Limit (極限): The value that a function approaches as the input approaches some value.
Infinity (無窮大): A concept that represents an unbounded quantity.
Finite (有限的): Having limits or bounds. A finite interval has a starting point and an ending point.
Infinite (無限的): Without limits or bounds. The set of all numbers goes on forever, so it's infinite.
Converging (收斂): Approaching a limit.
Approximation (近似值): A value or quantity that is nearly but not exactly correct.
L'Hôpital's Rule (羅必達法則): A rule that uses derivatives to calculate limits of fractions whose numerators and denominators both approach zero or infinity.
Continuous function (連續函數): A function is continuous if you can draw its graph without lifting your pen. This means there are no abrupt jumps or breaks in the graph.
Derivative (導數): A measure of how a function changes as its input changes.
Differential (微分): A function is differentiable if it possesses a derivative at every point in its domain. This indicates the function's graph is smooth, without sharp corners or breaks.
Implicit differentiation (隱微分): A technique used to find the derivative of a function that is not explicitly defined.
Dummy variable (啞變量): A variable that is used in a mathematical expression but whose value does not affect the outcome.
Mean Value Theorem (均值定理): For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c.
Extreme Values (極值): The extreme values of a function are its maximum and minimum values. These can be:
Local (局部): The highest or lowest value within a small region of the function's graph
Absolute (絕對): The highest or lowest value across the entire domain of the function.
Critical Point (臨界點): A critical point is where the derivative of a function is either zero or undefined. These points are essential for finding potential locations of maxima and minima.
Inflection Point/Point of inflection (反曲點): An inflection point is where the concavity of a function's graph changes from concave up (bowl-shaped) to concave down (upside-down bowl), or vice versa.
Concavity (凹凸性): Concavity refers to how a curve bends. A function is concave up if its graph curves upwards, and concave down if it curves downwards.89
Asymptote (漸近線): An asymptote is a line that a function's graph approaches as the input values get extremely large or extremely small. There are horizontal, vertical, and oblique asymptotes.