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 $\left[t,a\right]$ (if $t<a$ or $\left[a,t\right]$ if $t>a)$, with a position given by $s\left(t\right),$ that is $v_{\text{ave}}=\frac{s\left(t\right)-s\left(a\right)}{t-a}$

constant multiple law for limits

the limit law $\underset{x\to a}{\text{lim}}cf\left(x\right)=c·\underset{x\to a}{\text{lim}}f\left(x\right)=cL$

continuity at a point

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

continuity from the left

A function is continuous from the left at b if $\underset{x\to b^{-}}{\text{lim}}f\left(x\right)=f\left(b\right)$

continuity from the right

A function is continuous from the right at a if $\underset{x\to a^{+}}{\text{lim}}f(x)=f(a)$

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 $f\left(x\right)$ is continuous over a closed interval of the form $\left[a,b\right]$ if it is continuous at every point in $\left(a,b\right),$ and it is continuous from the right at a and from the left at b

difference law for limits

the limit law $\underset{x\to a}{\text{lim}}\left(f\left(x\right)-g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)-\underset{x\to a}{\text{lim}}g\left(x\right)=L-M$

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

$\underset{x\to a}{\text{lim}}f\left(x\right)=L$ if for every $\epsilon >0,$ there exists a $\delta >0$ such that if $0<\left|x-a\right|<\delta ,$ then $\left|f\left(x\right)-L\right|<\epsilon$

infinite discontinuity

An infinite discontinuity occurs at a point a if $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{±}\infty$ or $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{±}\infty$

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 $s\left(t\right)$ is the value that the average velocities on intervals of the form $\left[t,a\right]$ and $\left[a,t\right]$ 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 $\left[\text{a},\text{b}\right];$ if z is any real number between $f\left(a\right)$ and $f\left(b\right),$ then there is a number c in $\left[a,b\right]$ satisfying $f\left(c\right)=z$

intuitive definition of the limit

If all values of the function $f\left(x\right)$ approach the real number L as the values of $x\left(\ne a\right)$ approach a, $f\left(x\right)$ approaches L

jump discontinuity

A jump discontinuity occurs at a point a if $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)$ and $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)$ both exist, but $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)\ne \underset{x\to a^{+}}{\text{lim}}f\left(x\right)$

limit

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

limit laws

the individual properties of limits; for each of the individual laws, let $f\left(x\right)$ and $g\left(x\right)$ be defined for all $x\ne a$ over some open interval containing a; assume that L and M are real numbers so that $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=M;$ 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 $\underset{x\to a}{\text{lim}}{\left(f\left(x\right)\right)}^n={\left(\underset{x\to a}{\text{lim}}f\left(x\right)\right)}^n=L^n$ for every positive integer n

product law for limits

the limit law $\underset{x\to a}{\text{lim}}\left(f\left(x\right)·g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)·\underset{x\to a}{\text{lim}}g\left(x\right)=L·M$

quotient law for limits

the limit law $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{\text{lim}}f\left(x\right)}{\underset{x\to a}{\text{lim}}g\left(x\right)}=\frac{L}{M}$ for $M\ne 0$

removable discontinuity

A removable discontinuity occurs at a point a if $f\left(x\right)$ is discontinuous at a, but $\underset{x\to a}{\text{lim}}f\left(x\right)$ exists

root law for limits

the limit law $\underset{x\to a}{\text{lim}}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\text{lim}}f\left(x\right)}=\sqrt[n]{L}$ for all L if n is odd and for $L\ge 0$ if n is even

secant

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

squeeze theorem

states that if $f\left(x\right)\le g\left(x\right)\le h\left(x\right)$ for all $x\ne a$ over an open interval containing a and $\underset{x\to a}{\text{lim}}f\left(x\right)=L=\underset{x\to a}{\text{lim}}h\left(x\right)$ where L is a real number, then $\underset{x\to a}{\text{lim}}g\left(x\right)=L$

sum law for limits

The limit law $\underset{x\to a}{\text{lim}}\left(f\left(x\right)+g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)+\underset{x\to a}{\text{lim}}g\left(x\right)=L+M$

tangent

A tangent line to the graph of a function at a point $\left(a,f\left(a\right)\right)$ is the line that secant lines through $\left(a,f\left(a\right)\right)$ 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 $\left|a+b\right|\le \left|a\right|+\left|b\right|$

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