Calculus: Limits, Rates of Change, and Continuity Flashcards

Vocabulary flashcards covering core topics in calculus including secant and tangent lines, velocity, limits, limit laws, epsilon-delta definitions, squeeze theorem, and continuity concepts.

Secant Line

A line passing through two points $(a, f(a))$ and $(x, f(x))$ on the graph of a function $f(x)$, with slope given by $m_{\text{sec}} = \frac{f(x) - f(a)}{x - a}$.

Differential Calculus

The field of calculus concerned with the study of derivatives, tangent lines, and rates of change of functions.

Average Velocity

The change in position of an object divided by the length of the time period over a time interval $[a, t]$, calculated as $v_{\text{ave}} = \frac{s(t) - s(a)}{t - a}$.

Instantaneous Velocity

The value that average velocities approach on intervals of the form $[a, t]$ and $[t, a]$ as the values of $t$ become closer to $a$, provided such a value exists.

Integral Calculus

The field of calculus concerned with the study of integrals, areas under curves, and their applications.

Multivariable Calculus

The study of the calculus of functions of two or more variables.

Limit of a Function (Intuitive Definition)

The real number $L$ that the functional values $f(x)$ approach and stay close to as $x$ gets closer to $a$ (where $x \neq a$), expressed symbolically as \lim_{x \rightarrow a} f(x) = L.

Limit from the Left

A one-sided limit where the values of $f(x)$ approach the real number $L$ as $x$ approaches $a$ from values less than $a$ ($x < a$), written as \lim_{x \rightarrow a^-} f(x) = L.

Limit from the Right

A one-sided limit where the values of $f(x)$ approach the real number $L$ as $x$ approaches $a$ from values greater than $a$ ($x > a$), written as \lim_{x \rightarrow a^+} f(x) = L.

Vertical Asymptote

A vertical line $x = a$ where the functional values $f(x)$ increase or decrease without bound ($\pm\infty$) as $x$ approaches $a$ from the left or right.

Triangle Inequality

A mathematical property of absolute value stating that if $a$ and $b$ are any real numbers, then $|a + b| \le |a| + |b|$.

Epsilon-Delta Definition of a Limit

The precise definition stating that \lim_{x \rightarrow a} f(x) = L if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \varepsilon$.

Squeeze Theorem

A theorem stating that if $f(x) \le g(x) \le h(x)$ for all $x \neq a$ in an open interval containing $a$, and \lim_{x \rightarrow a} f(x) = L = \lim_{x \rightarrow a} h(x), then \lim_{x \rightarrow a} g(x) = L.

Continuity at a Point

A property of a function $f(x)$ at a point $a$ requiring three conditions: 1. $f(a)$ is defined; 2. \lim_{x \rightarrow a} f(x) exists; and 3. \lim_{x \rightarrow a} f(x) = f(a).

Removable Discontinuity

A discontinuity at $a$ for which there is a hole in the graph because \lim_{x \rightarrow a} f(x) exists, but either $f(a)$ is undefined or $f(a) \neq \lim_{x \rightarrow a} f(x)$.

Jump Discontinuity

A noninfinite discontinuity for which the sections of the function do not meet up because the left-hand and right-hand limits exist but are not equal (\lim_{x \rightarrow a^-} f(x) \neq \lim_{x \rightarrow a^+} f(x)).

Infinite Discontinuity

A discontinuity located at a vertical asymptote $x = a$ where one or both one-sided limits are infinite ($\pm\infty$).

Types of Discontinuities

The classification of point discontinuities into removable discontinuities (holes), jump discontinuities (steps), and infinite discontinuities (vertical asymptotes).