absolute convergence

if the series [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] converges, the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is said to converge absolutely

absolute error

if [latex]B[/latex] is an estimate of some quantity having an actual value of [latex]A[/latex], then the absolute error is given by [latex]|A-B|[/latex]

alternating series

a series of the form [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}[/latex] or [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}[/latex], where [latex]{b}_{n}\ge 0[/latex], is called an alternating series

alternating series test

for an alternating series of either form, if [latex]{b}_{n+1}\le {b}_{n}[/latex] for all integers [latex]n\ge 1[/latex] and [latex]{b}_{n}\to 0[/latex], then an alternating series converges

angular coordinate

[latex]\theta[/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise

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

arithmetic sequence

a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence

asymptotically semi-stable solution

[latex]y=k[/latex] if it is neither asymptotically stable nor asymptotically unstable

asymptotically stable solution

[latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] approaches [latex]k[/latex] as [latex]x[/latex] approaches infinity

asymptotically unstable solution

[latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] never approaches [latex]k[/latex] as [latex]x[/latex] approaches infinity

autonomous differential equation

an equation in which the right-hand side is a function of [latex]y[/latex] alone

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

binomial series

the Maclaurin series for [latex]f\left(x\right)={\left(1+x\right)}^{r}[/latex]; it is given by

[latex]{\left(1+x\right)}^{r}=\displaystyle\sum _{n=0}^{\infty }\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right){x}^{n}=1+rx+\frac{r\left(r - 1\right)}{2\text{!}}{x}^{2}+\cdots +\frac{r\left(r - 1\right)\cdots \left(r-n+1\right)}{n\text{!}}{x}^{n}+\cdots[/latex] for [latex]|x|<1[/latex]

bounded above

a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded above if there exists a constant [latex]M[/latex] such that [latex]{a}_{n}\le M[/latex] for all positive integers [latex]n[/latex]

bounded below

a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded below if there exists a constant [latex]M[/latex] such that [latex]M\le {a}_{n}[/latex] for all positive integers [latex]n[/latex]

bounded sequence

a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded if there exists a constant [latex]M[/latex] such that [latex]|{a}_{n}|\le M[/latex] for all positive integers [latex]n[/latex]

cardioid

a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\left(1+\sin\theta \right)[/latex] or [latex]r=a\left(1+\cos\theta \right)[/latex]

carrying capacity

the maximum population of an organism that the environment can sustain indefinitely

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

change of variables

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

comparison test

if [latex]0\le {a}_{n}\le {b}_{n}[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges; if [latex]{a}_{n}\ge {b}_{n}\ge 0[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges

computer algebra system (CAS)

technology used to perform many mathematical tasks, including integration

conditional convergence

if the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges, but the series [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] diverges, the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is said to converge conditionally

conic section

a conic section is any curve formed by the intersection of a plane with a cone of two nappes

convergence of a series

a series converges if the sequence of partial sums for that series converges

convergent sequence

a convergent sequence is a sequence [latex]\left\{{a}_{n}\right\}[/latex] for which there exists a real number [latex]L[/latex] such that [latex]{a}_{n}[/latex] is arbitrarily close to [latex]L[/latex] as long as [latex]n[/latex] is sufficiently large

cross-section

the intersection of a plane and a solid object

cusp

a pointed end or part where two curves meet

cycloid

the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage

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

differential equation

an equation involving a function [latex]y=y\left(x\right)[/latex] and one or more of its derivatives

direction field (slope field)

a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point

directrix

a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two

discriminant

the value [latex]4AC-{B}^{2}[/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[/latex], is called a discriminant

disk method

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

divergence of a series

a series diverges if the sequence of partial sums for that series diverges

divergence test

if [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0[/latex], then the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges

divergent sequence

a sequence that is not convergent is divergent

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]

eccentricity

the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix

equilibrium solution

any solution to the differential equation of the form [latex]y=c[/latex], where [latex]c[/latex] is a constant

Euler’s Method

a numerical technique used to approximate solutions to an initial-value problem

explicit formula

a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\left(n\right)[/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]

focal parameter

the focal parameter is the distance from a focus of a conic section to the nearest directrix

focus

a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two

frustum

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

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

general form

an equation of a conic section written as a general second-degree equation

general solution (or family of solutions)

the entire set of solutions to a given differential equation

geometric sequence

a sequence [latex]\left\{{a}_{n}\right\}[/latex] in which the ratio [latex]\frac{{a}_{n+1}}{{a}_{n}}[/latex] is the same for all positive integers [latex]n[/latex] is called a geometric sequence

geometric series

a geometric series is a series that can be written in the form

[latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\cdots[/latex]

growth rate

the constant [latex]r>0[/latex] in the exponential growth function [latex]P\left(t\right)={P}_{0}{e}^{rt}[/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]

harmonic series

the harmonic series takes the form

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots[/latex]

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

hydrostatic pressure

the pressure exerted by water on a submerged object

improper integral

an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges

index variable

the subscript used to define the terms in a sequence is called the index

infinite series

an infinite series is an expression of the form

[latex]{a}_{1}+{a}_{2}+{a}_{3}+\cdots =\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex]

initial population

the population at time [latex]t=0[/latex]

initial value(s)

a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable

initial velocity

the velocity at time [latex]t=0[/latex]

initial-value problem

a differential equation together with an initial value or values

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

integral test

for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms [latex]{a}_{n}[/latex], if there exists a continuous, decreasing function [latex]f[/latex] such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], then

[latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{ and }{\displaystyle\int }_{1}^{\infty }f\left(x\right)dx[/latex]

either both converge or both diverge

integrand

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

integrating factor

any function [latex]f\left(x\right)[/latex] that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions

integration by parts

a technique of integration that allows the exchange of one integral for another using the formula [latex]{\displaystyle\int}udv=uv-{\displaystyle\int}vdu[/latex]

integration by substitution

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

integration table

a table that lists integration formulas

interval of convergence

the set of real numbers x for which a power series converges

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

limaçon

the graph of the equation [latex]r=a+b\sin\theta[/latex] or [latex]r=a+b\cos\theta[/latex]. If [latex]a=b[/latex] then the graph is a cardioid

limit comparison test

suppose [latex]{a}_{n},{b}_{n}\ge 0[/latex] for all [latex]n\ge 1[/latex]. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to L\ne 0[/latex], then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] both converge or both diverge; if [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to 0[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to \infty[/latex], and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges

limit of a sequence

the real number [latex]L[/latex] to which a sequence converges is called the limit of the sequence

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

description of a first-order differential equation that can be written in the form [latex]a\left(x\right){y}^{\prime }+b\left(x\right)y=c\left(x\right)[/latex]

logistic differential equation

a differential equation that incorporates the carrying capacity [latex]K[/latex] and growth rate [latex]r[/latex] into a population model

lower sum

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

Maclaurin polynomial

a Taylor polynomial centered at 0; the [latex]n[/latex]th Taylor polynomial for [latex]f[/latex] at 0 is the [latex]n[/latex]th Maclaurin polynomial for [latex]f[/latex]

Maclaurin series

a Taylor series for a function [latex]f[/latex] at [latex]x=0[/latex] is known as a Maclaurin series for [latex]f[/latex]

major axis

the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis

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

midpoint rule

a rule that uses a Riemann sum of the form [latex]{M}_{n}=\displaystyle\sum _{i=1}^{n}f\left({m}_{i}\right)\Delta x[/latex], where [latex]{m}_{i}[/latex] is the midpoint of the ith subinterval to approximate [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex]

minor axis

the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis

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

monotone sequence

an increasing or decreasing sequence

nappe

a nappe is one half of a double cone

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

nonelementary integral

an integral for which the antiderivative of the integrand cannot be expressed as an elementary function

numerical integration

the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule

order of a differential equation

the highest order of any derivative of the unknown function that appears in the equation

orientation

the direction that a point moves on a graph as the parameter increases

p-series

a series of the form [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}[/latex]

parameter

an independent variable that both x and y depend on in a parametric curve; usually represented by the variable t

parametric curve

the graph of the parametric equations [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] over an interval [latex]a\le t\le b[/latex] combined with the equations

parametric equations

the equations [latex]x=x\left(t\right)[/latex] and [latex]y=y\left(t\right)[/latex] that define a parametric curve

parameterization of a curve

rewriting the equation of a curve defined by a function [latex]y=f\left(x\right)[/latex] as parametric equations

partial fraction decomposition

a technique used to break down a rational function into the sum of simple rational functions

partial sum

the [latex]k\text{th}[/latex] partial sum of the infinite series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is the finite sum

[latex]{S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}[/latex]

particular solution

member of a family of solutions to a differential equation that satisfies a particular initial condition

partition

a set of points that divides an interval into subintervals

phase line

a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions

polar axis

the horizontal axis in the polar coordinate system corresponding to [latex]r\ge 0[/latex]

polar coordinate system

a system for locating points in the plane. The coordinates are [latex]r[/latex], the radial coordinate, and [latex]\theta[/latex], the angular coordinate

polar equation

an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system

pole

the central point of the polar coordinate system, equivalent to the origin of a Cartesian system

power reduction formula

a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power

power series

a series of the form [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}[/latex] is a power series centered at [latex]x=0[/latex]; a series of the form [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] is a power series centered at [latex]x=a[/latex]

radial coordinate

[latex]r[/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole

radius of convergence

if there exists a real number [latex]R>0[/latex] such that a power series centered at [latex]x=a[/latex] converges for [latex]|x-a|R[/latex], then R is the radius of convergence; if the power series only converges at [latex]x=a[/latex], the radius of convergence is [latex]R=0[/latex]; if the power series converges for all real numbers x, the radius of convergence is [latex]R=\infty[/latex]

ratio test

for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with nonzero terms, let [latex]\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|[/latex]; if [latex]0\le \rho <1[/latex], the series converges absolutely; if [latex]\rho >1[/latex], the series diverges; if [latex]\rho =1[/latex], the test is inconclusive

recurrence relation

a recurrence relation is a relationship in which a term [latex]{a}_{n}[/latex] in a sequence is defined in terms of earlier terms in the sequence

regular partition

a partition in which the subintervals all have the same width

relative error

error as a percentage of the absolute value, given by [latex]|\frac{A-B}{A}|=|\frac{A-B}{A}|\cdot 100\text{%}[/latex]

remainder estimate

for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms [latex]{a}_{n}[/latex] and a continuous, decreasing function [latex]f[/latex] such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], the remainder [latex]{R}_{N}=\displaystyle\sum _{n=1}^{\infty }{a}_{n}-\displaystyle\sum _{n=1}^{N}{a}_{n}[/latex] satisfies the following estimate:

[latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/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

root test

for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex], let [latex]\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}[/latex]; if [latex]0\le \rho <1[/latex], the series converges absolutely; if [latex]\rho >1[/latex], the series diverges; if [latex]\rho =1[/latex], the test is inconclusive

rose

graph of the polar equation [latex]r=a\cos{n}\theta[/latex] or [latex]r=a\sin{n}\theta[/latex] for a positive constant [latex]a[/latex] and an integer [latex]n \ge 2[/latex]

separable differential equation

any equation that can be written in the form [latex]y^{\prime} =f\left(x\right)g\left(y\right)[/latex]

separation of variables

a method used to solve a separable differential equation

sequence

an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\text{,}\ldots[/latex] is a sequence

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

Simpson’s rule

a rule that approximates [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex] using the integrals of a piecewise quadratic function. The approximation [latex]{S}_{n}[/latex] to [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex] is given by [latex]{S}_{n}=\frac{\Delta x}{3}\left(\begin{array}{c}f\left({x}_{0}\right)+4f\left({x}_{1}\right)+2f\left({x}_{2}\right)+4f\left({x}_{3}\right)+2f\left({x}_{4}\right)+4f\left({x}_{5}\right)\\ +\cdots +2f\left({x}_{n - 2}\right)+4f\left({x}_{n - 1}\right)+f\left({x}_{n}\right)\end{array}\right)[/latex] trapezoidal rule a rule that approximates [latex]{\displaystyle\int }_{a}^{b}f\left(x\right)dx[/latex] using trapezoids

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

solid of revolution

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

solution curve

a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field

solution to a differential equation

a function [latex]y=f\left(x\right)[/latex] that satisfies a given differential equation

space-filling curve

a curve that completely occupies a two-dimensional subset of the real plane

standard form

the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex]

standard form

an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes

step size

the increment [latex]h[/latex] that is added to the [latex]x[/latex] value at each step in Euler’s Method

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 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]

Taylor polynomials

the [latex]n[/latex]th Taylor polynomial for [latex]f[/latex] at [latex]x=a[/latex] is [latex]{p}_{n}\left(x\right)=f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)+\frac{f^{\prime\prime}\left(a\right)}{2\text{!}}{\left(x-a\right)}^{2}+\cdots +\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}[/latex]

Taylor series

a power series at [latex]a[/latex] that converges to a function [latex]f[/latex] on some open interval containing [latex]a[/latex]

Taylor’s theorem with remainder

for a function [latex]f[/latex] and the nth Taylor polynomial for [latex]f[/latex] at [latex]x=a[/latex], the remainder [latex]{R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)[/latex] satisfies [latex]{R}_{n}\left(x\right)=\frac{{f}^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}[/latex]

for some [latex]c[/latex] between [latex]x[/latex] and [latex]a[/latex]; if there exists an interval [latex]I[/latex] containing [latex]a[/latex] and a real number [latex]M[/latex] such that [latex]|{f}^{\left(n+1\right)}\left(x\right)|\le M[/latex] for all [latex]x[/latex] in [latex]I[/latex], then [latex]|{R}_{n}\left(x\right)|\le \frac{M}{\left(n+1\right)\text{!}}{|x-a|}^{n+1}[/latex]

telescoping series

a telescoping series is one in which most of the terms cancel in each of the partial sums

term

the number [latex]{a}_{n}[/latex] in the sequence [latex]\left\{{a}_{n}\right\}[/latex] is called the [latex]n\text{th}[/latex] term of the sequence

term-by-term differentiation of a power series

a technique for evaluating the derivative of a power series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] by evaluating the derivative of each term separately to create the new power series [latex]\displaystyle\sum _{n=1}^{\infty }n{c}_{n}{\left(x-a\right)}^{n - 1}[/latex]

term-by-term integration of a power series

a technique for integrating a power series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] by integrating each term separately to create the new power series [latex]C+\displaystyle\sum _{n=0}^{\infty }{c}_{n}\frac{{\left(x-a\right)}^{n+1}}{n+1}[/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

threshold population

the minimum population that is necessary for a species to survive

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

trigonometric integral

an integral involving powers and products of trigonometric functions

trigonometric substitution

an integration technique that converts an algebraic integral containing expressions of the form [latex]\sqrt{{a}^{2}-{x}^{2}}[/latex], [latex]\sqrt{{a}^{2}+{x}^{2}}[/latex], or [latex]\sqrt{{x}^{2}-{a}^{2}}[/latex] into a trigonometric integral

unbounded sequence

a sequence that is not bounded is called unbounded

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]

vertex

a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch

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