| 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. |