Calculus I: Limits, Continuity, and Derivatives Flashcards

Vocabulary flashcards covering core calculus concepts from homework on limits, continuity, vertical and horizontal asymptotes, difference quotients, derivatives, and rates of change.

Difference Quotient

The expression $\frac{f(x+h) - f(x)}{h}$ (where $h \neq 0$), which measures the average rate of change of $f(x)$ over the interval $[x, x+h]$.

Definition of the Derivative

The limit of the difference quotient as $h$ approaches $0$, given by $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, representing the instantaneous rate of change or the slope of the tangent line.

Removable Discontinuity

A point $x = a$ where $\lim_{x \to a} f(x)$ exists, but $f(a)$ is either undefined or not equal to the limit.

Jump Discontinuity

A point $x = a$ where both one-sided limits $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ exist and are finite, but are not equal to each other.

Infinite Discontinuity

A point $x = a$ where at least one of the one-sided limits, $\lim_{x \to a^-} f(x)$ or $\lim_{x \to a^+} f(x)$, approaches $\infty$ or $-\infty$.

Continuity at a Point

A condition satisfied when $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

Intermediate Value Theorem

A theorem stating that if $f(x)$ is continuous on a closed interval $[a, b]$, then for any value $y$ between $f(a)$ and $f(b)$, there exists at least one number $c \in [a, b]$ such that $f(c) = y$.

Vertical Asymptote

A vertical line $x = a$ where the function $f(x)$ approaches $\infty$ or $-\infty$ as $x$ approaches $a$ from either the left or the right.

Limit at Infinity

The value approached by a function $f(x)$ as $x$ grows arbitrarily large in the positive or negative direction, denoted $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$.

Average Rate of Change

The ratio of the change in the function value to the change in the independent variable over an interval $[a, b]$, calculated as $\frac{f(b) - f(a)}{b - a}$.

Instantaneous Rate of Change

The exact rate at which a function $f(x)$ changes at a single point $x = a$, calculated as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Tangent Line Equation

The equation of the line passing through $(a, f(a))$ with slope $f'(a)$, given by $y - f(a) = f'(a)(x - a)$.

Conjugate Method for Limits

An algebraic technique involving multiplying the numerator and denominator by a conjugate expression to rationalize square roots in a difference quotient or limit.

Particle at Rest

The state of a moving particle whose position is given by $s(t)$ when its instantaneous velocity function $v(t) = s'(t)$ equals $0$.

One-Sided Limit

The value that a function $f(x)$ approaches as $x$ approaches a specified value strictly from the left ($x \to a^-$) or strictly from the right ($x \to a^+$).