Ch. 2 Key Terms - Calculus Volume 1 | OpenStax

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Key Terms

average velocity: the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval $[t, a]$ (if $t < a$ or $[a, t]$ if $t > a)$ , with a position given by $s (t),$ that is $v_{ave} = \frac{s (t) - s (a)}{t - a}$

constant multiple law for limits: the limit law $lim_{x \to a} c f (x) = c \cdot lim_{x \to a} f (x) = c L$

continuity at a point: A function $f (x)$ is continuous at a point a if and only if the following three conditions are satisfied: (1) $f (a)$ is defined, (2) $lim_{x \to a} f (x)$ exists, and (3) $lim_{x \to a} f (x) = f (a)$

continuity from the left: A function is continuous from the left at b if $lim_{x \to b^{-}} f (x) = f (b)$

continuity from the right: A function is continuous from the right at a if $lim_{x \to a^{+}} f (x) = f (a)$

continuity over an interval: a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function $f (x)$ is continuous over a closed interval of the form $[a, b]$ if it is continuous at every point in $(a, b),$ and it is continuous from the right at a and from the left at b

difference law for limits: the limit law $lim_{x \to a} (f (x) - g (x)) = lim_{x \to a} f (x) - lim_{x \to a} g (x) = L - M$

differential calculus: the field of calculus concerned with the study of derivatives and their applications

discontinuity at a point: A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point

epsilon-delta definition of the limit: $lim_{x \to a} f (x) = L$ if for every $ε > 0,$ there exists a $δ > 0$ such that if $0 < | x - a | < δ,$ then $| f (x) - L | < ε$

infinite discontinuity: An infinite discontinuity occurs at a point a if $lim_{x \to a^{-}} f (x) = ± \infty$ or $lim_{x \to a^{+}} f (x) = ± \infty$

infinite limit: A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a

instantaneous velocity: The instantaneous velocity of an object with a position function that is given by $s (t)$ is the value that the average velocities on intervals of the form $[t, a]$ and $[a, t]$ approach as the values of t move closer to $a,$ provided such a value exists

integral calculus: the study of integrals and their applications

Intermediate Value Theorem: Let f be continuous over a closed bounded interval $[a, b];$ if z is any real number between $f (a)$ and $f (b),$ then there is a number c in $[a, b]$ satisfying $f (c) = z$

intuitive definition of the limit: If all values of the function $f (x)$ approach the real number L as the values of $x (\neq a)$ approach a, $f (x)$ approaches L

jump discontinuity: A jump discontinuity occurs at a point a if $lim_{x \to a^{-}} f (x)$ and $lim_{x \to a^{+}} f (x)$ both exist, but $lim_{x \to a^{-}} f (x) \neq lim_{x \to a^{+}} f (x)$

limit: the process of letting x or t approach a in an expression; the limit of a function $f (x)$ as x approaches a is the value that $f (x)$ approaches as x approaches a

limit laws: the individual properties of limits; for each of the individual laws, let $f (x)$ and $g (x)$ be defined for all $x \neq a$ over some open interval containing a; assume that L and M are real numbers so that $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M;$ let c be a constant

multivariable calculus: the study of the calculus of functions of two or more variables

one-sided limit: A one-sided limit of a function is a limit taken from either the left or the right

power law for limits: the limit law $lim_{x \to a} {(f (x))}^{n} = {(lim_{x \to a} f (x))}^{n} = L^{n}$ for every positive integer n

product law for limits: the limit law $lim_{x \to a} (f (x) \cdot g (x)) = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x) = L \cdot M$

quotient law for limits: the limit law $lim_{x \to a} \frac{f (x)}{g (x)} = \frac{lim_{x \to a} f (x)}{lim_{x \to a} g (x)} = \frac{L}{M}$ for $M \neq 0$

removable discontinuity: A removable discontinuity occurs at a point a if $f (x)$ is discontinuous at a, but $lim_{x \to a} f (x)$ exists

root law for limits: the limit law $lim_{x \to a} \sqrt[n]{f (x)} = \sqrt[n]{lim_{x \to a} f (x)} = \sqrt[n]{L}$ for all L if n is odd and for $L \geq 0$ if n is even

secant: A secant line to a function $f (x)$ at a is a line through the point $(a, f (a))$ and another point on the function; the slope of the secant line is given by $m_{sec} = \frac{f (x) - f (a)}{x - a}$

squeeze theorem: states that if $f (x) \leq g (x) \leq h (x)$ for all $x \neq a$ over an open interval containing a and $lim_{x \to a} f (x) = L = lim_{x \to a} h (x)$ where L is a real number, then $lim_{x \to a} g (x) = L$

sum law for limits: The limit law $lim_{x \to a} (f (x) + g (x)) = lim_{x \to a} f (x) + lim_{x \to a} g (x) = L + M$

tangent: A tangent line to the graph of a function at a point $(a, f (a))$ is the line that secant lines through $(a, f (a))$ approach as they are taken through points on the function with x-values that approach a; the slope of the tangent line to a graph at a measures the rate of change of the function at a

triangle inequality: If a and b are any real numbers, then $| a + b | \leq | a | + | b |$

vertical asymptote: A function has a vertical asymptote at $x = a$ if the limit as x approaches a from the right or left is infinite

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Authors: Gilbert Strang, Edwin “Jed” Herman
Publisher/website: OpenStax
Book title: Calculus Volume 1
Publication date: Mar 30, 2016
Location: Houston, Texas
Book URL: https://openstax.org/books/calculus-volume-1/pages/1-introduction
Section URL: https://openstax.org/books/calculus-volume-1/pages/2-key-terms

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