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The value that a function or sequence approaches as the input or index approaches some value.

A measure of how a function changes as its input changes; it represents an instantaneous rate of change.

A mathematical concept that represents the area under a curve or the accumulation of quantities.

A function that has no breaks, holes, or jumps; it can be drawn without lifting the pencil from the paper.

A function that has at least one point where it is not continuous, which can be a jump, hole, or other form of disruption in the graph.

A formula used to find the derivative of a product of two functions. It states that $(uv)' = u'v + uv'$.

A formula to compute the derivative of a quotient of two functions. It states that $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.

A method for finding the derivative of the composition of two or more functions: if $h(x) = f(g(x))$, then $h'(x) = f'(g(x))g'(x)$.

A basic rule for differentiating functions of the form $x^n$, which states that $\frac{d}{dx}x^n = nx^{n-1}$.

A theorem that links the concept of differentiation and integration, and states that if $F$ is an antiderivative of $f$ on an interval, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

A theorem stating that for a function continuous on [a, b] and differentiable on (a, b), there exists some $c$ in (a, b) such that $f'(c) = \frac{f(b) - f(a)}{b-a}$.

A theorem that if a function $f$ is continuous on the closed interval [a, b], then for any value $L$ between $f(a)$ and $f(b)$, there exists some $c$ in [a, b] where $f(c) = L$.

A function $F$ is called an antiderivative of $f$ on an interval if $F'(x) = f(x)$ for all $x$ in the interval.

The integral of a function over a specified interval, which gives the net area between the function and the x-axis on that interval.

An integral without upper and lower limits that represents a family of functions and includes an arbitrary constant of integration, typically denoted as $C$.

A method for approximating the definite integral of a function, by summing up the areas of multiple simple shapes, such as rectangles, that cover the region under the curve.

A point on the graph of a function where the derivative is either zero or undefined. It is where the function may have a local maximum, local minimum, or a point of inflection.

A point on the graph of a function where the function's value is higher than at any neighboring points in a small interval around it.

A point on the graph of a function where the function's value is lower than at any neighboring points in a small interval around it.

A point on the graph of a function at which the curvature changes sign, indicating a change in concavity.

A function is concave up on an interval if the function's graph lies above its tangent lines, and the second derivative is positive.

A function is concave down on an interval if the function's graph lies below its tangent lines, and the second derivative is negative.

A function is monotonic on an interval if it is either entirely non-increasing or non-decreasing throughout that interval.

A function is uniformly continuous on an interval if for any small distance, a common distance can be found such that any two points within this common distance will have their function values within the small distance.

The derivative of a function of multiple variables with respect to one variable, treating the other variables as constants.

The rate at which a function changes at a point in the direction of a given vector.

A vector that points in the direction of greatest increase of a function and whose magnitude is the rate of increase in that direction

An infinite sum of terms calculated from the values of a function's derivatives at a single point. It approximates the function.