{"id":369,"date":"2024-10-01T18:45:50","date_gmt":"2024-10-01T18:45:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/glossary-of-terms\/"},"modified":"2025-12-17T15:18:37","modified_gmt":"2025-12-17T15:18:37","slug":"glossary-of-terms","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/glossary-of-terms\/","title":{"raw":"Glossary of Terms","rendered":"Glossary of Terms"},"content":{"raw":"\n
\n \t
absolute convergence<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
absolute error<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
alternating series<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
alternating series test<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
angular coordinate<\/dt>\n \t
[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<\/em>) axis, measured counterclockwise<\/dd>\n<\/dl>\n
\n \t
arc length<\/dt>\n \t
the arc length of a curve can be thought of as the distance a person would travel along the path of the curve<\/dd>\n<\/dl>\n
\n \t
arithmetic sequence<\/dt>\n \t
a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence<\/dd>\n<\/dl>\n
\n \t
asymptotically semi-stable solution<\/dt>\n \t
[latex]y=k[\/latex] if it is neither asymptotically stable nor asymptotically unstable<\/dd>\n<\/dl>\n
\n \t
asymptotically stable solution<\/dt>\n \t
[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<\/dd>\n<\/dl>\n
\n \t
asymptotically unstable solution<\/dt>\n \t
[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<\/dd>\n<\/dl>\n
\n \t
autonomous differential equation<\/dt>\n \t
an equation in which the right-hand side is a function of [latex]y[\/latex] alone<\/dd>\n<\/dl>\n
\n \t
average value of a function<\/dt>\n \t
(or [latex]f_{\\text{ave}}[\/latex]<\/strong>) 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<\/dd>\n<\/dl>\n
\n \t
binomial series<\/dt>\n \t
the Maclaurin series for [latex]f\\left(x\\right)={\\left(1+x\\right)}^{r}[\/latex]; it is given by\n<\/span>\n[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]<\/dd>\n<\/dl>\n
\n \t
bounded above<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
bounded below<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
bounded sequence<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
cardioid<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
carrying capacity<\/dt>\n \t
the maximum population of an organism that the environment can sustain indefinitely<\/dd>\n<\/dl>\n
\n \t
catenary<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
center of mass<\/dt>\n \t
the point at which the total mass of the system could be concentrated without changing the moment<\/dd>\n<\/dl>\n
\n \t
centroid<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
change of variables<\/dt>\n \t
the substitution of a variable, such as [latex]u[\/latex], for an expression in the integrand<\/dd>\n<\/dl>\n
\n \t
comparison test<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
computer algebra system (CAS)<\/dt>\n \t
technology used to perform many mathematical tasks, including integration<\/dd>\n<\/dl>\n
\n \t
conditional convergence<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
conic section<\/dt>\n \t
a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\n<\/dl>\n
\n \t
convergence of a series<\/dt>\n \t
a series converges if the sequence of partial sums for that series converges<\/dd>\n<\/dl>\n
\n \t
convergent sequence<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
cross-section<\/dt>\n \t
the intersection of a plane and a solid object<\/dd>\n<\/dl>\n
\n \t
cusp<\/dt>\n \t
a pointed end or part where two curves meet<\/dd>\n<\/dl>\n
\n \t
cycloid<\/dt>\n \t
the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\n<\/dl>\n
\n \t
definite integral<\/dt>\n \t
a primary operation of calculus; the area between the curve and the [latex]x[\/latex]-axis over a given interval is a definite integral<\/dd>\n<\/dl>\n
\n \t
density function<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
differential equation<\/dt>\n \t
an equation involving a function [latex]y=y\\left(x\\right)[\/latex] and one or more of its derivatives<\/dd>\n<\/dl>\n
\n \t
direction field (slope field)<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
directrix<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
discriminant<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
disk method<\/dt>\n \t
a special case of the slicing method used with solids of revolution when the slices are disks<\/dd>\n<\/dl>\n
\n \t
divergence of a series<\/dt>\n \t
a series diverges if the sequence of partial sums for that series diverges<\/dd>\n<\/dl>\n
\n \t
divergence test<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
divergent sequence<\/dt>\n \t
a sequence that is not convergent is divergent<\/dd>\n<\/dl>\n
\n \t
doubling time<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
eccentricity<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
equilibrium solution<\/dt>\n \t
any solution to the differential equation of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/dd>\n<\/dl>\n
\n \t
Euler\u2019s Method<\/dt>\n \t
a numerical technique used to approximate solutions to an initial-value problem<\/dd>\n<\/dl>\n
\n \t
explicit formula<\/dt>\n \t
a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\\left(n\\right)[\/latex]<\/dd>\n<\/dl>\n
\n \t
exponential decay<\/dt>\n \t
systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}[\/latex]<\/dd>\n<\/dl>\n
\n \t
exponential growth<\/dt>\n \t
systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[\/latex]<\/dd>\n<\/dl>\n
\n \t
focal parameter<\/dt>\n \t
the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\n<\/dl>\n
\n \t
focus<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
frustum<\/dt>\n \t
a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base<\/dd>\n<\/dl>\n
\n \t
fundamental theorem of calculus<\/dt>\n \t
the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration<\/dd>\n<\/dl>\n
\n \t
fundamental theorem of calculus, part 1<\/dt>\n \t
uses a definite integral to define an antiderivative of a function<\/dd>\n<\/dl>\n
\n \t
fundamental theorem of calculus, part 2<\/dt>\n \t
(also, evaluation theorem<\/strong>) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting<\/dd>\n<\/dl>\n
\n \t
general form<\/dt>\n \t
an equation of a conic section written as a general second-degree equation<\/dd>\n<\/dl>\n
\n \t
general solution (or family of solutions)<\/dt>\n \t
the entire set of solutions to a given differential equation<\/dd>\n<\/dl>\n
\n \t
geometric sequence<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
geometric series<\/dt>\n \t
a geometric series is a series that can be written in the form\n<\/span>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\\cdots [\/latex]<\/div><\/dd>\n<\/dl>\n
\n \t
growth rate<\/dt>\n \t
the constant [latex]r>0[\/latex] in the exponential growth function [latex]P\\left(t\\right)={P}_{0}{e}^{rt}[\/latex]<\/dd>\n<\/dl>\n
\n \t
half-life<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
harmonic series<\/dt>\n \t
the harmonic series takes the form\n<\/span>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots [\/latex]<\/div><\/dd>\n<\/dl>\n
\n \t
Hooke\u2019s law<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
hydrostatic pressure<\/dt>\n \t
the pressure exerted by water on a submerged object<\/dd>\n<\/dl>\n
\n \t
improper integral<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
index variable<\/dt>\n \t
the subscript used to define the terms in a sequence is called the index<\/dd>\n<\/dl>\n
\n \t
infinite series<\/dt>\n \t
an infinite series is an expression of the form\n<\/span>\n
[latex]{a}_{1}+{a}_{2}+{a}_{3}+\\cdots =\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex]<\/div><\/dd>\n<\/dl>\n
\n \t
initial population<\/dt>\n \t
the population at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n
\n \t
initial value(s)<\/dt>\n \t
a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable<\/dd>\n<\/dl>\n
\n \t
initial velocity<\/dt>\n \t
the velocity at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n
\n \t
initial-value problem<\/dt>\n \t
a differential equation together with an initial value or values<\/dd>\n<\/dl>\n
\n \t
integrable function<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
integral test<\/dt>\n \t
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\n<\/span>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ and }{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n
either both converge or both diverge<\/div><\/dd>\n<\/dl>\n
\n \t
integrand<\/dt>\n \t
the function to the right of the integration symbol; the integrand includes the function being integrated<\/dd>\n<\/dl>\n
\n \t
integrating factor<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
integration by parts<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
integration by substitution<\/dt>\n \t
a technique for integration that allows integration of functions that are the result of a chain-rule derivative<\/dd>\n<\/dl>\n
\n \t
integration table<\/dt>\n \t
a table that lists integration formulas<\/dd>\n<\/dl>\n
\n \t
interval of convergence<\/dt>\n \t
the set of real numbers x<\/em> for which a power series converges<\/dd>\n<\/dl>\n
\n \t
lamina<\/dt>\n \t
a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional<\/dd>\n<\/dl>\n
\n \t
left-endpoint approximation<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
lima\u00e7on<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
limit comparison test<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
limit of a sequence<\/dt>\n \t
the real number [latex]L[\/latex] to which a sequence converges is called the limit of the sequence<\/dd>\n<\/dl>\n
\n \t
limits of integration<\/dt>\n \t
these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated<\/dd>\n<\/dl>\n
\n \t
linear<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
logistic differential equation<\/dt>\n \t
a differential equation that incorporates the carrying capacity [latex]K[\/latex] and growth rate [latex]r[\/latex] into a population model<\/dd>\n<\/dl>\n
\n \t
lower sum<\/dt>\n \t
a sum obtained by using the minimum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n
\n \t
Maclaurin polynomial<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
Maclaurin series<\/dt>\n \t
a Taylor series for a function [latex]f[\/latex] at [latex]x=0[\/latex] is known as a Maclaurin series for [latex]f[\/latex]<\/dd>\n<\/dl>\n
\n \t
major axis<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
mean value theorem for integrals<\/dt>\n \t
guarantees that a point [latex]c[\/latex] exists such that [latex]f(c)[\/latex] is equal to the average value of the function<\/dd>\n<\/dl>\n
\n \t
method of cylindrical shells<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
midpoint rule<\/dt>\n \t
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<\/em>th subinterval to approximate [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex]<\/dd>\n<\/dl>\n
\n \t
minor axis<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
moment<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
monotone sequence<\/dt>\n \t
an increasing or decreasing sequence<\/dd>\n<\/dl>\n
\n \t
nappe<\/dt>\n \t
a nappe is one half of a double cone<\/dd>\n<\/dl>\n
\n \t
net change theorem<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
net signed area<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
nonelementary integral<\/dt>\n \t
an integral for which the antiderivative of the integrand cannot be expressed as an elementary function<\/dd>\n<\/dl>\n
\n \t
numerical integration<\/dt>\n \t
the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson\u2019s rule<\/dd>\n<\/dl>\n
\n \t
order of a differential equation<\/dt>\n \t
the highest order of any derivative of the unknown function that appears in the equation<\/dd>\n<\/dl>\n
\n \t
orientation<\/dt>\n \t
the direction that a point moves on a graph as the parameter increases<\/dd>\n<\/dl>\n
\n \t
p<\/em>-series<\/dt>\n \t
a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex]<\/dd>\n<\/dl>\n
\n \t
parameter<\/dt>\n \t
an independent variable that both x<\/em> and y<\/em> depend on in a parametric curve; usually represented by the variable t<\/em><\/dd>\n<\/dl>\n
\n \t
parametric curve<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
parametric equations<\/dt>\n \t
the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\n<\/dl>\n
\n \t
parameterization of a curve<\/dt>\n \t
rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\n<\/dl>\n
\n \t
partial fraction decomposition<\/dt>\n \t
a technique used to break down a rational function into the sum of simple rational functions<\/dd>\n<\/dl>\n
\n \t
partial sum<\/dt>\n \t
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\n<\/span>\n
[latex]{S}_{k}=\\displaystyle\\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots +{a}_{k}[\/latex]<\/div><\/dd>\n<\/dl>\n
\n \t
particular solution<\/dt>\n \t
member of a family of solutions to a differential equation that satisfies a particular initial condition<\/dd>\n<\/dl>\n
\n \t
partition<\/dt>\n \t
a set of points that divides an interval into subintervals<\/dd>\n<\/dl>\n
\n \t
phase line<\/dt>\n \t
a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions<\/dd>\n<\/dl>\n
\n \t
polar axis<\/dt>\n \t
the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\n<\/dl>\n
\n \t
polar coordinate system<\/dt>\n \t
a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta [\/latex], the angular coordinate<\/dd>\n<\/dl>\n
\n \t
polar equation<\/dt>\n \t
an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\n<\/dl>\n
\n \t
pole<\/dt>\n \t
the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\n<\/dl>\n
\n \t
power reduction formula<\/dt>\n \t
a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power<\/dd>\n<\/dl>\n
\n \t
power series<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
radial coordinate<\/dt>\n \t
[latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\n<\/dl>\n
\n \t
radius of convergence<\/dt>\n \t
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<\/em> 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<\/em>, the radius of convergence is [latex]R=\\infty [\/latex]<\/dd>\n<\/dl>\n
\n \t
ratio test<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
recurrence relation<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
regular partition<\/dt>\n \t
a partition in which the subintervals all have the same width<\/dd>\n<\/dl>\n
\n \t
relative error<\/dt>\n \t
error as a percentage of the absolute value, given by [latex]|\\frac{A-B}{A}|=|\\frac{A-B}{A}|\\cdot 100\\text{%}[\/latex]<\/dd>\n<\/dl>\n
\n \t
remainder estimate<\/dt>\n \t
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:\n<\/span>\n
[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div><\/dd>\n<\/dl>\n
\n \t
riemann sum<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
right-endpoint approximation<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
root test<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
rose<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
separable differential equation<\/dt>\n \t
any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex]<\/dd>\n<\/dl>\n
\n \t
separation of variables<\/dt>\n \t
a method used to solve a separable differential equation<\/dd>\n<\/dl>\n
\n \t
sequence<\/dt>\n \t
an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\\text{,}\\ldots[\/latex] is a sequence<\/dd>\n<\/dl>\n
\n \t
sigma notation<\/dt>\n \t
(also, summation notation<\/strong>) 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<\/dd>\n<\/dl>\n
\n \t
Simpson\u2019s rule<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
slicing method<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
solid of revolution<\/dt>\n \t
a solid generated by revolving a region in a plane around a line in that plane<\/dd>\n<\/dl>\n
\n \t
solution curve<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
solution to a differential equation<\/dt>\n \t
a function [latex]y=f\\left(x\\right)[\/latex] that satisfies a given differential equation<\/dd>\n<\/dl>\n
\n \t
space-filling curve<\/dt>\n \t
a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\n<\/dl>\n
\n \t
standard form<\/dt>\n \t
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]<\/dd><\/dl>\n
\n \t
standard form<\/dt>\n \t
an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\n<\/dl>\n
\n \t
step size<\/dt>\n \t
the increment [latex]h[\/latex] that is added to the [latex]x[\/latex] value at each step in Euler\u2019s Method<\/dd>\n<\/dl>\n
\n \t
surface area<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
symmetry principle<\/dt>\n \t
the symmetry principle states that if a region R<\/em> is symmetric about a line [latex]l[\/latex], then the centroid of R<\/em> lies on [latex]l[\/latex]<\/dd>\n<\/dl>\n
\n \t
Taylor polynomials<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
Taylor series<\/dt>\n \t
a power series at [latex]a[\/latex] that converges to a function [latex]f[\/latex] on some open interval containing [latex]a[\/latex]<\/dd>\n<\/dl>\n
\n \t
Taylor\u2019s theorem with remainder<\/dt>\n \t
for a function [latex]f[\/latex] and the n<\/em>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] \n<\/span>\nfor 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]<\/dd>\n<\/dl>\n
\n \t
telescoping series<\/dt>\n \t
a telescoping series is one in which most of the terms cancel in each of the partial sums<\/dd>\n<\/dl>\n
\n \t
term<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
term-by-term differentiation of a power series<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
term-by-term integration of a power series<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
theorem of Pappus for volume<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
threshold population<\/dt>\n \t
the minimum population that is necessary for a species to survive<\/dd>\n<\/dl>\n
\n \t
total area<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
trigonometric integral<\/dt>\n \t
an integral involving powers and products of trigonometric functions<\/dd>\n<\/dl>\n
\n \t
trigonometric substitution<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
unbounded sequence<\/dt>\n \t
a sequence that is not bounded is called unbounded<\/dd>\n<\/dl>\n
\n \t
upper sum<\/dt>\n \t
a sum obtained by using the maximum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n
\n \t
variable of integration<\/dt>\n \t
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]<\/dd>\n<\/dl>\n
\n \t
vertex<\/dt>\n \t
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<\/dd>\n<\/dl>\n
\n \t
washer method<\/dt>\n \t
a special case of the slicing method used with solids of revolution when the slices are washers<\/dd>\n<\/dl>\n
\n \t
work<\/dt>\n \t
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<\/dd>\n<\/dl>\n","rendered":"
\n
absolute convergence<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
absolute error<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
alternating series<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
alternating series test<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
angular coordinate<\/dt>\n
[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<\/em>) axis, measured counterclockwise<\/dd>\n<\/dl>\n
\n
arc length<\/dt>\n
the arc length of a curve can be thought of as the distance a person would travel along the path of the curve<\/dd>\n<\/dl>\n
\n
arithmetic sequence<\/dt>\n
a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence<\/dd>\n<\/dl>\n
\n
asymptotically semi-stable solution<\/dt>\n
[latex]y=k[\/latex] if it is neither asymptotically stable nor asymptotically unstable<\/dd>\n<\/dl>\n
\n
asymptotically stable solution<\/dt>\n
[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<\/dd>\n<\/dl>\n
\n
asymptotically unstable solution<\/dt>\n
[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<\/dd>\n<\/dl>\n
\n
autonomous differential equation<\/dt>\n
an equation in which the right-hand side is a function of [latex]y[\/latex] alone<\/dd>\n<\/dl>\n
\n
average value of a function<\/dt>\n
(or [latex]f_{\\text{ave}}[\/latex]<\/strong>) 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<\/dd>\n<\/dl>\n
\n
binomial series<\/dt>\n
the Maclaurin series for [latex]f\\left(x\\right)={\\left(1+x\\right)}^{r}[\/latex]; it is given by
\n<\/span>
\n[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]<\/dd>\n<\/dl>\n
\n
bounded above<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
bounded below<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
bounded sequence<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
cardioid<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
carrying capacity<\/dt>\n
the maximum population of an organism that the environment can sustain indefinitely<\/dd>\n<\/dl>\n
\n
catenary<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
center of mass<\/dt>\n
the point at which the total mass of the system could be concentrated without changing the moment<\/dd>\n<\/dl>\n
\n
centroid<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
change of variables<\/dt>\n
the substitution of a variable, such as [latex]u[\/latex], for an expression in the integrand<\/dd>\n<\/dl>\n
\n
comparison test<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
computer algebra system (CAS)<\/dt>\n
technology used to perform many mathematical tasks, including integration<\/dd>\n<\/dl>\n
\n
conditional convergence<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
conic section<\/dt>\n
a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\n<\/dl>\n
\n
convergence of a series<\/dt>\n
a series converges if the sequence of partial sums for that series converges<\/dd>\n<\/dl>\n
\n
convergent sequence<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
cross-section<\/dt>\n
the intersection of a plane and a solid object<\/dd>\n<\/dl>\n
\n
cusp<\/dt>\n
a pointed end or part where two curves meet<\/dd>\n<\/dl>\n
\n
cycloid<\/dt>\n
the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\n<\/dl>\n
\n
definite integral<\/dt>\n
a primary operation of calculus; the area between the curve and the [latex]x[\/latex]-axis over a given interval is a definite integral<\/dd>\n<\/dl>\n
\n
density function<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
differential equation<\/dt>\n
an equation involving a function [latex]y=y\\left(x\\right)[\/latex] and one or more of its derivatives<\/dd>\n<\/dl>\n
\n
direction field (slope field)<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
directrix<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
discriminant<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
disk method<\/dt>\n
a special case of the slicing method used with solids of revolution when the slices are disks<\/dd>\n<\/dl>\n
\n
divergence of a series<\/dt>\n
a series diverges if the sequence of partial sums for that series diverges<\/dd>\n<\/dl>\n
\n
divergence test<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
divergent sequence<\/dt>\n
a sequence that is not convergent is divergent<\/dd>\n<\/dl>\n
\n
doubling time<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
eccentricity<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
equilibrium solution<\/dt>\n
any solution to the differential equation of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/dd>\n<\/dl>\n
\n
Euler\u2019s Method<\/dt>\n
a numerical technique used to approximate solutions to an initial-value problem<\/dd>\n<\/dl>\n
\n
explicit formula<\/dt>\n
a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\\left(n\\right)[\/latex]<\/dd>\n<\/dl>\n
\n
exponential decay<\/dt>\n
systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}[\/latex]<\/dd>\n<\/dl>\n
\n
exponential growth<\/dt>\n
systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[\/latex]<\/dd>\n<\/dl>\n
\n
focal parameter<\/dt>\n
the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\n<\/dl>\n
\n
focus<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
frustum<\/dt>\n
a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base<\/dd>\n<\/dl>\n
\n
fundamental theorem of calculus<\/dt>\n
the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration<\/dd>\n<\/dl>\n
\n
fundamental theorem of calculus, part 1<\/dt>\n
uses a definite integral to define an antiderivative of a function<\/dd>\n<\/dl>\n
\n
fundamental theorem of calculus, part 2<\/dt>\n
(also, evaluation theorem<\/strong>) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting<\/dd>\n<\/dl>\n
\n
general form<\/dt>\n
an equation of a conic section written as a general second-degree equation<\/dd>\n<\/dl>\n
\n
general solution (or family of solutions)<\/dt>\n
the entire set of solutions to a given differential equation<\/dd>\n<\/dl>\n
\n
geometric sequence<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
geometric series<\/dt>\n
a geometric series is a series that can be written in the form
\n<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\\cdots[\/latex]<\/div>\n<\/dd>\n<\/dl>\n
\n
growth rate<\/dt>\n
the constant [latex]r>0[\/latex] in the exponential growth function [latex]P\\left(t\\right)={P}_{0}{e}^{rt}[\/latex]<\/dd>\n<\/dl>\n
\n
half-life<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
harmonic series<\/dt>\n
the harmonic series takes the form
\n<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots[\/latex]<\/div>\n<\/dd>\n<\/dl>\n
\n
Hooke\u2019s law<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
hydrostatic pressure<\/dt>\n
the pressure exerted by water on a submerged object<\/dd>\n<\/dl>\n
\n
improper integral<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
index variable<\/dt>\n
the subscript used to define the terms in a sequence is called the index<\/dd>\n<\/dl>\n
\n
infinite series<\/dt>\n
an infinite series is an expression of the form
\n<\/span><\/p>\n
[latex]{a}_{1}+{a}_{2}+{a}_{3}+\\cdots =\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}[\/latex]<\/div>\n<\/dd>\n<\/dl>\n
\n
initial population<\/dt>\n
the population at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n
\n
initial value(s)<\/dt>\n
a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable<\/dd>\n<\/dl>\n
\n
initial velocity<\/dt>\n
the velocity at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n
\n
initial-value problem<\/dt>\n
a differential equation together with an initial value or values<\/dd>\n<\/dl>\n
\n
integrable function<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
integral test<\/dt>\n
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
\n<\/span><\/p>\n
[latex]\\displaystyle\\sum _{n=1}^{\\infty }{a}_{n}\\text{ and }{\\displaystyle\\int }_{1}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n
either both converge or both diverge<\/div>\n<\/dd>\n<\/dl>\n
\n
integrand<\/dt>\n
the function to the right of the integration symbol; the integrand includes the function being integrated<\/dd>\n<\/dl>\n
\n
integrating factor<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
integration by parts<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
integration by substitution<\/dt>\n
a technique for integration that allows integration of functions that are the result of a chain-rule derivative<\/dd>\n<\/dl>\n
\n
integration table<\/dt>\n
a table that lists integration formulas<\/dd>\n<\/dl>\n
\n
interval of convergence<\/dt>\n
the set of real numbers x<\/em> for which a power series converges<\/dd>\n<\/dl>\n
\n
lamina<\/dt>\n
a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional<\/dd>\n<\/dl>\n
\n
left-endpoint approximation<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
lima\u00e7on<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
limit comparison test<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
limit of a sequence<\/dt>\n
the real number [latex]L[\/latex] to which a sequence converges is called the limit of the sequence<\/dd>\n<\/dl>\n
\n
limits of integration<\/dt>\n
these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated<\/dd>\n<\/dl>\n
\n
linear<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
logistic differential equation<\/dt>\n
a differential equation that incorporates the carrying capacity [latex]K[\/latex] and growth rate [latex]r[\/latex] into a population model<\/dd>\n<\/dl>\n
\n
lower sum<\/dt>\n
a sum obtained by using the minimum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n
\n
Maclaurin polynomial<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
Maclaurin series<\/dt>\n
a Taylor series for a function [latex]f[\/latex] at [latex]x=0[\/latex] is known as a Maclaurin series for [latex]f[\/latex]<\/dd>\n<\/dl>\n
\n
major axis<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
mean value theorem for integrals<\/dt>\n
guarantees that a point [latex]c[\/latex] exists such that [latex]f(c)[\/latex] is equal to the average value of the function<\/dd>\n<\/dl>\n
\n
method of cylindrical shells<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
midpoint rule<\/dt>\n
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<\/em>th subinterval to approximate [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex]<\/dd>\n<\/dl>\n
\n
minor axis<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
moment<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
monotone sequence<\/dt>\n
an increasing or decreasing sequence<\/dd>\n<\/dl>\n
\n
nappe<\/dt>\n
a nappe is one half of a double cone<\/dd>\n<\/dl>\n
\n
net change theorem<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
net signed area<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
nonelementary integral<\/dt>\n
an integral for which the antiderivative of the integrand cannot be expressed as an elementary function<\/dd>\n<\/dl>\n
\n
numerical integration<\/dt>\n
the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson\u2019s rule<\/dd>\n<\/dl>\n
\n
order of a differential equation<\/dt>\n
the highest order of any derivative of the unknown function that appears in the equation<\/dd>\n<\/dl>\n
\n
orientation<\/dt>\n
the direction that a point moves on a graph as the parameter increases<\/dd>\n<\/dl>\n
\n
p<\/em>-series<\/dt>\n
a series of the form [latex]\\displaystyle\\sum _{n=1}^{\\infty }\\frac{1}{{n}^{p}}[\/latex]<\/dd>\n<\/dl>\n
\n
parameter<\/dt>\n
an independent variable that both x<\/em> and y<\/em> depend on in a parametric curve; usually represented by the variable t<\/em><\/dd>\n<\/dl>\n
\n
parametric curve<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
parametric equations<\/dt>\n
the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\n<\/dl>\n
\n
parameterization of a curve<\/dt>\n
rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\n<\/dl>\n
\n
partial fraction decomposition<\/dt>\n
a technique used to break down a rational function into the sum of simple rational functions<\/dd>\n<\/dl>\n
\n
partial sum<\/dt>\n
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
\n<\/span><\/p>\n
[latex]{S}_{k}=\\displaystyle\\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\\cdots +{a}_{k}[\/latex]<\/div>\n<\/dd>\n<\/dl>\n
\n
particular solution<\/dt>\n
member of a family of solutions to a differential equation that satisfies a particular initial condition<\/dd>\n<\/dl>\n
\n
partition<\/dt>\n
a set of points that divides an interval into subintervals<\/dd>\n<\/dl>\n
\n
phase line<\/dt>\n
a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions<\/dd>\n<\/dl>\n
\n
polar axis<\/dt>\n
the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\n<\/dl>\n
\n
polar coordinate system<\/dt>\n
a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta[\/latex], the angular coordinate<\/dd>\n<\/dl>\n
\n
polar equation<\/dt>\n
an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\n<\/dl>\n
\n
pole<\/dt>\n
the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\n<\/dl>\n
\n
power reduction formula<\/dt>\n
a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power<\/dd>\n<\/dl>\n
\n
power series<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
radial coordinate<\/dt>\n
[latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\n<\/dl>\n
\n
radius of convergence<\/dt>\n
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<\/em> 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<\/em>, the radius of convergence is [latex]R=\\infty[\/latex]<\/dd>\n<\/dl>\n
\n
ratio test<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
recurrence relation<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
regular partition<\/dt>\n
a partition in which the subintervals all have the same width<\/dd>\n<\/dl>\n
\n
relative error<\/dt>\n
error as a percentage of the absolute value, given by [latex]|\\frac{A-B}{A}|=|\\frac{A-B}{A}|\\cdot 100\\text{%}[\/latex]<\/dd>\n<\/dl>\n
\n
remainder estimate<\/dt>\n
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:
\n<\/span><\/p>\n
[latex]{\\displaystyle\\int }_{N+1}^{\\infty }f\\left(x\\right)dx<{R}_{N}<{\\displaystyle\\int }_{N}^{\\infty }f\\left(x\\right)dx[\/latex]<\/div>\n<\/dd>\n<\/dl>\n
\n
riemann sum<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
right-endpoint approximation<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
root test<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
rose<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
separable differential equation<\/dt>\n
any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex]<\/dd>\n<\/dl>\n
\n
separation of variables<\/dt>\n
a method used to solve a separable differential equation<\/dd>\n<\/dl>\n
\n
sequence<\/dt>\n
an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\\text{,}\\ldots[\/latex] is a sequence<\/dd>\n<\/dl>\n
\n
sigma notation<\/dt>\n
(also, summation notation<\/strong>) 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<\/dd>\n<\/dl>\n
\n
Simpson\u2019s rule<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
slicing method<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
solid of revolution<\/dt>\n
a solid generated by revolving a region in a plane around a line in that plane<\/dd>\n<\/dl>\n
\n
solution curve<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
solution to a differential equation<\/dt>\n
a function [latex]y=f\\left(x\\right)[\/latex] that satisfies a given differential equation<\/dd>\n<\/dl>\n
\n
space-filling curve<\/dt>\n
a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\n<\/dl>\n
\n
standard form<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
standard form<\/dt>\n
an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\n<\/dl>\n
\n
step size<\/dt>\n
the increment [latex]h[\/latex] that is added to the [latex]x[\/latex] value at each step in Euler\u2019s Method<\/dd>\n<\/dl>\n
\n
surface area<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
symmetry principle<\/dt>\n
the symmetry principle states that if a region R<\/em> is symmetric about a line [latex]l[\/latex], then the centroid of R<\/em> lies on [latex]l[\/latex]<\/dd>\n<\/dl>\n
\n
Taylor polynomials<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
Taylor series<\/dt>\n
a power series at [latex]a[\/latex] that converges to a function [latex]f[\/latex] on some open interval containing [latex]a[\/latex]<\/dd>\n<\/dl>\n
\n
Taylor\u2019s theorem with remainder<\/dt>\n
for a function [latex]f[\/latex] and the n<\/em>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]
\n<\/span>
\nfor 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]<\/dd>\n<\/dl>\n
\n
telescoping series<\/dt>\n
a telescoping series is one in which most of the terms cancel in each of the partial sums<\/dd>\n<\/dl>\n
\n
term<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
term-by-term differentiation of a power series<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
term-by-term integration of a power series<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
theorem of Pappus for volume<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
threshold population<\/dt>\n
the minimum population that is necessary for a species to survive<\/dd>\n<\/dl>\n
\n
total area<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
trigonometric integral<\/dt>\n
an integral involving powers and products of trigonometric functions<\/dd>\n<\/dl>\n
\n
trigonometric substitution<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
unbounded sequence<\/dt>\n
a sequence that is not bounded is called unbounded<\/dd>\n<\/dl>\n
\n
upper sum<\/dt>\n
a sum obtained by using the maximum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n
\n
variable of integration<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
vertex<\/dt>\n
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<\/dd>\n<\/dl>\n
\n
washer method<\/dt>\n
a special case of the slicing method used with solids of revolution when the slices are washers<\/dd>\n<\/dl>\n
\n
work<\/dt>\n
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<\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":22,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/369"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/369\/revisions"}],"predecessor-version":[{"id":2510,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/369\/revisions\/2510"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/369\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=369"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=369"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=369"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}