| 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. |