- 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
*i*th 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] and diverges 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
*n*th 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