AP Calculus Unit 1 Vocabulary

A collection of vocabulary flashcards covering the key definitions, theorems, and types of discontinuities introduced in AP Calculus Unit 1.

Calculus

The study of continuous change.

Limit

A fundamental concept based on approach, describing the value a function gets close to as the input approaches a specific number.

Left-Hand Limit

The value a function approaches as $x$ approaches a target value from the left side (denoted with a minus sign).

Right-Hand Limit

The value a function approaches as $x$ approaches a target value from the right side (denoted with a plus sign).

Limit of a Constant Property

A foundational property of limits stating that the limit of a constant is equal to that constant.

Direct Substitution

The basic method used to evaluate a limit by plugging the value $x$ is approaching directly into the expression for $x$.

Squeeze Theorem

A theorem used to find the limit of an inner function squeezed between two outer functions; if the outer functions approach the same limit, the inner function must approach that same value.

Continuity (Graphical Definition)

A property of a graph meaning it can be traced with a pencil without lifting it, having no breaks, jumps, or holes.

Removable Discontinuity

A type of discontinuity representing holes in a graph, typically created in rational functions when a factor in the numerator cancels with an identical factor in the denominator.

Vertical Asymptote Discontinuity

A type of discontinuity occurring in rational functions when factors in the denominator do not cancel out, or in trigonometric functions like tangent.

Jump Discontinuity

A discontinuity where a graph jumps from one function value to another, identified in piecewise equations when plugging boundary values into sub-functions yields different results.

Continuity at a Point

The mathematical definition requiring three conditions at $x = c$: $f(c)$ is defined, $\bar{\bar{\text{limit}}}_{x \rightarrow c} f(x)$ exists (where $\bar{\bar{\text{limit}}}$ represents the limit as $x \rightarrow c$), and $\bar{\bar{\text{limit}}}_{x \rightarrow c} f(x) = f(c)$.

Intermediate Value Theorem

A theorem stating that if a function $f$ is continuous on the interval $[a, b]$, $f(a) \neq f(b)$, and a target value $k$ lies between $f(a)$ and $f(b)$, then $f(x) = k$ for at least one $x$ between $a$ and $b$.