed by combining two functions where the output of one function becomes the input of another. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| differentiable function | Functions that have a derivative at every point in their domain, meaning they are smooth and continuous without sharp corners or breaks. |
| Term | Definition |
|---|---|
| chain rule | A differentiation rule that provides a method for finding the derivative of a composite function by multiplying the derivative of the outer function by the derivative of the inner function. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| implicit differentiation | A technique for finding the derivative of a function defined implicitly by differentiating both sides of an equation with respect to the independent variable. |
| implicitly defined function | A function defined by an equation relating x and y, where y is not explicitly solved in terms of x. |
| Term | Definition |
|---|---|
| chain rule | A differentiation rule that provides a method for finding the derivative of a composite function by multiplying the derivative of the outer function by the derivative of the inner function. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| inverse function | A function that reverses the effect of another function, such that if f(a) = b, then the inverse function f⁻¹(b) = a. |
| inverse trigonometric functions | Functions that reverse the action of trigonometric functions, such as arcsine, arccosine, and arctangent, which return an angle given a trigonometric ratio. |
| Term | Definition |
|---|---|
| chain rule | A differentiation rule that provides a method for finding the derivative of a composite function by multiplying the derivative of the outer function by the derivative of the inner function. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| inverse function | A function that reverses the effect of another function, such that if f(a) = b, then the inverse function f⁻¹(b) = a. |
| inverse trigonometric functions | Functions that reverse the action of trigonometric functions, such as arcsine, arccosine, and arctangent, which return an angle given a trigonometric ratio. |
| Term | Definition |
|---|---|
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| higher-order derivatives | Derivatives of derivatives obtained by repeatedly differentiating a function; the second derivative, third derivative, and beyond. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| independent variable | The input variable of a function, typically represented as x, with respect to which the rate of change is measured. |
| instantaneous rate of change | The rate at which a function is changing at a specific point, represented by the derivative at that point. |
| Term | Definition |
|---|---|
| acceleration | The derivative of the velocity function with respect to time, representing the rate of change of velocity for a moving particle. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| position | The location of an object along a straight line, typically represented as a function of time. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| rectilinear motion | Motion of a particle along a straight line, characterized by changes in position, velocity, and acceleration. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving without regard to direction. |
| velocity | The derivative of a position function with respect to time, representing the rate and direction of change of position for a moving particle. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| chain rule | A differentiation rule that provides a method for finding the derivative of a composite function by multiplying the derivative of the outer function by the derivative of the inner function. |
| product rule | A differentiation rule that states the derivative of a product of two functions equals the first function times the derivative of the second plus the second function times the derivative of the first. |
| quotient rule | A differentiation rule used to find the derivative of a quotient of two differentiable functions. |
| related rates | Problems in which the rates of change of two or more related quantities are connected, and the derivative is used to find an unknown rate of change from known rates. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| related rates | Problems in which the rates of change of two or more related quantities are connected, and the derivative is used to find an unknown rate of change from known rates. |
| Term | Definition |
|---|---|
| locally linear approximation | An approximation of a function's behavior in a small region around a point using a linear function, typically the tangent line at that point. |
| overestimate | An approximation that is greater than the actual value of a function. |
| point of tangency | The point where a tangent line touches a curve. |
| tangent line | A line that touches a curve at a single point and has a slope equal to the derivative of the function at that point. |
| underestimate | An approximation that is less than the actual value of a function. |
| Term | Definition |
|---|---|
| indeterminate forms | Limit expressions that do not have a determinate value without further analysis, such as 0/0 or ∞/∞. |
| L'Hospital's Rule | A method for evaluating limits of indeterminate forms by taking the derivative of the numerator and denominator separately. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| maximum value | The largest output value that a function attains on a given interval. |
| minimum value | The smallest output value that a function attains on a given interval. |
| optimization | The process of finding the minimum or maximum value of a function on a given interval. |
| Term | Definition |
|---|---|
| average rate of change | The change in the value of a function divided by the change in the input over an interval [a, b], calculated as (f(b) - f(a))/(b - a). |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| differentiable | A property of a function that has a derivative at every point in an interval, meaning the function is smooth with no sharp corners or cusps. |
| instantaneous rate of change | The rate at which a function is changing at a specific point, represented by the derivative at that point. |
| Mean Value Theorem | A theorem stating that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change equals the average rate of change over that interval. |
| Term | Definition |
|---|---|
| applied contexts | Real-world situations or practical problems where mathematical functions are used to model and solve problems. |
| maximum value | The largest output value that a function attains on a given interval. |
| minimum value | The smallest output value that a function attains on a given interval. |
| Term | Definition |
|---|---|
| critical point | A point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| implicit differentiation | A technique for finding the derivative of a function defined implicitly by differentiating both sides of an equation with respect to the independent variable. |
| implicit relation | A relation defined by an equation in which the dependent variable is not explicitly solved in terms of the independent variable. |
| implicitly defined function | A function defined by an equation relating x and y, where y is not explicitly solved in terms of x. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| critical point | A point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema. |
| Extreme Value Theorem | A theorem stating that if a function is continuous on a closed interval [a, b], then the function must attain both a minimum and maximum value on that interval. |
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| maximum value | The largest output value that a function attains on a given interval. |
| minimum value | The smallest output value that a function attains on a given interval. |
| relative extrema | Maximum or minimum values of a function at a point relative to nearby points. |
| Term | Definition |
|---|---|
| decreasing | An interval on which a function's output values are getting smaller as the input values increase, corresponding to where the first derivative is negative. |
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| increasing | An interval on which a function's output values are getting larger as the input values increase, corresponding to where the first derivative is positive. |
| Term | Definition |
|---|---|
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| relative extrema | Maximum or minimum values of a function at a point relative to nearby points. |
| Term | Definition |
|---|---|
| absolute extrema | The maximum or minimum values of a function over its entire domain or a specified interval. |
| critical point | A point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| endpoints | The boundary points of a closed interval where a function's absolute extrema may occur. |
| Term | Definition |
|---|---|
| concave down | A property of a function where the graph curves downward, occurring when the function's derivative is decreasing on an interval. |
| concave up | A property of a function where the graph curves upward, occurring when the function's derivative is increasing on an interval. |
| points of inflection | Points on the graph of a function where the concavity changes from concave up to concave down or vice versa. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| absolute maximum | The highest value of a function over its entire domain or a specified interval. |
| absolute minimum | The lowest value of a function over its entire domain or a specified interval. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| critical point | A point in the domain of a function where the derivative is zero or undefined, which are candidates for local and absolute extrema. |
| global extremum | The absolute maximum or minimum value of a function over its entire domain or a specified interval. |
| relative maximum | A point where a function reaches a highest value in a neighborhood around that point. |
| relative minimum | A point where a function reaches a lowest value in a neighborhood around that point. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| analytical representation | The representation of a function or its derivatives using equations and algebraic expressions. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| function behavior | The characteristics of a function including its increasing/decreasing intervals, concavity, extrema, and end behavior. |
| graphical representation | The visual display of a function or its derivatives on a coordinate plane. |
| key features | Important characteristics of a function including extrema, inflection points, intervals of increase/decrease, and concavity. |
| numerical representation | The representation of a function or its derivatives using tables of values or numerical data. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| function behavior | The characteristics of a function including its increasing/decreasing intervals, concavity, extrema, and end behavior. |
| key features | Important characteristics of a function including extrema, inflection points, intervals of increase/decrease, and concavity. |
| Term | Definition |
|---|---|
| accumulation of change | The total amount of change in a quantity over an interval, represented by the area between a rate of change function and the x-axis. |
| area under a curve | The region between the graph of a function and the x-axis over a specified interval, which represents the accumulation of change when the function is a rate of change. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| completing the square | An algebraic technique for rearranging quadratic expressions into perfect square form to simplify integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| integrands | The function being integrated in an integral expression. |
| long division | An algebraic technique for dividing polynomials to rearrange rational functions into equivalent forms suitable for integration. |
| substitution | An integration technique where a variable is replaced with another expression to simplify the integrand into a more manageable form. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| integrands | The function being integrated in an integral expression. |
| integration by parts | A technique for finding antiderivatives of products of functions, based on the product rule for derivatives. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| linear factors | First-degree polynomial expressions of the form (ax + b) used as denominators in partial fraction decomposition. |
| linear partial fractions | A decomposition technique that expresses a rational function as a sum of simpler fractions with linear denominators, used to simplify integration. |
| nonrepeating factors | Linear factors that appear only once in the denominator of a rational function, used in partial fraction decomposition. |
| rational function | A function expressed as the ratio of two polynomial functions. |
| Term | Definition |
|---|---|
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| improper integral | An integral with one or both limits infinite, or with an unbounded integrand in the interval of integration. |
| infinite limit | Limits that describe the behavior of a function as it approaches infinity or negative infinity, or as the function values grow without bound. |
| limits of definite integrals | A method for evaluating improper integrals by expressing them as limits of definite integrals with finite bounds. |
| unbounded integrand | A function that approaches infinity at one or more points within the interval of integration. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| left Riemann sum | An approximation method for a definite integral using rectangles whose heights are determined by the function values at the left endpoints of subintervals. |
| midpoint Riemann sum | An approximation method for a definite integral using rectangles whose heights are determined by the function values at the midpoints of subintervals. |
| nonuniform partition | A division of an interval into subintervals of varying widths. |
| numerical methods | Computational techniques used to approximate definite integrals when exact analytical solutions are difficult or impossible to obtain. |
| overestimate | An approximation that is greater than the actual value of a function. |
| right Riemann sum | An approximation method for a definite integral using rectangles whose heights are determined by the function values at the right endpoints of subintervals. |
| trapezoidal sum | An approximation method for a definite integral using trapezoids to estimate the area under a curve. |
| underestimate | An approximation that is less than the actual value of a function. |
| uniform partition | A division of an interval into subintervals of equal width. |
| Term | Definition |
|---|---|
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| limiting case | The value or behavior that a mathematical expression approaches as a parameter (such as the width of subintervals) approaches zero. |
| partition | A division of an interval into subintervals used to construct a Riemann sum. |
| Riemann sum | A sum of the form ∑f(x_i*)Δx_i used to approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles. |
| subinterval | One of the smaller intervals created by dividing a larger interval [a,b] into n parts. |
| Term | Definition |
|---|---|
| accumulation function | Functions that represent the accumulated total of a quantity over an interval, typically defined as g(x) = ∫[a to x] f(t) dt. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| accumulation function | Functions that represent the accumulated total of a quantity over an interval, typically defined as g(x) = ∫[a to x] f(t) dt. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| area | In the context of definite integrals, the region between a curve and the x-axis over a specified interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| integral of a constant times a function | The property stating that the integral of a constant multiplied by a function equals the constant times the integral of the function. |
| integral of a function over adjacent intervals | The property stating that the integral of a function over a combined interval equals the sum of integrals over each subinterval. |
| integral of the sum of two functions | The property stating that the integral of a sum of functions equals the sum of the integrals of the individual functions. |
| jump discontinuity | A type of discontinuity where the left-hand and right-hand limits of a function exist but are not equal, causing the function to jump from one value to another. |
| properties of definite integrals | Rules that govern how definite integrals behave, including linearity, reversal of limits, and additivity over adjacent intervals. |
| removable discontinuity | A discontinuity that can be eliminated by defining or redefining the function value at that point to equal the limit. |
| reversal of limits of integration | The property stating that reversing the upper and lower limits of a definite integral changes the sign of the result. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Fundamental Theorem of Calculus | The central theorem linking differentiation and integration, stating that if f is continuous on [a, b] and F is an antiderivative of f, then ∫(a to b) f(x) dx = F(b) - F(a). |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| closed-form antiderivative | An antiderivative that can be expressed using elementary functions and standard mathematical operations. |
| constant of integration | The arbitrary constant C added to an antiderivative to represent the family of all possible antiderivatives of a function. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| Term | Definition |
|---|---|
| antiderivative | Functions whose derivative equals a given function; the reverse process of differentiation. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| indefinite integral | Antiderivatives of a function, represented as ∫f(x)dx = F(x) + C, where C is an arbitrary constant. |
| integrands | The function being integrated in an integral expression. |
| limits of integration | The upper and lower bounds of a definite integral that must be adjusted when using substitution of variables. |
| substitution of variables | A technique for finding antiderivatives by replacing a variable or expression with a new variable to simplify the integrand. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| function | A mathematical relationship that assigns exactly one output value to each input value of an independent variable. |
| independent variable | The input variable of a function, typically represented as x, with respect to which the rate of change is measured. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| solution | A function that satisfies a differential equation when substituted into it along with its derivatives. |
| verify | To confirm that a proposed function satisfies a differential equation by substituting it and its derivatives into the equation. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| first-order differential equations | Differential equations that involve only the first derivative of a function. |
| slope field | A graphical representation of a differential equation showing the slope of solution curves at a finite set of points in the plane. |
| solutions to differential equations | Functions that satisfy a given differential equation when substituted into it. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| solution | A function that satisfies a differential equation when substituted into it along with its derivatives. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| Euler's method | A numerical procedure for approximating solutions to differential equations by using tangent line segments to estimate values at successive points along a solution curve. |
| solution curve | A graph representing the solution to a differential equation, showing how the dependent variable changes with respect to the independent variable. |
| Term | Definition |
|---|---|
| antidifferentiation | The process of finding a function whose derivative is a given function; the reverse operation of differentiation, also known as integration. |
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| separation of variables | A method for solving differential equations by rearranging the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the other side. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| domain restrictions | Limitations on the set of input values for which a solution to a differential equation is valid or defined. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| particular solution | A specific solution to a differential equation obtained by using initial conditions to determine the values of arbitrary constants. |
| Term | Definition |
|---|---|
| differential equation | An equation that relates a function to its derivatives, describing how a quantity changes in relation to one or more variables. |
| exponential decay | A process in which a quantity decreases at a rate proportional to its current size, modeled by dy/dt = ky where k < 0. |
| exponential growth | A process in which a quantity increases at a rate proportional to its current size, modeled by dy/dt = ky where k > 0. |
| exponential growth and decay model | A differential equation of the form dy/dt = ky that models quantities that increase or decrease at a rate proportional to their current amount. |
| general solution | The complete family of solutions to a differential equation, containing arbitrary constants that represent all possible particular solutions. |
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| particular solution | A specific solution to a differential equation obtained by using initial conditions to determine the values of arbitrary constants. |
| proportional | A relationship between two quantities where one is a constant multiple of the other. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| carrying capacity | The maximum value that a population or quantity can sustain in a logistic growth model, represented by the limiting value as time approaches infinity. |
| dependent variable | The variable in a differential equation whose value depends on the independent variable and whose rate of change is being described. |
| independent variable | The input variable of a function, typically represented as x, with respect to which the rate of change is measured. |
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| jointly proportional | A relationship where one quantity is proportional to the product of two or more other quantities. |
| limiting value | The value that a function approaches as the independent variable approaches infinity, representing the long-term behavior of the system. |
| logistic differential equation | A differential equation of the form dy/dt = ky(a - y) that models logistic growth, where the rate of change depends on both the current quantity and the difference from carrying capacity. |
| logistic growth model | A mathematical model describing population or quantity growth that accounts for limited resources, where growth rate depends on both the current size and the difference from carrying capacity. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| average value of a function | The mean value of a function over a specified interval, calculated using the formula (1/(b-a)) ∫[a to b] f(x) dx. |
| continuous | A function that has no breaks, jumps, or holes in its graph over a given interval. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| interval | A connected set of real numbers, typically expressed as a range between two endpoints. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| disc method | A technique for finding the volume of a solid of revolution by integrating the cross-sectional areas of circular discs perpendicular to the axis of rotation. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| Term | Definition |
|---|---|
| cross section | Two-dimensional slices of a three-dimensional solid, perpendicular to an axis, used to build up the volume through integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| ring shaped | Cross sections that have the shape of a washer or annulus, with an outer radius and an inner radius, used in volume calculations. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| washer method | A technique for finding the volume of a solid of revolution by integrating the areas of ring-shaped (washer-shaped) cross sections perpendicular to the axis of rotation. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| ring-shaped cross sections | Annular (donut-shaped) slices of a solid of revolution formed when the region being rotated has a gap between the axis of rotation and the outer boundary. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| washer method | A technique for finding the volume of a solid of revolution by integrating the areas of ring-shaped (washer-shaped) cross sections perpendicular to the axis of rotation. |
| Term | Definition |
|---|---|
| arc length | The distance along a curve between two points, calculated using a definite integral. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| planar curve | A curve that exists in a two-dimensional plane and can be defined by a function or parametric equations. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| displacement | The net change in position of a particle over a time interval, found by integrating the velocity vector. |
| rectilinear motion | Motion of a particle along a straight line, characterized by changes in position, velocity, and acceleration. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving without regard to direction. |
| total distance traveled | The total length of the path traveled by a particle over a time interval, found by integrating the speed. |
| velocity | The derivative of a position function with respect to time, representing the rate and direction of change of position for a moving particle. |
| Term | Definition |
|---|---|
| accumulation | The process of gathering or building up a quantity over time or over an interval, which can be expressed and calculated using definite integrals. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| net change | The total change in a quantity over an interval, calculated as the difference between final and initial values, often found using definite integrals. |
| rate of change | The measure of how quickly a quantity changes with respect to another variable, often time. |
| Term | Definition |
|---|---|
| areas in the plane | Regions bounded by curves and axes in a coordinate system whose measurements can be determined using integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| areas in the plane | Regions bounded by curves and axes in a coordinate system whose measurements can be determined using integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| absolute value of the difference | The absolute value of the difference between two functions, used to calculate area between curves regardless of which function is on top. |
| area between curves | The region enclosed between two or more curves, calculated using definite integrals. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| Term | Definition |
|---|---|
| cross section | Two-dimensional slices of a three-dimensional solid, perpendicular to an axis, used to build up the volume through integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| rectangular cross sections | Two-dimensional rectangular slices of a solid whose areas can be integrated to find the total volume. |
| square cross sections | Two-dimensional square slices of a solid whose areas can be integrated to find the total volume. |
| volumes of solids | The measure of three-dimensional space occupied by a solid object, calculated using integration techniques. |
| Term | Definition |
|---|---|
| area formulas | Mathematical expressions used to calculate the area of two-dimensional shapes, which are applied to cross sections in volume calculations. |
| cross section | Two-dimensional slices of a three-dimensional solid, perpendicular to an axis, used to build up the volume through integration. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| semicircular cross sections | Three-dimensional solids whose perpendicular slices are semicircular in shape. |
| triangular cross sections | Three-dimensional solids whose perpendicular slices are triangular in shape. |
| volumes of solids | The measure of three-dimensional space occupied by a solid object, calculated using integration techniques. |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| disc method | A technique for finding the volume of a solid of revolution by integrating the cross-sectional areas of circular discs perpendicular to the axis of rotation. |
| solids of revolution | Three-dimensional solids formed by rotating a two-dimensional region around an axis. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| dx/dt | The derivative of x with respect to the parameter t; the rate of change of the x-coordinate as the parameter changes. |
| dy/dt | The derivative of y with respect to the parameter t; the rate of change of the y-coordinate as the parameter changes. |
| dy/dx | Leibniz notation for the derivative of y with respect to x. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| tangent line | A line that touches a curve at a single point and has a slope equal to the derivative of the function at that point. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| second derivative | The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down. |
| Term | Definition |
|---|---|
| arc length | The distance along a curve between two points, calculated using a definite integral. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| vector-valued function | Functions that output vectors rather than scalar values, where each component is a function of the same independent variable. |
| Term | Definition |
|---|---|
| initial condition | Specified values of a function at particular points that determine which particular solution to a differential equation is selected. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| rate vector | A vector-valued function that describes the rate of change of position with respect to time, representing velocity or acceleration. |
| vector-valued function | Functions that output vectors rather than scalar values, where each component is a function of the same independent variable. |
| Term | Definition |
|---|---|
| acceleration | The derivative of the velocity function with respect to time, representing the rate of change of velocity for a moving particle. |
| displacement | The net change in position of a particle over a time interval, found by integrating the velocity vector. |
| parametric function | Functions where x and y coordinates are each expressed as separate functions of a third variable, typically time (t), rather than y as a function of x. |
| planar motion | The movement of a particle in a two-dimensional plane, described using parametric or vector-valued functions. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving without regard to direction. |
| total distance traveled | The total length of the path traveled by a particle over a time interval, found by integrating the speed. |
| vector-valued function | Functions that output vectors rather than scalar values, where each component is a function of the same independent variable. |
| velocity | The derivative of a position function with respect to time, representing the rate and direction of change of position for a moving particle. |
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| polar coordinates | A coordinate system in which points are located by their distance from a fixed point (the pole) and an angle measured from a fixed direction (the polar axis). |
| polar equation | An equation that describes a curve using polar coordinates, typically in the form r = f(θ). |
| Term | Definition |
|---|---|
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| polar coordinates | A coordinate system in which points are located by their distance from a fixed point (the pole) and an angle measured from a fixed direction (the polar axis). |
| polar curve | Curves defined by equations in polar coordinates, where points are located by a distance r from the origin and an angle θ from the positive x-axis. |
| rectangular coordinates | A coordinate system in which points are located using perpendicular x and y axes, also known as Cartesian coordinates. |
| Term | Definition |
|---|---|
| areas of regions | The measure of the two-dimensional space enclosed by one or more curves. |
| definite integral | The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. |
| polar curve | Curves defined by equations in polar coordinates, where points are located by a distance r from the origin and an angle θ from the positive x-axis. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| nth partial sum | The sum of the first n terms of a series. |
| sequence of partial sums | The sequence formed by successive partial sums of a series, where each term is the sum of the first n terms. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| alternating series | A series whose terms alternate in sign, typically written in the form Σ(-1)^n * a_n where a_n > 0. |
| alternating series error bound | A method for estimating the maximum error between a partial sum and the actual sum of a convergent alternating series, equal to the absolute value of the first omitted term. |
| alternating series test | A convergence test that determines whether an alternating series converges based on whether its terms decrease in absolute value and approach zero. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| partial sum | The sum of the first n terms of a series, denoted S_n. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| centered at | The point around which a Taylor polynomial is constructed; the point where the polynomial and function share the same value and derivatives. |
| coefficient | The numerical factor in front of a term in a polynomial, calculated as the nth derivative of the function divided by n factorial. |
| degree | The highest power of the variable in a polynomial term. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| factorial | The product of all positive integers up to a given number, denoted as n!, used in the denominator of Taylor polynomial coefficients. |
| function approximation | Using a simpler function, such as a polynomial, to estimate the values of a more complex function. |
| interval | A connected set of real numbers, typically expressed as a range between two endpoints. |
| Taylor polynomial | A finite polynomial that approximates a function, formed by taking a partial sum of the Taylor series for that function. |
| Term | Definition |
|---|---|
| alternating series error bound | A method for estimating the maximum error between a partial sum and the actual sum of a convergent alternating series, equal to the absolute value of the first omitted term. |
| error bound | A maximum value that represents how far a Taylor polynomial approximation can deviate from the actual function value. |
| Lagrange error bound | A formula that provides the maximum possible error when using a Taylor polynomial to approximate a function value. |
| Taylor polynomial approximation | A polynomial function used to approximate the value of a function near a specific point. |
| Term | Definition |
|---|---|
| interval of convergence | The set of all x-values for which a power series converges, determined by testing the radius of convergence and checking the endpoints. |
| power series | An infinite series of the form Σ(aₙ(x-c)ⁿ) where aₙ are coefficients, x is a variable, and c is the center of the series. |
| radius of convergence | The value that determines the distance from the center of a power series within which the series converges. |
| ratio test | A convergence test used to determine the radius of convergence of a power series by examining the limit of the ratio of consecutive terms. |
| Taylor series | A power series representation of a function that converges to that function over an open interval with positive radius of convergence. |
| term-by-term differentiation | The process of differentiating a power series by differentiating each term individually, which preserves the radius of convergence. |
| term-by-term integration | The process of integrating a power series by integrating each term individually, which preserves the radius of convergence. |
| Term | Definition |
|---|---|
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| Maclaurin series | A special case of a Taylor series where the function is expanded around the point x = 0. |
| Taylor polynomial | A finite polynomial that approximates a function, formed by taking a partial sum of the Taylor series for that function. |
| Taylor series | A power series representation of a function that converges to that function over an open interval with positive radius of convergence. |
| Term | Definition |
|---|---|
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| power series | An infinite series of the form Σ(aₙ(x-c)ⁿ) where aₙ are coefficients, x is a variable, and c is the center of the series. |
| term-by-term differentiation | The process of differentiating a power series by differentiating each term individually, which preserves the radius of convergence. |
| term-by-term integration | The process of integrating a power series by integrating each term individually, which preserves the radius of convergence. |
| Term | Definition |
|---|---|
| constant ratio | The fixed multiplicative factor between successive terms in a geometric series, denoted as r. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| nth term test | A test for divergence that examines whether the limit of the nth term of a series equals zero; if the limit is not zero, the series diverges. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| integral test | A method for determining whether an infinite series converges or diverges by comparing it to an improper integral. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| alternating harmonic series | The infinite series 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln(2). |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| harmonic series | The infinite series 1 + 1/2 + 1/3 + 1/4 + ..., which diverges despite having terms that approach zero. |
| p-series | An infinite series of the form 1 + 1/2^p + 1/3^p + 1/4^p + ..., which converges when p > 1 and diverges when p ≤ 1. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| comparison test | A method for determining convergence or divergence of a series by comparing it to another series whose convergence is known. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| limit comparison test | A method for determining convergence or divergence of a series by comparing the limit of the ratio of its terms to those of another series. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| alternating series | A series whose terms alternate in sign, typically written in the form Σ(-1)^n * a_n where a_n > 0. |
| alternating series test | A convergence test that determines whether an alternating series converges based on whether its terms decrease in absolute value and approach zero. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| ratio test | A convergence test used to determine the radius of convergence of a power series by examining the limit of the ratio of consecutive terms. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| absolutely convergent | A series that converges when all terms are replaced by their absolute values. |
| conditionally convergent | A series that converges but does not converge absolutely; the series converges only because of the signs of its terms. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
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