absolute extremum

if [latex]f[/latex] has an absolute maximum or absolute minimum at [latex]c[/latex], we say [latex]f[/latex] has an absolute extremum at [latex]c[/latex]

absolute maximum

if [latex]f(c)\ge f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], we say [latex]f[/latex] has an absolute maximum at [latex]c[/latex]

absolute minimum

if [latex]f(c)\le f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], we say [latex]f[/latex] has an absolute minimum at [latex]c[/latex]

absolute value function

[latex]f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}[/latex]

acceleration

is the rate of change of the velocity, that is, the derivative of velocity

algebraic function

a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable [latex]x[/latex]

amount of change

the amount of a function [latex]f(x)[/latex] over an interval [latex][x,x+h][/latex] is [latex]f(x+h)-f(x)[/latex]

antiderivative

a function [latex]F[/latex] such that [latex]F^{\prime}(x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex] is an antiderivative of [latex]f[/latex]

arc length

the arc length of a curve can be thought of as the distance a person would travel along the path of the curve

average rate of change

is a function [latex]f(x)[/latex] over an interval [latex][x,x+h][/latex] is [latex]\dfrac{f(x+h)-f(a)}{b-a}[/latex]

average value of a function

(or [latex]f_{\text{ave}}[/latex]) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval

base

the number [latex]b[/latex] in the exponential function [latex]f(x)=b^x[/latex] and the logarithmic function [latex]f(x)=\log_b x[/latex]

catenary

a curve in the shape of the function [latex]y=a\text{cosh}(x\text{/}a)[/latex] is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary

center of mass

the point at which the total mass of the system could be concentrated without changing the moment

centroid

the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region

chain rule

the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function

change of variables

the substitution of a variable, such as [latex]u[/latex], for an expression in the integrand

composite function

given two functions [latex]f[/latex] and [latex]g[/latex], a new function, denoted [latex]g\circ f[/latex], such that [latex](g\circ f)(x)=g(f(x))[/latex]

concave down

if [latex]f[/latex] is differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}[/latex] is decreasing over [latex]I[/latex], then [latex]f[/latex] is concave down over [latex]I[/latex]

concave up

if [latex]f[/latex] is differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}[/latex] is increasing over [latex]I[/latex], then [latex]f[/latex] is concave up over [latex]I[/latex]

concavity

the upward or downward curve of the graph of a function

concavity test

suppose [latex]f[/latex] is twice differentiable over an interval [latex]I[/latex]; if [latex]f^{\prime \prime}>0[/latex] over [latex]I[/latex], then [latex]f[/latex] is concave up over [latex]I[/latex]; if [latex]f^{\prime \prime}<0[/latex] over [latex]I[/latex], then [latex]f[/latex] is concave down over [latex]I[/latex]

constant multiple rule

the derivative of a constant [latex]c[/latex] multiplied by a function [latex]f[/latex] is the same as the constant multiplied by the derivative: [latex]\frac{d}{dx}(cf(x))=cf^{\prime}(x)[/latex]

constant rule

the derivative of a constant function is zero: [latex]\frac{d}{dx}(c)=0[/latex], where [latex]c[/latex] is a constant

continuity at a point

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

continuity from the left

A function is continuous from the left at [latex]b[/latex] if [latex]\underset{x\to b^-}{\lim}f(x)=f(b)[/latex]

continuity from the right

A function is continuous from the right at [latex]a[/latex] if [latex]\underset{x\to a^+}{\lim}f(x)=f(a)[/latex]

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

critical point

if [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined, we say that [latex]c[/latex] is a critical point of [latex]f[/latex]

cross-section

the intersection of a plane and a solid object

cubic function

a polynomial of degree 3; that is, a function of the form [latex]f(x)=ax^3+bx^2+cx+d[/latex], where [latex]a \ne 0[/latex]

decreasing on the interval [latex]I[/latex]

a function decreasing on the interval [latex]I[/latex] if, for all [latex]x_1, \, x_2\in I, \, f(x_1)\ge f(x_2)[/latex] if [latex]x_1

definite integral

a primary operation of calculus; the area between the curve and the [latex]x[/latex]-axis over a given interval is a definite integral

density function

a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume

dependent variable

the output variable for a function

derivative

the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

derivative function

gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

difference quotient

of a function [latex]f(x)[/latex] at [latex]a[/latex] is given by

[latex]\dfrac{f(a+h)-f(a)}{h}[/latex] or [latex]\dfrac{f(x)-f(a)}{x-a}[/latex]

difference rule

the derivative of the difference of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the difference of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex]: [latex]\frac{d}{dx}(f(x)-g(x))=f^{\prime}(x)-g^{\prime}(x)[/latex]

differentiable at [latex]a[/latex]

a function for which [latex]f^{\prime}(a)[/latex] exists is differentiable at [latex]a[/latex]

differentiable on [latex]S[/latex]

a function for which [latex]f^{\prime}(x)[/latex] exists for each [latex]x[/latex] in the open set [latex]S[/latex] is differentiable on [latex]S[/latex]

differentiable function

a function for which [latex]f^{\prime}(x)[/latex] exists is a differentiable function

differential

the differential [latex]dx[/latex] is an independent variable that can be assigned any nonzero real number; the differential [latex]dy[/latex] is defined to be [latex]dy=f^{\prime}(x) \, dx[/latex]

differential form

given a differentiable function [latex]y=f^{\prime}(x)[/latex], the equation [latex]dy=f^{\prime}(x) \, dx[/latex] is the differential form of the derivative of [latex]y[/latex] with respect to [latex]x[/latex]

differentiation

the process of taking a derivative

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

disk method

a special case of the slicing method used with solids of revolution when the slices are disks

domain

the set of inputs for a function

doubling time

if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\frac{(\text{ln}2)}{k}[/latex]

end behavior

the behavior of a function as [latex]x\to \infty[/latex] and [latex]x\to −\infty[/latex]

epsilon-delta definition of the limit

[latex]\underset{x\to a}{\lim}f(x)=L[/latex] if for every [latex]\varepsilon >0[/latex], there exists a [latex]\delta >0[/latex] such that if [latex]0<|x-a|<\delta[/latex], then [latex]|f(x)-L|<\varepsilon[/latex]

even function

a function is even if [latex]f(−x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]

exponent

the value [latex]x[/latex] in the expression [latex]b^x[/latex]

exponential decay

systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\text{−}kt}[/latex]

exponential growth

systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[/latex]

extreme value theorem

if [latex]f[/latex] is a continuous function over a finite, closed interval, then [latex]f[/latex] has an absolute maximum and an absolute minimum

Fermat’s theorem

if [latex]f[/latex] has a local extremum at [latex]c[/latex], then [latex]c[/latex] is a critical point of [latex]f[/latex]

first derivative test

let [latex]f[/latex] be a continuous function over an interval [latex]I[/latex] containing a critical point [latex]c[/latex] such that [latex]f[/latex] is differentiable over [latex]I[/latex] except possibly at [latex]c[/latex]; if [latex]f^{\prime}[/latex] changes sign from positive to negative as [latex]x[/latex] increases through [latex]c[/latex], then [latex]f[/latex] has a local maximum at [latex]c[/latex]; if [latex]f^{\prime}[/latex] changes sign from negative to positive as [latex]x[/latex] increases through [latex]c[/latex], then [latex]f[/latex] has a local minimum at [latex]c[/latex]; if [latex]f^{\prime}[/latex] does not change sign as [latex]x[/latex] increases through [latex]c[/latex], then [latex]f[/latex] does not have a local extremum at [latex]c[/latex]

frustum

a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base

function

a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

fundamental theorem of calculus

the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration

fundamental theorem of calculus, part 1

uses a definite integral to define an antiderivative of a function

fundamental theorem of calculus, part 2

(also, evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting

graph of a function

the set of points [latex](x,y)[/latex] such that [latex]x[/latex] is in the domain of [latex]f[/latex] and [latex]y=f(x)[/latex]

half-life

if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\frac{(\text{ln}2)}{k}[/latex]

higher-order derivative

a derivative of a derivative, from the second derivative to the [latex]n[/latex]th derivative, is called a higher-order derivative

Hooke’s law

this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, [latex]F=kx,[/latex] where [latex]k[/latex] is a constant

horizontal asymptote

if [latex]\underset{x\to \infty }{\lim}f(x)=L[/latex] or [latex]\underset{x\to −\infty }{\lim}f(x)=L[/latex], then [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex]

horizontal line test

a function [latex]f[/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[/latex], at most, once

hydrostatic pressure

the pressure exerted by water on a submerged object

hyperbolic functions

the functions denoted [latex]\sinh, \, \cosh, \, \tanh, \, \text{csch}, \, \text{sech}[/latex], and [latex]\coth[/latex], which involve certain combinations of [latex]e^x[/latex] and [latex]e^{−x}[/latex]

implicit differentiation

is a technique for computing [latex]\frac{dy}{dx}[/latex] for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable [latex]y[/latex] as a function) and solving for [latex]\frac{dy}{dx}[/latex]

increasing on the interval [latex]I[/latex]

a function increasing on the interval [latex]I[/latex] if for all [latex]x_1, \, x_2\in I, \, f(x_1)\le f(x_2)[/latex] if [latex]x_1

indefinite integral

the most general antiderivative of [latex]f(x)[/latex] is the indefinite integral of [latex]f[/latex]; we use the notation [latex]\displaystyle\int f(x) dx[/latex] to denote the indefinite integral of [latex]f[/latex]

independent variable

the input variable for a function

indeterminate forms

when evaluating a limit, the forms [latex]0/0[/latex], [latex]\infty / \infty[/latex], [latex]0 \cdot \infty[/latex], [latex]\infty -\infty[/latex], [latex]0^0[/latex], [latex]\infty^0[/latex], and [latex]1^{\infty}[/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is

infinite discontinuity

An infinite discontinuity occurs at a point [latex]a[/latex] if [latex]\underset{x\to a^-}{\lim}f(x)=\pm \infty[/latex] or [latex]\underset{x\to a^+}{\lim}f(x)=\pm \infty[/latex]

infinite limit

A function has an infinite limit at a point [latex]a[/latex] if it either increases or decreases without bound as it approaches [latex]a[/latex]

infinite limit at infinity

a function that becomes arbitrarily large as [latex]x[/latex] becomes large

inflection point

if [latex]f[/latex] is continuous at [latex]c[/latex] and [latex]f[/latex] changes concavity at [latex]c[/latex], the point [latex](c,f(c))[/latex] is an inflection point of [latex]f[/latex]

initial value problem

a problem that requires finding a function [latex]y[/latex] that satisfies the differential equation [latex]\frac{dy}{dx}=f(x)[/latex] together with the initial condition [latex]y(x_0)=y_0[/latex]

instantaneous rate of change

the rate of change of a function at any point along the function [latex]a[/latex], also called [latex]f^{\prime}(a)[/latex], or the derivative of the function at [latex]a[/latex]

integrable function

a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as [latex]n[/latex] goes to infinity exists

integrand

the function to the right of the integration symbol; the integrand includes the function being integrated

integration by substitution

a technique for integration that allows integration of functions that are the result of a chain-rule derivative

Intermediate Value Theorem

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

intuitive definition of the limit

If all values of the function [latex]f(x)[/latex] approach the real number [latex]L[/latex] as the values of [latex]x(\ne a)[/latex] approach [latex]a[/latex], [latex]f(x)[/latex] approaches [latex]L[/latex]

inverse function

for a function [latex]f[/latex], the inverse function [latex]f^{-1}[/latex] satisfies [latex]f^{-1}(y)=x[/latex] if [latex]f(x)=y[/latex]

inverse hyperbolic functions

the inverses of the hyperbolic functions where [latex]\cosh[/latex] and [latex]\text{sech}[/latex] are restricted to the domain [latex][0,\infty)[/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function

inverse trigonometric functions

the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

iterative process

process in which a list of numbers [latex]x_0,x_1,x_2,x_3, \cdots[/latex] is generated by starting with a number [latex]x_0[/latex] and defining [latex]x_n=F(x_{n-1})[/latex] for [latex]n \ge 1[/latex]

jump discontinuity

A jump discontinuity occurs at a point [latex]a[/latex] if [latex]\underset{x\to a^-}{\lim}f(x)[/latex] and [latex]\underset{x\to a^+}{\lim}f(x)[/latex] both exist, but [latex]\underset{x\to a^-}{\lim}f(x) \ne \underset{x\to a^+}{\lim}f(x)[/latex]

lamina

a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional

left-endpoint approximation

an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle

L’Hôpital’s rule

if [latex]f[/latex] and [latex]g[/latex] are differentiable functions over an interval [latex]a[/latex], except possibly at [latex]a[/latex], and [latex]\underset{x\to a}{\lim} f(x)=0=\underset{x\to a}{\lim} g(x)[/latex] or [latex]\underset{x\to a}{\lim} f(x)[/latex] and [latex]\underset{x\to a}{\lim} g(x)[/latex] are infinite, then [latex]\underset{x\to a}{\lim}\dfrac{f(x)}{g(x)}=\underset{x\to a}{\lim}\dfrac{f^{\prime}(x)}{g^{\prime}(x)}[/latex], assuming the limit on the right exists or is [latex]\infty[/latex] or [latex]−\infty[/latex]

limit at infinity

the limiting value, if it exists, of a function as [latex]x\to \infty[/latex] or [latex]x\to −\infty[/latex]

limits of integration

these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated

linear approximation

the linear function [latex]L(x)=f(a)+f^{\prime}(a)(x-a)[/latex] is the linear approximation of [latex]f[/latex] at [latex]x=a[/latex]

linear function

a function that can be written in the form [latex]f(x)=mx+b[/latex]

local extremum

if [latex]f[/latex] has a local maximum or local minimum at [latex]c[/latex], we say [latex]f[/latex] has a local extremum at [latex]c[/latex]

local maximum

if there exists an interval [latex]I[/latex] such that [latex]f(c)\ge f(x)[/latex] for all [latex]x\in I[/latex], we say [latex]f[/latex] has a local maximum at [latex]c[/latex]

local minimum

if there exists an interval [latex]I[/latex] such that [latex]f(c)\le f(x)[/latex] for all [latex]x\in I[/latex], we say [latex]f[/latex] has a local minimum at [latex]c[/latex]

logarithmic differentiation

is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly

logarithmic function

a function of the form [latex]f(x)=\log_b(x)[/latex] for some base [latex]b>0, \, b \ne 1[/latex] such that [latex]y=\log_b(x)[/latex] if and only if [latex]b^y=x[/latex]

lower sum

a sum obtained by using the minimum value of [latex]f(x)[/latex] on each subinterval

marginal cost

is the derivative of the cost function, or the approximate cost of producing one more item

marginal revenue

is the derivative of the revenue function, or the approximate revenue obtained by selling one more item

marginal profit

is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item

mathematical model

A method of simulating real-life situations with mathematical equations

mean value theorem

if [latex]f[/latex] is continuous over [latex][a,b][/latex] and differentiable over [latex](a,b)[/latex], then there exists [latex]c \in (a,b)[/latex] such that

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex]

mean value theorem for integrals

guarantees that a point [latex]c[/latex] exists such that [latex]f(c)[/latex] is equal to the average value of the function

method of cylindrical shells

a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable

moment

if [latex]n[/latex] masses are arranged on a number line, the moment of the system with respect to the origin is given by [latex]M=\displaystyle\sum_{i=1}{n} {m}_{i}{x}_{i};[/latex] if, instead, we consider a region in the plane, bounded above by a function [latex]f(x)[/latex] over an interval [latex]\left[a,b\right],[/latex] then the moments of the region with respect to the [latex]x[/latex]– and [latex]y[/latex]-axes are given by [latex]{M}_{x}=\rho {\displaystyle\int }_{a}^{b}\frac{{\left[f(x)\right]}^{2}}{2}dx[/latex] and [latex]{M}_{y}=\rho {\displaystyle\int }_{a}^{b}xf(x)dx,[/latex] respectively

natural exponential function

the function [latex]f(x)=e^x[/latex]

natural logarithm

the function [latex]\ln x=\log_e x[/latex]

net change theorem

if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity

net signed area

the area between a function and the [latex]x[/latex]-axis such that the area below the [latex]x[/latex]-axis is subtracted from the area above the [latex]x[/latex]-axis; the result is the same as the definite integral of the function

Newton’s method

method for approximating roots of [latex]f(x)=0[/latex]; using an initial guess [latex]x_0[/latex], each subsequent approximation is defined by the equation [latex]x_n=x_{n-1}-\dfrac{f(x_{n-1})}{f^{\prime}(x_{n-1})}[/latex]

number e

as [latex]m[/latex] gets larger, the quantity [latex](1+(1/m))^m[/latex] gets closer to some real number; we define that real number to be [latex]e[/latex]; the value of [latex]e[/latex] is approximately 2.718282

oblique asymptote

the line [latex]y=mx+b[/latex] if [latex]f(x)[/latex] approaches it as [latex]x\to \infty[/latex] or [latex]x\to −\infty[/latex]

odd function

a function is odd if [latex]f(−x)=−f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]

one-sided limit

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

one-to-one function

a function [latex]f[/latex] is one-to-one if [latex]f(x_1) \ne f(x_2)[/latex] if [latex]x_1 \ne x_2[/latex]

optimization problems

problems that are solved by finding the maximum or minimum value of a function

partition

a set of points that divides an interval into subintervals

percentage error

the relative error expressed as a percentage

periodic function

a function is periodic if it has a repeating pattern as the values of [latex]x[/latex] move from left to right

piecewise-defined function

a function that is defined differently on different parts of its domain

point-slope equation

equation of a linear function indicating its slope and a point on the graph of the function

polynomial function

a function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \cdots +a_1x+a_0[/latex]

population growth rate

is the derivative of the population with respect to time

power function

a function of the form [latex]f(x)=x^n[/latex] for any positive integer [latex]n \ge 1[/latex]

power rule

the derivative of a power function is a function in which the power on [latex]x[/latex] becomes the coefficient of the term and the power on [latex]x[/latex] in the derivative decreases by 1: If [latex]n[/latex] is an integer, then [latex]\frac{d}{dx}(x^n)=nx^{n-1}[/latex]

product rule

the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: [latex]\frac{d}{dx}(f(x)g(x))=f^{\prime}(x)g(x)+g^{\prime}(x)f(x)[/latex]

propagated error

the error that results in a calculated quantity [latex]f(x)[/latex] resulting from a measurement error [latex]dx[/latex]

quadratic function

a polynomial of degree 2; that is, a function of the form [latex]f(x)=ax^2+bx+c[/latex] where [latex]a \ne 0[/latex]

quotient rule

the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: [latex]\frac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f^{\prime}(x)g(x)-g^{\prime}(x)f(x)}{(g(x))^2}[/latex]

radians

for a circular arc of length [latex]s[/latex] on a circle of radius 1, the radian measure of the associated angle [latex]\theta[/latex] is [latex]s[/latex]

range

the set of outputs for a function

rational function

a function of the form [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials

regular partition

a partition in which the subintervals all have the same width

related rates

are rates of change associated with two or more related quantities that are changing over time

relative error

given an absolute error [latex]\Delta q[/latex] for a particular quantity, [latex]\dfrac{\Delta q}{q}[/latex] is the relative error.

removable discontinuity

A removable discontinuity occurs at a point [latex]a[/latex] if [latex]f(x)[/latex] is discontinuous at [latex]a[/latex], but [latex]\underset{x\to a}{\lim}f(x)[/latex] exists

restricted domain

a subset of the domain of a function [latex]f[/latex]

riemann sum

an estimate of the area under the curve of the form [latex]A\approx \underset{i=1}{\overset{n}{\Sigma}}f(x_i^*)\Delta x[/latex]

right-endpoint approximation

the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle

rolle’s theorem

if [latex]f[/latex] is continuous over [latex][a,b][/latex] and differentiable over [latex](a,b)[/latex], and if [latex]f(a)=f(b)[/latex], then there exists [latex]c \in (a,b)[/latex] such that [latex]f^{\prime}(c)=0[/latex]

root function

a function of the form [latex]f(x)=x^{1/n}[/latex] for any integer [latex]n \ge 2[/latex]

second derivative test

suppose [latex]f^{\prime}(c)=0[/latex] and [latex]f^{\prime \prime}[/latex] is continuous over an interval containing [latex]c[/latex]; if [latex]f^{\prime \prime}(c)>0[/latex], then [latex]f[/latex] has a local minimum at [latex]c[/latex]; if [latex]f^{\prime \prime}(c)<0[/latex], then [latex]f[/latex] has a local maximum at [latex]c[/latex]; if [latex]f^{\prime \prime}(c)=0[/latex], then the test is inconclusive

sigma notation

(also, summation notation) the Greek letter sigma ([latex]\Sigma[/latex]) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it

slicing method

a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume

slope

the change in [latex]y[/latex] for each unit change in [latex]x[/latex]

slope-intercept form

equation of a linear function indicating its slope and [latex]y[/latex]-intercept

solid of revolution

a solid generated by revolving a region in a plane around a line in that plane

speed

is the absolute value of velocity, that is, [latex]|v(t)|[/latex] is the speed of an object at time [latex]t[/latex] whose velocity is given by [latex]v(t)[/latex]

standard form

equation of a linear function with both variable terms set equal to a constant, [latex]ax+by=c[/latex].

sum rule

the derivative of the sum of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the sum of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex]: [latex]\frac{d}{dx}(f(x)+g(x))=f^{\prime}(x)+g^{\prime}(x)[/latex]

surface area

the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces

symmetry about the origin

the graph of a function [latex]f[/latex] is symmetric about the origin if [latex](−x,−y)[/latex] is on the graph of [latex]f[/latex] whenever [latex](x,y)[/latex] is on the graph

symmetry about the [latex]y[/latex]-axis

the graph of a function [latex]f[/latex] is symmetric about the [latex]y[/latex]-axis if [latex](−x,y)[/latex] is on the graph of [latex]f[/latex] whenever [latex](x,y)[/latex] is on the graph

symmetry principle

the symmetry principle states that if a region R is symmetric about a line [latex]l[/latex], then the centroid of R lies on [latex]l[/latex]

table of values

a table containing a list of inputs and their corresponding outputs

tangent line approximation (linearization)

since the linear approximation of [latex]f[/latex] at [latex]x=a[/latex] is defined using the equation of the tangent line, the linear approximation of [latex]f[/latex] at [latex]x=a[/latex] is also known as the tangent line approximation to [latex]f[/latex] at [latex]x=a[/latex]

theorem of Pappus for volume

this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region

total area

total area between a function and the [latex]x[/latex]-axis is calculated by adding the area above the [latex]x[/latex]-axis and the area below the [latex]x[/latex]-axis; the result is the same as the definite integral of the absolute value of the function

transcendental function

a function that cannot be expressed by a combination of basic arithmetic operations

transformation of a function

a shift, scaling, or reflection of a function

triangle inequality

If [latex]a[/latex] and [latex]b[/latex] are any real numbers, then [latex]|a+b|\le |a|+|b|[/latex]

trigonometric functions

functions of an angle defined as ratios of the lengths of the sides of a right triangle

trigonometric identity

an equation involving trigonometric functions that is true for all angles [latex]\theta[/latex] for which the functions in the equation are defined

upper sum

a sum obtained by using the maximum value of [latex]f(x)[/latex] on each subinterval

variable of integration

indicates which variable you are integrating with respect to; if it is [latex]x[/latex], then the function in the integrand is followed by [latex]dx[/latex]

vertical asymptote

A function has a vertical asymptote at [latex]x=a[/latex] if the limit as [latex]x[/latex] approaches [latex]a[/latex] from the right or left is infinite

vertical line test

given the graph of a function, every vertical line intersects the graph, at most, once

washer method

a special case of the slicing method used with solids of revolution when the slices are washers

work

the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance

zeros of a function

when a real number [latex]x[/latex] is a zero of a function [latex]f[/latex], [latex]f(x)=0[/latex]