# Glossary

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Words (or words that have the same definition) | The definition is case sensitive | (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] | (Optional) Caption for Image | (Optional) External or Internal Link | (Optional) Source for Definition |
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(Eg. "Genetic, Hereditary, DNA ...") | (Eg. "Relating to genes or heredity") | The infamous double helix | https://bio.libretexts.org/ | CC-BY-SA; Delmar Larsen |

Word(s) | Definition | Image | Caption | Link | Source |
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zeros of a function | when a real number \(x\) is a zero of a function \(f,\;f(x)=0\) | ||||

zero vector | the vector with both initial point and terminal point \((0,0)\) | ||||

work done by a force | work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector \(\vecs{ F}\) and the displacement of an object by a vector \(\vecs{ s}\), then the work done by the force is the dot product of \(\vecs{ F}\) and \(\vecs{ s}\). | ||||

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 | ||||

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

vertical trace | the set of ordered triples \((c,y,z)\) that solves the equation \(f(c,y)=z\) for a given constant \(x=c\) or the set of ordered triples \((x,d,z)\) that solves the equation \(f(x,d)=z\) for a given constant \(y=d\) | ||||

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

vertical asymptote | A function has a vertical asymptote at \(x=a\) if the limit as \(x\) approaches \(a\) from the right or left is infinite | ||||

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 | ||||

velocity vector | the derivative of the position vector | ||||

vector-valued function | a function of the form \(\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}\) or \(\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}+h(t)\hat{\mathbf{k}}\),where the component functions \(f\), \(g\), and \(h\) are real-valued functions of the parameter \(t\). |
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vector sum | the sum of two vectors, \(\vecs{v}\) and \(\vecs{w}\), can be constructed graphically by placing the initial point of \(\vecs{w}\) at the terminal point of \(\vecs{v}\); then the vector sum \(\vecs{v}+\vecs{w}\) is the vector with an initial point that coincides with the initial point of \(\vecs{v}\), and with a terminal point that coincides with the terminal point of \(\vecs{w}\) | ||||

vector projection | the component of a vector that follows a given direction | ||||

vector parameterization | any representation of a plane or space curve using a vector-valued function | ||||

vector line integral | the vector line integral of vector field \(\vecs F\) along curve \(C\) is the integral of the dot product of \(\vecs F\) with unit tangent vector \(\vecs T\) of \(C\) with respect to arc length, \(∫_C \vecs F·\vecs T\, ds\); such an integral is defined in terms of a Riemann sum, similar to a single-variable integral | ||||

vector field | measured in \(ℝ^2\), an assignment of a vector \(\vecs{F}(x,y)\) to each point \((x,y)\) of a subset \(D\) of \(ℝ^2\); in \(ℝ^3\), an assignment of a vector \(\vecs{F}(x,y,z)\) to each point \((x,y,z)\) of a subset \(D\) of \(ℝ^3\) | ||||

vector equation of a plane | the equation \(\vecs n⋅\vecd{PQ}=0,\) where \(P\) is a given point in the plane, \(Q\) is any point in the plane, and \(\vecs n\) is a normal vector of the plane | ||||

vector equation of a line | the equation \(\vecs r=\vecs r_0+t\vecs v\) used to describe a line with direction vector \(\vecs v=⟨a,b,c⟩\) passing through point \(P=(x_0,y_0,z_0)\), where \(\vecs r_0=⟨x_0,y_0,z_0⟩\), is the position vector of point \(P\) | ||||

vector difference | the vector difference \(\vecs{v}−\vecs{w}\) is defined as \(\vecs{v}+(−\vecs{w})=\vecs{v}+(−1)\vecs{w}\) | ||||

vector addition | a vector operation that defines the sum of two vectors | ||||

vector | a mathematical object that has both magnitude and direction | ||||

variable of integration | indicates which variable you are integrating with respect to; if it is \(x\), then the function in the integrand is followed by \(dx\) | ||||

upper sum | a sum obtained by using the maximum value of \(f(x)\) on each subinterval | ||||

unit vector field | a vector field in which the magnitude of every vector is 1 | ||||

unit vector | a vector with magnitude \(1\) | ||||

unbounded sequence | a sequence that is not bounded is called unbounded | ||||

Type II | a region \(D\) in the \(xy\)-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions \(h_1(y)\) and \(h_2(h)\) | ||||

Type I | a region \(D\) in the \(xy\)- plane is Type I if it lies between two vertical lines and the graphs of two continuous functions \(g_1(x)\) and \(g_2(x)\) | ||||

triple integral in spherical coordinates | the limit of a triple Riemann sum, provided the following limit exists: \[lim_{l,m,n\rightarrow\infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\rho_{ijk}^*, \theta_{ijk}^*, \varphi_{ijk}^*) (\rho_{ijk}^*)^2 \sin \, \varphi \Delta \rho \Delta \theta \Delta \varphi \nonumber \] | ||||

triple integral in cylindrical coordinates | the limit of a triple Riemann sum, provided the following limit exists: \[lim_{l,m,n\rightarrow\infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(r_{ijk}^*, \theta_{ijk}^*, s_{ijk}^*) r_{ijk}^* \Delta r \Delta \theta \Delta z \nonumber \] | ||||

triple integral | the triple integral of a continuous function \(f(x,y,z)\) over a rectangular solid box \(B\) is the limit of a Riemann sum for a function of three variables, if this limit exists | ||||

trigonometric substitution | an integration technique that converts an algebraic integral containing expressions of the form \(\sqrt{a^2−x^2}\), \(\sqrt{a^2+x^2}\), or \(\sqrt{x^2−a^2}\) into a trigonometric integral | ||||

trigonometric integral | an integral involving powers and products of trigonometric functions | ||||

trigonometric identity | an equation involving trigonometric functions that is true for all angles \(θ\) for which the functions in the equation are defined | ||||

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

triangle method | a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector | ||||

triangle inequality | If \(a\) and \(b\) are any real numbers, then \(|a+b|≤|a|+|b|\) | ||||

triangle inequality | the length of any side of a triangle is less than the sum of the lengths of the other two sides | ||||

tree diagram | illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for | ||||

trapezoidal rule | a rule that approximates \(\displaystyle ∫^b_af(x)\,dx\) using the area of trapezoids. The approximation \(T_n\) to \(\displaystyle ∫^b_af(x)\,dx\) is given by \[T_n=\frac{Δx}{2}\big(f(x_0)+2\, f(x_1)+2\, f(x_2)+⋯+2\, f(x_{n−1})+f(x_n)\big). \nonumber \] | ||||

transformation of a function | a shift, scaling, or reflection of a function | ||||

transformation | a function that transforms a region GG in one plane into a region RR in another plane by a change of variables | ||||

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

trace | the intersection of a three-dimensional surface with a coordinate plane | ||||

total differential | the total differential of the function \( f(x,y)\) at \( (x_0,y_0)\) is given by the formula \( dz=f_x(x_0,y_0)dx+fy(x_0,y_0)dy\) | ||||

total area | total area between a function and the \(x\)-axis is calculated by adding the area above the \(x\)-axis and the area below the \(x\)-axis; the result is the same as the definite integral of the absolute value of the function | ||||

threshold population | the minimum population that is necessary for a species to survive | ||||

three-dimensional rectangular coordinate system | a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple \((x,y,z)\) that plots its location relative to the defining axes | ||||

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 | ||||

terminal point | the endpoint of a vector | ||||

term-by-term integration of a power series | a technique for integrating a power series \(\displaystyle \sum_{n=0}^∞c_n(x−a)^n\) by integrating each term separately to create the new power series \(\displaystyle C+\sum_{n=0}^∞c_n\dfrac{(x−a)^{n+1}}{n+1}\) | ||||

term-by-term differentiation of a power series | a technique for evaluating the derivative of a power series \(\displaystyle \sum_{n=0}^∞c_n(x−a)^n\) by evaluating the derivative of each term separately to create the new power series \(\displaystyle \sum_{n=1}^∞nc_n(x−a)^{n−1}\) | ||||

term | the number \(\displaystyle a_n\) in the sequence \(\displaystyle {a_n}\) is called the \(\displaystyle nth\) term of the sequence | ||||

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

Taylor’s theorem with remainder | for a function \(f\) and the \(n^{\text{th}}\)-degree Taylor polynomial for \(f\) at \(x=a\), the remainder \(R_n(x)=f(x)−p_n(x)\) satisfies \(R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1}\) for some\(c\) between \(x\) and \(a\); if there exists an interval \(I\) containing \(a\) and a real number \(M\) such that \(∣f^{(n+1)}(x)∣≤M\) for all \(x\) in \(I\), then \(|R_n(x)|≤\dfrac{M}{(n+1)!}|x−a|^{n+1}\) | ||||

Taylor series | a power series at \(a\) that converges to a function \(f\) on some open interval containing \(a\). | ||||

Taylor polynomials | the \(n^{\text{th}}\)-degree Taylor polynomial for \(f\) at \(x=a\) is \(p_n(x)=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n\) | ||||

tangential component of acceleration | the coefficient of the unit tangent vector \(\vecs T\) when the acceleration vector is written as a linear combination of \(\vecs T\) and \(\vecs N\) | ||||

tangent vector | to \(\vecs{r}(t)\) at \(t=t_0\) any vector \(\vecs v\) such that, when the tail of the vector is placed at point \(\vecs r(t_0)\) on the graph, vector \(\vecs{v}\) is tangent to curve C |
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tangent plane | given a function \( f(x,y)\) that is differentiable at a point \( (x_0,y_0)\), the equation of the tangent plane to the surface \( z=f(x,y)\) is given by \( z=f(x_0,y_0)+f_x(x_0,y_0)(x−x_0)+f_y(x_0,y_0)(y−y_0)\) | ||||

tangent line approximation (linearization) | since the linear approximation of \(f\) at \(x=a\) is defined using the equation of the tangent line, the linear approximation of \(f\) at \(x=a\) is also known as the tangent line approximation to \(f\) at \(x=a\) | ||||

tangent | A tangent line to the graph of a function at a point (\(a,f(a)\)) is the line that secant lines through (\(a,f(a)\)) approach as they are taken through points on the function with \(x\)-values that approach \(a\); the slope of the tangent line to a graph at \(a\) measures the rate of change of the function at \(a\) | ||||

table of values | a table containing a list of inputs and their corresponding outputs | ||||

symmetry principle | the symmetry principle states that if a region \(R\) is symmetric about a line \(I\), then the centroid of \(R\) lies on \(I\) | ||||

symmetry about the origin | the graph of a function \(f\) is symmetric about the origin if \((−x,−y)\) is on the graph of \(f\) whenever \((x,y)\) is on the graph | ||||

symmetry about the \(y\)-axis | the graph of a function \(f\) is symmetric about the \(y\)-axis if \((−x,y)\) is on the graph of \(f\) whenever \((x,y)\) is on the graph | ||||

symmetric equations of a line | the equations \(\dfrac{x−x_0}{a}=\dfrac{y−y_0}{b}=\dfrac{z−z_0}{c}\) describing the line with direction vector \(v=⟨a,b,c⟩\) passing through point \((x_0,y_0,z_0)\) | ||||

surface integral of a vector field | a surface integral in which the integrand is a vector field | ||||

surface integral of a scalar-valued function | a surface integral in which the integrand is a scalar function | ||||

surface integral | an integral of a function over a surface | ||||

surface independent | flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface | ||||

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 | ||||

surface area | the area of surface \(S\) given by the surface integral \[\iint_S \,dS \nonumber \] | ||||

surface | the graph of a function of two variables, \(z=f(x,y)\) | ||||

sum rule | the derivative of the sum of a function \(f\) and a function \(g\) is the same as the sum of the derivative of \(f\) and the derivative of \(g\): \(\dfrac{d}{dx}\big(f(x)+g(x)\big)=f′(x)+g′(x)\) | ||||

sum law for limits | The limit law \(\lim_{x→a}(f(x)+g(x))=\lim_{x→a}f(x)+\lim_{x→a}g(x)=L+M\) | ||||

stream function | if \(\vecs F=⟨P,Q⟩\) is a source-free vector field, then stream function \(g\) is a function such that \(P=g_y\) and \(Q=−g_x\) | ||||

Stokes’ theorem | relates the flux integral over a surface \(S\) to a line integral around the boundary \(C\) of the surface \(S\) | ||||

step size | the increment hh that is added to the xx value at each step in Euler’s Method | ||||

standard-position vector | a vector with initial point \((0,0)\) | ||||

standard unit vectors | unit vectors along the coordinate axes: \(\hat{\mathbf i}=⟨1,0⟩,\, \hat{\mathbf j}=⟨0,1⟩\) | ||||

standard form | the form of a first-order linear differential equation obtained by writing the differential equation in the form \( y'+p(x)y=q(x)\) | ||||

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

standard equation of a sphere | \((x−a)^2+(y−b)^2+(z−c)^2=r^2\) describes a sphere with center \((a,b,c)\) and radius \(r\) | ||||

squeeze theorem | states that if \(f(x)≤g(x)≤h(x)\) for all \(x≠a\) over an open interval containing a and \(\lim_{x→a}f(x)=L=\lim_ {x→a}h(x)\) where L is a real number, then \(\lim_{x→a}g(x)=L\) | ||||

spherical coordinate system | a way to describe a location in space with an ordered triple \((ρ,θ,φ),\) where \(ρ\) is the distance between \(P\) and the origin \((ρ≠0), θ\) is the same angle used to describe the location in cylindrical coordinates, and \(φ\) is the angle formed by the positive \(z\)-axis and line segment \(\bar{OP}\), where \(O\) is the origin and \(0≤φ≤π\) | ||||

sphere | the set of all points equidistant from a given point known as the center |
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speed | is the absolute value of velocity, that is, \(|v(t)|\) is the speed of an object at time \(t\) whose velocity is given by \(v(t)\) | ||||

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

space curve | the set of ordered triples \((f(t),g(t),h(t))\) together with their defining parametric equations \(x=f(t)\), \(y=g(t)\) and \(z=h(t)\) | ||||

solution to a differential equation | a function \(y=f(x)\) that satisfies a given differential equation | ||||

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 | ||||

solid of revolution | a solid generated by revolving a region in a plane around a line in that plane | ||||

smooth | curves where the vector-valued function \(\vecs r(t)\) is differentiable with a non-zero derivative | ||||

slope-intercept form | equation of a linear function indicating its slope and \(y\)-intercept | ||||

slope | the change in \(y\) for each unit change in \(x\) | ||||

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 | ||||

skew lines | two lines that are not parallel but do not intersect | ||||

Simpson’s rule | a rule that approximates \(\displaystyle ∫^b_af(x)\,dx\) using the area under a piecewise quadratic function. The approximation \(S_n\) to \(\displaystyle ∫^b_af(x)\,dx\) is given by \[S_n=\frac{Δx}{3}\big(f(x_0)+4\,f(x_1)+2\,f(x_2)+4\,f(x_3)+2\,f(x_4)+⋯+2\,f(x_{n−2})+4\,f(x_{n−1})+f(x_n)\big). \nonumber \] | ||||

simply connected region | a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region | ||||

simple harmonic motion | motion described by the equation \(x(t)=c_1 \cos (ωt)+c_2 \sin (ωt)\), as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely | ||||

simple curve | a curve that does not cross itself | ||||

sigma notation | (also, summation notation) the Greek letter sigma (\(Σ\)) 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 | ||||

sequence | an ordered list of numbers of the form \(\displaystyle a_1,a_2,a_3,…\) is a sequence | ||||

separation of variables | a method used to solve a separable differential equation | ||||

separable differential equation | any equation that can be written in the form \(y'=f(x)g(y)\) | ||||

second derivative test | suppose \(f'(c)=0\) and \(f'\)' is continuous over an interval containing \(c\); if \(f''(c)>0\), then \(f\) has a local minimum at \(c\); if \(f''(c)<0\), then \(f\) has a local maximum at \(c\); if \(f''(c)=0\), then the test is inconclusive | ||||

secant | A secant line to a function \(f(x)\) at \(a\) is a line through the point (\(a,f(a)\)) and another point on the function; the slope of the secant line is given by \(m_{sec}=\dfrac{f(x)−f(a)}{x−a}\) | ||||

scalar projection | the magnitude of the vector projection of a vector | ||||

scalar multiplication | a vector operation that defines the product of a scalar and a vector | ||||

scalar line integral | the scalar line integral of a function \(f\) along a curve \(C\) with respect to arc length is the integral \(\displaystyle \int_C f\,ds\), it is the integral of a scalar function \(f\) along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral | ||||

scalar equation of a plane | the equation \(a(x−x_0)+b(y−y_0)+c(z−z_0)=0\) used to describe a plane containing point \(P=(x_0,y_0,z_0)\) with normal vector \(n=⟨a,b,c⟩\) or its alternate form \(ax+by+cz+d=0\), where \(d=−ax_0−by_0−cz_0\) | ||||

scalar | a real number | ||||

saddle point | given the function \(z=f(x,y),\) the point \((x_0,y_0,f(x_0,y_0))\) is a saddle point if both \(f_x(x_0,y_0)=0\) and \(f_y(x_0,y_0)=0\), but \(f\) does not have a local extremum at \((x_0,y_0)\) | ||||

rulings | parallel lines that make up a cylindrical surface | ||||

rotational field | a vector field in which the vector at point \((x,y)\) is tangent to a circle with radius \(r=\sqrt{x^2+y^2}\); in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin | ||||

rose | graph of the polar equation \(r=a\cos 2θ\) or \(r=a\sin 2θ\)for a positive constant \(a\) | ||||

root test | for a series \(\displaystyle \sum^∞_{n=1}a_n,\) let \( \displaystyle ρ=\lim_{n→∞}\sqrt[n]{|a_n|}\); if \( 0≤ρ<1\), the series converges absolutely; if \( ρ>1\), the series diverges; if \( ρ=1\), the test is inconclusive | ||||

root law for limits | the limit law \(\lim_{x→a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x→a}f(x)}=\sqrt[n]{L}\) for all L if n is odd and for \(L≥0\) if n is even | ||||

root function | a function of the form \(f(x)=x^{1/n}\) for any integer \(n≥2\) | ||||

rolle’s theorem | if \(f\) is continuous over \([a,b]\) and differentiable over \((a,b)\), and if \(f(a)=f(b)\), then there exists \(c∈(a,b)\) such that \(f′(c)=0\) | ||||

RLC series circuit |
a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit |
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right-hand rule | a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the \(z\)-axis in such a way that the fingers curl from the positive \(x\)-axis to the positive \(y\)-axis, the thumb points in the direction of the positive \(z\)-axis | ||||

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 | ||||

riemann sum | an estimate of the area under the curve of the form \(A≈\displaystyle \sum_{i=1}^nf(x^∗_i)Δx\) | ||||

restricted domain | a subset of the domain of a function \(f\) | ||||

reparameterization | an alternative parameterization of a given vector-valued function | ||||

removable discontinuity | A removable discontinuity occurs at a point \(a\) if \(f(x)\) is discontinuous at \(a\), but \(\displaystyle \lim_{x→a}f(x)\) exists | ||||

remainder estimate | for a series \(\displaystyle \sum^∞_{n=}1a_n\) with positive terms \( a_n\) and a continuous, decreasing function \( f\) such that \( f(n)=a_n\) for all positive integers \( n\), the remainder \(\displaystyle R_N=\sum^∞_{n=1}a_n−\sum^N_{n=1}a_n\) satisfies the following estimate: \[∫^∞_{N+1}f(x)\,dx<R_N<∫^∞_Nf(x)\,dx \nonumber \] | ||||

relative error | given an absolute error \(Δq\) for a particular quantity, \(\frac{Δq}{q}\) is the relative error. | ||||

relative error | error as a percentage of the actual value, given by \[\text{relative error}=\left|\frac{A−B}{A}\right|⋅100\% \nonumber \] | ||||

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

regular partition | a partition in which the subintervals all have the same width | ||||

regular parameterization | parameterization \(\vecs r(u,v) = \langle x(u,v), \, y(u,v), \, z(u,v)\rangle\) such that \(r_u \times r_v\) is not zero for point \((u,v)\) in the parameter domain | ||||

region | an open, connected, nonempty subset of \(\mathbb{R}^2\) | ||||

recurrence relation | a recurrence relation is a relationship in which a term \(a_n\) in a sequence is defined in terms of earlier terms in the sequence | ||||

rational function | a function of the form \(f(x)=p(x)/q(x)\), where \(p(x)\) and \(q(x)\) are polynomials | ||||

ratio test | for a series \(\displaystyle \sum^∞_{n=1}a_n\) with nonzero terms, let \( \displaystyle ρ=\lim_{n→∞}|a_{n+1}/a_n|\); if \( 0≤ρ<1\), the series converges absolutely; if \( ρ>1\), the series diverges; if \( ρ=1\), the test is inconclusive | ||||

range | the set of outputs for a function | ||||

radius of gyration | the distance from an object’s center of mass to its axis of rotation | ||||

radius of curvature | the reciprocal of the curvature | ||||

radius of convergence | if there exists a real number \(R>0\) such that a power series centered at \(x=a\) converges for \(|x−a|<R\) and diverges for \(|x−a|>R\), then \(R\) is the radius of convergence; if the power series only converges at \(x=a\), the radius of convergence is \(R=0\); if the power series converges for all real numbers \(x\), the radius of convergence is \(R=∞\) | ||||

radians | for a circular arc of length \(s\) on a circle of radius 1, the radian measure of the associated angle \(θ\) is \(s\) | ||||

radial field | a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin | ||||

radial coordinate | \(r\) the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole | ||||

quotient rule | the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: \(\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f′(x)g(x)−g′(x)f(x)}{\big(g(x)\big)^2}\) | ||||

quotient law for limits | the limit law \(\lim_{x→a}\dfrac{f(x)}{g(x)}=\dfrac{\lim_{x→a}f(x)}{\lim_{x→a}g(x)}=\dfrac{L}{M}\) for M≠0 | ||||

quadric surfaces | surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas) | ||||

quadratic function | a polynomial of degree 2; that is, a function of the form \(f(x)=ax^2+bx+c\) where \(a≠0\) | ||||

propagated error | the error that results in a calculated quantity \(f(x)\) resulting from a measurement error \(dx\) | ||||

projectile motion | motion of an object with an initial velocity but no force acting on it other than gravity | ||||

product rule | the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: \(\dfrac{d}{dx}\big(f(x)g(x)\big)=f′(x)g(x)+g′(x)f(x)\) | ||||

product law for limits | the limit law \[\lim_{x→a}(f(x)⋅g(x))=\lim_{x→a}f(x)⋅\lim_{x→a}g(x)=L⋅M \nonumber \] | ||||

principal unit tangent vector | a unit vector tangent to a curve C |
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principal unit normal vector | a vector orthogonal to the unit tangent vector, given by the formula \(\frac{\vecs T′(t)}{‖\vecs T′(t)‖}\) | ||||

power series | a series of the form \(\sum_{n=0}^∞c_nx^n\) is a power series centered at \(x=0\); a series of the form \(\sum_{n=0}^∞c_n(x−a)^n\) is a power series centered at \(x=a\) | ||||

power rule | the derivative of a power function is a function in which the power on \(x\) becomes the coefficient of the term and the power on \(x\) in the derivative decreases by 1: If \(n\) is an integer, then \(\dfrac{d}{dx}\left(x^n\right)=nx^{n−1}\) | ||||

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 law for limits | the limit law \[\lim_{x→a}(f(x))^n=(\lim_{x→a}f(x))^n=L^n \nonumber \]for every positive integer n | ||||

power function | a function of the form \(f(x)=x^n\) for any positive integer \(n≥1\) | ||||

potential function | a scalar function \(f\) such that \(\vecs ∇f=\vecs{F}\) | ||||

population growth rate | is the derivative of the population with respect to time | ||||

polynomial function | a function of the form \(f(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0\) | ||||

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

polar rectangle | the region enclosed between the circles \(r = a\) and \(r = b\) and the angles \(\theta = \alpha\) and \(\theta = \beta\); it is described as \(R = \{(r, \theta)\,|\,a \leq r \leq b, \, \alpha \leq \theta \leq \beta\}\) | ||||

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

polar coordinate system | a system for locating points in the plane. The coordinates are \(r\), the radial coordinate, and \(θ\), the angular coordinate | ||||

polar axis | the horizontal axis in the polar coordinate system corresponding to \(r≥0\) | ||||

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

plane curve | the set of ordered pairs \((f(t),g(t))\) together with their defining parametric equations \(x=f(t)\) and \(y=g(t)\) | ||||

planar transformation | a function \(T\) that transforms a region \(G\) in one plane into a region \(R\) in another plane by a change of variables | ||||

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

piecewise smooth curve | an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves | ||||

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

periodic function | a function is periodic if it has a repeating pattern as the values of \(x\) move from left to right | ||||

percentage error | the relative error expressed as a percentage | ||||

partition | a set of points that divides an interval into subintervals | ||||

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

particular solution | a solution \(y_p(x)\) of a differential equation that contains no arbitrary constants | ||||

partial sum | the \( kth\) partial sum of the infinite series \(\displaystyle \sum^∞_{n=1}a_n\) is the finite sum \(\displaystyle S_k=\sum_{n=1}^ka_n=a_1+a_2+a_3+⋯+a_k\) | ||||

partial fraction decomposition | a technique used to break down a rational function into the sum of simple rational functions | ||||

partial differential equation | an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives | ||||

partial derivative | a derivative of a function of more than one independent variable in which all the variables but one are held constant | ||||

parametric equations of a line | the set of equations \(x=x_0+ta, y=y_0+tb,\) and \(z=z_0+tc\) describing the line with direction vector \(v=⟨a,b,c⟩\) passing through point \((x_0,y_0,z_0)\) | ||||

parametric equations | the equations \(x=x(t)\) and \(y=y(t)\) that define a parametric curve | ||||

parametric curve | the graph of the parametric equations \(x(t)\) and \(y(t)\) over an interval \(a≤t≤b\) combined with the equations | ||||

parameterized surface (parametric surface) | a surface given by a description of the form \(\vecs r(u,v) = \langle x(u,v), \, y(u,v), \, z(u,v)\rangle\), where the parameters \(u\) and \(v\) vary over a parameter domain in the \(uv\)-plane | ||||

parameterization of a curve | rewriting the equation of a curve defined by a function \(y=f(x)\) as parametric equations | ||||

parameter domain (parameter space) | the region of the \(uv\)-plane over which the parameters \(u\) and \(v\) vary for parameterization \(\vecs r(u,v) = \langle x(u,v), \, y(u,v), \, z(u,v)\rangle\) | ||||

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

parallelogram method | a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram | ||||

p-series |
a series of the form \(\displaystyle \sum^∞_{n=1}1/n^p\) | ||||

osculating plane | the plane determined by the unit tangent and the unit normal vector | ||||

osculating circle | a circle that is tangent to a curve \(C\) at a point \(P\) and that shares the same curvature | ||||

orthogonal vectors | vectors that form a right angle when placed in standard position | ||||

orientation of a surface | if a surface has an “inner” side and an “outer” side, then an orientation is a choice of the inner or the outer side; the surface could also have “upward” and “downward” orientations | ||||

orientation of a curve | the orientation of a curve \(C\) is a specified direction of \(C\) | ||||

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

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

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

optimization problem | calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers | ||||

open set | a set \(S\) that contains none of its boundary points | ||||

one-to-one transformation | a transformation \(T : G \rightarrow R\) defined as \(T(u,v) = (x,y)\) is said to be one-to-one if no two points map to the same image point | ||||

one-to-one function | a function \(f\) is one-to-one if \(f(x_1)≠f(x_2)\) if \(x_1≠x_2\) | ||||

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

odd function | a function is odd if \(f(−x)=−f(x)\) for all \(x\) in the domain of \(f\) | ||||

octants | the eight regions of space created by the coordinate planes | ||||

oblique asymptote | the line \(y=mx+b\) if \(f(x)\) approaches it as \(x→∞\) or\( x→−∞\) | ||||

objective function | the function that is to be maximized or minimized in an optimization problem | ||||

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 | ||||

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

normalization | using scalar multiplication to find a unit vector with a given direction | ||||

normal vector | a vector perpendicular to a plane | ||||

normal plane | a plane that is perpendicular to a curve at any point on the curve | ||||

normal component of acceleration | the coefficient of the unit normal vector \(\vecs N\) when the acceleration vector is written as a linear combination of \(\vecs T\) and \(\vecs N\) | ||||

nonhomogeneous linear equation | a second-order differential equation that can be written in the form \(a_2(x)y″+a_1(x)y′+a_0(x)y=r(x)\), but \(r(x)≠0\) for some value of \(x\) | ||||

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

Newton’s method | method for approximating roots of \(f(x)=0;\) using an initial guess \(x_0\); each subsequent approximation is defined by the equation \(x_n=x_{n−1}−\frac{f(x_{n−1})}{f'(x_{n−1})}\) | ||||

net signed area | the area between a function and the \(x\)-axis such that the area below the \(x\)-axis is subtracted from the area above the \(x\)-axis; the result is the same as the definite integral of the function | ||||

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 | ||||

natural logarithm | the function \(\ln x=\log_ex\) | ||||

natural exponential function | the function \(f(x)=e^x\) | ||||

nappe | a nappe is one half of a double cone | ||||

multivariable calculus | the study of the calculus of functions of two or more variables | ||||

monotone sequence | an increasing or decreasing sequence | ||||

moment | if n masses are arranged on a number line, the moment of the system with respect to the origin is given by \(\displaystyle M=\sum^n_{i=1}m_ix_i\); if, instead, we consider a region in the plane, bounded above by a function \(f(x)\) over an interval \([a,b]\), then the moments of the region with respect to the \(x\)- and \(y\)-axes are given by \(\displaystyle M_x=ρ∫^b_a\dfrac{[f(x)]^2}{2}\,dx\) and \(\displaystyle M_y=ρ∫^b_axf(x)\,dx\), respectively | ||||

mixed partial derivatives | second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables | ||||

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 | ||||

midpoint rule | a rule that uses a Riemann sum of the form \(\displaystyle M_n=\sum^n_{i=1}f(m_i)Δx\), where \( m_i\) is the midpoint of the \(i^{\text{th}}\) subinterval to approximate \(\displaystyle ∫^b_af(x)\,dx\) | ||||

method of variation of parameters | a method that involves looking for particular solutions in the form \(y_p(x)=u(x)y_1(x)+v(x)y_2(x)\), where \(y_1\) and \(y_2\) are linearly independent solutions to the complementary equations, and then solving a system of equations to find \(u(x)\) and \(v(x)\) | ||||

method of undetermined coefficients | a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess | ||||

method of Lagrange multipliers | a method of solving an optimization problem subject to one or more constraints | ||||

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 | ||||

mean value theorem for integrals | guarantees that a point \(c\) exists such that \(f(c)\) is equal to the average value of the function | ||||

mean value theorem | if \(f\) is continuous over \([a,b]\) and differentiable over \((a,b)\), then there exists \(c∈(a,b)\) such that \(f′(c)=\frac{f(b)−f(a)}{b−a}\) | ||||

mathematical model | A method of simulating real-life situations with mathematical equations | ||||

mass flux | the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area | ||||

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

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

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

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 | ||||

magnitude | the length of a vector | ||||

Maclaurin series | a Taylor series for a function \(f\) at \(x=0\) is known as a Maclaurin series for \(f\) | ||||

Maclaurin polynomial | a Taylor polynomial centered at \(0\); the \(n^{\text{th}}\)-degree Taylor polynomial for \(f\) at \(0\) is the \(n^{\text{th}}\)-degree Maclaurin polynomial for \(f\) | ||||

lower sum | a sum obtained by using the minimum value of \(f(x)\) on each subinterval | ||||

logistic differential equation | a differential equation that incorporates the carrying capacity \(K\) and growth rate rr into a population model | ||||

logarithmic function | a function of the form \(f(x)=\log_b(x)\) for some base \(b>0,\,b≠1\) such that \(y=\log_b(x)\) if and only if \(b^y=x\) | ||||

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

local minimum | if there exists an interval \(I\) such that \(f(c)≤f(x)\) for all \(x∈I\), we say \(f\) has a local minimum at \(c\) | ||||

local maximum | if there exists an interval \(I\) such that \(f(c)≥f(x)\) for all \(x∈I\), we say \(f\) has a local maximum at \(c\) | ||||

local extremum | if \(f\) has a local maximum or local minimum at \(c\), we say \(f\) has a local extremum at \(c\) | ||||

linearly independent | a set of functions \(f_1(x),f_2(x),…,f_n(x)\) for which there are no constants \(c_1,c_2,…c_n\), such that \(c_1f_1(x)+c_2f_2(x)+⋯+c_nf_n(x)=0\) for all \(x\) in the interval of interest |
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linearly dependent | a set of functions \(f_1(x),f_2(x),…,f_n(x)\) for whichthere are constants \(c_1,c_2,…c_n\), not all zero, such that \(c_1f_1(x)+c_2f_2(x)+⋯+c_nf_n(x)=0\) for all \(x\) in the interval of interest |
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linear function | a function that can be written in the form \(f(x)=mx+b\) | ||||

linear approximation | the linear function \(L(x)=f(a)+f'(a)(x−a)\) is the linear approximation of \(f\) at \(x=a\) | ||||

linear approximation | given a function \( f(x,y)\) and a tangent plane to the function at a point \( (x_0,y_0)\), we can approximate \( f(x,y)\) for points near \( (x_0,y_0)\) using the tangent plane formula | ||||

linear | description of a first-order differential equation that can be written in the form \( a(x)y′+b(x)y=c(x)\) | ||||

line integral | the integral of a function along a curve in a plane or in space | ||||

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 | ||||

limit of a vector-valued function | a vector-valued function \(\vecs r(t)\) has a limit \(\vecs L\) as \(t\) approaches \(a\) if \(\lim \limits{t \to a} \left| \vecs r(t) - \vecs L \right| = 0\) | ||||

limit of a sequence | the real number LL to which a sequence converges is called the limit of the sequence | ||||

limit laws | the individual properties of limits; for each of the individual laws, let \(f(x)\) and \(g(x)\) be defined for all \(x≠a\) over some open interval containing a; assume that L and M are real numbers so that \(\lim_{x→a}f(x)=L\) and \(\lim_{x→a}g(x)=M\); let c be a constant | ||||

limit comparison test | Suppose \(a_n,b_n≥0\) for all \(n≥1\). If \(\displaystyle \lim_{n→∞}a_n/b_n→L≠0\), then \(\displaystyle \sum^∞_{n=1}a_n\) and \(\displaystyle \sum^∞_{n=1}b_n\) both converge or both diverge; if \(\displaystyle \lim_{n→∞}a_n/b_n→0\) and \(\displaystyle \sum^∞_{n=1}b_n\) converges, then \(\displaystyle \sum^∞_{n=1}a_n\) converges. If \(\displaystyle \lim_{n→∞}a_n/b_n→∞\), and \(\displaystyle \sum^∞_{n=1}b_n\) diverges, then \(\displaystyle \sum^∞_{n=1}a_n\) diverges. | ||||

limit at infinity | a function that approaches a limit value \(L\) as \(x\) becomes large | ||||

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

limaçon | the graph of the equation \(r=a+b\sin θ\) or \(r=a+b\cos θ.\) If \(a=b\) then the graph is a cardioid | ||||

level surface of a function of three variables | the set of points satisfying the equation \(f(x,y,z)=c\) for some real number \(c\) in the range of \(f\) | ||||

level curve of a function of two variables | the set of points satisfying the equation \(f(x,y)=c\) for some real number \(c\) in the range of \(f\) | ||||

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 | ||||

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

Lagrange multiplier | the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable \(λ\) | ||||

L’Hôpital’s rule | If \(f\) and \(g\) are differentiable functions over an interval \(a\), except possibly at \(a\), and \(\displaystyle \lim_{x→a}f(x)=0=\lim_{x→a}g(x)\) or \(\displaystyle \lim_{x→a}f(x)\) and \(\displaystyle \lim_{x→a}g(x)\) are infinite, then \(\displaystyle \lim_{x→a}\dfrac{f(x)}{g(x)}=\lim_{x→a}\dfrac{f′(x)}{g′(x)}\), assuming the limit on the right exists or is \(∞\) or \(−∞\). | ||||

Kepler’s laws of planetary motion | three laws governing the motion of planets, asteroids, and comets in orbit around the Sun | ||||

jump discontinuity | A jump discontinuity occurs at a point \(a\) if \(\displaystyle \lim_{x→a^−}f(x)\) and \(\displaystyle \lim_{x→a^+}f(x)\) both exist, but \(\displaystyle \lim_{x→a^−}f(x)≠\lim_{x→a^+}f(x)\) | ||||

Jacobian | the Jacobian \(J (u,v)\) in two variables is a \(2 \times 2\) determinant: \[J(u,v) = \begin{vmatrix} \frac{\partial x}{\partial u} \frac{\partial y}{\partial u} \nonumber \ \frac{\partial x}{\partial v} \frac{\partial y}{\partial v} \end{vmatrix}; \nonumber \] the Jacobian \(J (u,v,w)\) in three variables is a \(3 \times 3\) determinant: \[J(u,v,w) = \begin{vmatrix} \frac{\partial x}{\partial u} \frac{\partial y}{\partial u} \frac{\partial z}{\partial u} \nonumber \ \frac{\partial x}{\partial v} \frac{\partial y}{\partial v} \frac{\partial z}{\partial v} \nonumber \ \frac{\partial x}{\partial w} \frac{\partial y}{\partial w} \frac{\partial z}{\partial w}\end{vmatrix} \nonumber \] | ||||

iterative process | process in which a list of numbers \(x_0,x_1,x_2,x_3…\) is generated by starting with a number \(x_0\) and defining \(x_n=F(x_{n−1})\) for \(n≥1\) | ||||

iterated integral | for a function \(f(x,y)\) over the region \(R\) is a. \(\displaystyle \int_a^b \int_c^d f(x,y) \,dx \, dy = \int_a^b \left[\int_c^d f(x,y) \, dy\right] \, dx,\) b. \(\displaystyle \int_c^d \int_a^b f(x,y) \, dx \, dy = \int_c^d \left[\int_a^b f(x,y) \, dx\right] \, dy,\) where \(a,b,c\), and \(d\) are any real numbers and \(R = [a,b] \times [c,d]\) | ||||

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

inverse hyperbolic functions | the inverses of the hyperbolic functions where \(\cosh\) and \( \operatorname{sech}\) are restricted to the domain \([0,∞)\);each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function | ||||

inverse function | for a function \(f\), the inverse function \(f^{−1}\) satisfies \(f^{−1}(y)=x\) if \(f(x)=y\) | ||||

intuitive definition of the limit | If all values of the function \(f(x)\) approach the real number \(L\) as the values of \(x(≠a)\) approach a, \(f(x)\) approaches L | ||||

interval of convergence | the set of real numbers \(x\) for which a power series converges | ||||

intermediate variable | given a composition of functions (e.g., \(\displaystyle f(x(t),y(t)))\), the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function \(\displaystyle f(x(t),y(t)),\) the variables \(\displaystyle x\) and \(\displaystyle y\) are examples of intermediate variables | ||||

Intermediate Value Theorem | Let \(f\) be continuous over a closed bounded interval [\(a,b\)] if \(z\) is any real number between \(f(a)\) and \(f(b)\), then there is a number c in [\(a,b\)] satisfying \(f(c)=z\) | ||||

interior point | a point \(P_0\) of \(\mathbb{R}\) is a boundary point if there is a \(δ\) disk centered around \(P_0\) contained completely in \(\mathbb{R}\) | ||||

integration table | a table that lists integration formulas | ||||

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

integration by parts | a technique of integration that allows the exchange of one integral for another using the formula \(\displaystyle ∫u\,dv=uv−∫v\,du\) | ||||

integrating factor | any function \(f(x)\) 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 | ||||

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

integral test | for a series \(\displaystyle \sum^∞_{n=1}a_n\) with positive terms \( a_n\), if there exists a continuous, decreasing function \( f\) such that \( f(n)=a_n\) for all positive integers \( n\), then \[\sum_{n=1}^∞a_n \nonumber \] and \[∫^∞_1f(x)\,dx \nonumber \] either both converge or both diverge | ||||

integral calculus | the study of integrals and their applications | ||||

integrable function | a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as \(n\) goes to infinity exists | ||||

instantaneous velocity | The instantaneous velocity of an object with a position function that is given by \(s(t)\) is the value that the average velocities on intervals of the form [\(t,a\)] and [\(a,t\)] approach as the values of \(t\) move closer to \(a\), provided such a value exists | ||||

instantaneous rate of change | the rate of change of a function at any point along the function \(a\), also called \(f′(a)\), or the derivative of the function at \(a\) | ||||

initial-value problem | a differential equation together with an initial value or values | ||||

initial velocity | the velocity at time \(t=0\) | ||||

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 value problem | a problem that requires finding a function \(y\) that satisfies the differential equation \(\dfrac{dy}{dx}=f(x)\) together with the initial condition \(y(x_0)=y_0\) | ||||

initial population | the population at time \(t=0\) | ||||

initial point | the starting point of a vector | ||||

inflection point | if \(f\) is continuous at \(c\) and \(f\) changes concavity at \(c\), the point \((c,f(c))\) is an inflection point of \(f\) | ||||

infinite series | an infinite series is an expression of the form \(\displaystyle a_1+a_2+a_3+⋯=\sum_{n=1}^∞a_n\) | ||||

infinite limit at infinity | a function that becomes arbitrarily large as \(x\) becomes large | ||||

infinite limit | A function has an infinite limit at a point \(a\) if it either increases or decreases without bound as it approaches \(a\) | ||||

infinite discontinuity | An infinite discontinuity occurs at a point \(a\) if \(\displaystyle \lim_{x→a^−}f(x)=±∞\) or \(\displaystyle \lim_{x→a^+}f(x)=±∞\) | ||||

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

indeterminate forms | When evaluating a limit, the forms \(\dfrac{0}{0}\),\(∞/∞, 0⋅∞, ∞−∞, 0^0, ∞^0\), and \(1^∞\) are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is. | ||||

independent variable | the input variable for a function | ||||

independence of path | a vector field \(\vecs{F}\) has path independence if \(\displaystyle \int_{C_1} \vecs F⋅d\vecs r=\displaystyle \int_{C_2} \vecs F⋅d\vecs r\) for any curves \(C_1\) and \(C_2\) in the domain of \(\vecs{F}\) with the same initial points and terminal points | ||||

indefinite integral of a vector-valued function | a vector-valued function with a derivative that is equal to a given vector-valued function | ||||

indefinite integral | the most general antiderivative of \(f(x)\) is the indefinite integral of \(f\); we use the notation \(\displaystyle \int f(x)\,dx\) to denote the indefinite integral of \(f\) | ||||

increasing on the interval \(I\) | a function increasing on the interval \(I\) if for all \(x_1,\,x_2∈I,\;f(x_1)≤f(x_2)\) if \(x_1<x_2\) | ||||

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 | ||||

improper double integral | a double integral over an unbounded region or of an unbounded function | ||||

implicit differentiation | is a technique for computing \(\dfrac{dy}{dx}\) for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable \(y\) as a function) and solving for \(\dfrac{dy}{dx}\) | ||||

hyperboloid of two sheets | a three-dimensional surface described by an equation of the form \( \dfrac{z^2}{c^2}−\dfrac{x^2}{a^2}−\dfrac{y^2}{b^2}=1\); traces of this surface include ellipses and hyperbolas | ||||

hyperboloid of one sheet | a three-dimensional surface described by an equation of the form \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=1;\) traces of this surface include ellipses and hyperbolas | ||||

hyperbolic functions | the functions denoted \(\sinh,\,\cosh,\,\operatorname{tanh},\,\operatorname{csch},\,\operatorname{sech},\) and \(\coth\), which involve certain combinations of \(e^x\) and \(e^{−x}\) | ||||

hydrostatic pressure | the pressure exerted by water on a submerged object | ||||

horizontal line test | a function \(f\) is one-to-one if and only if every horizontal line intersects the graph of \(f\), at most, once | ||||

horizontal asymptote | if \(\displaystyle \lim_{x→∞}f(x)=L\) or \(\displaystyle \lim_{x→−∞}f(x)=L\), then \(y=L\) is a horizontal asymptote of \(f\) | ||||

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, \(F=kx\), where \(k\) is a constant | ||||

homogeneous linear equation | a second-order differential equation that can be written in the form \(a_2(x)y″+a_1(x)y′+a_0(x)y=r(x)\), but \(r(x)=0\) for every value of \(x\) | ||||

higher-order partial derivatives | second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives | ||||

higher-order derivative | a derivative of a derivative, from the second derivative to the \(n^{\text{th}}\) derivative, is called a higher-order derivative | ||||

helix | a three-dimensional curve in the shape of a spiral | ||||

heat flow | a vector field proportional to the negative temperature gradient in an object | ||||

harmonic series | the harmonic series takes the form \(\displaystyle \sum_{n=1}^∞\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+⋯\) | ||||

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 \((\ln 2)/k\) | ||||

growth rate | the constant \(r>0\) in the exponential growth function \(P(t)=P_0e^{rt}\) | ||||

grid curves | curves on a surface that are parallel to grid lines in a coordinate plane | ||||

Green’s theorem | relates the integral over a connected region to an integral over the boundary of the region | ||||

graph of a function of two variables | a set of ordered triples \((x,y,z)\) that satisfies the equation \(z=f(x,y)\) plotted in three-dimensional Cartesian space | ||||

graph of a function | the set of points \((x,y)\) such that \(x\) is in the domain of \(f\) and \(y=f(x)\) | ||||

gradient field | a vector field \(\vecs{F}\) for which there exists a scalar function \(f\) such that \(\vecs ∇f=\vecs{F}\); in other words, a vector field that is the gradient of a function; such vector fields are also called conservative |
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geometric series | a geometric series is a series that can be written in the form \(\displaystyle \sum_{n=1}^∞ar^{n−1}=a+ar+ar^2+ar^3+⋯\) | ||||

geometric sequence | a sequence \(\displaystyle {a_n}\) in which the ratio \(\displaystyle a_{n+1}/a_n\) is the same for all positive integers \(\displaystyle n\) is called a geometric sequence | ||||

generalized chain rule | the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables | ||||

general solution (or family of solutions) | the entire set of solutions to a given differential equation | ||||

general form of the equation of a plane | an equation in the form \(ax+by+cz+d=0,\) where \(\vecs n=⟨a,b,c⟩\) is a normal vector of the plane, \(P=(x_0,y_0,z_0)\) is a point on the plane, and \(d=−ax_0−by_0−cz_0\) | ||||

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

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 | ||||

fundamental theorem of calculus, part 1 | uses a definite integral to define an antiderivative of a function | ||||

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

Fundamental Theorem for Line Integrals | the value of line integral \(\displaystyle \int_C\vecs ∇f⋅d\vecs r\) depends only on the value of \(f\) at the endpoints of \(C: \displaystyle \int_C \vecs ∇f⋅d\vecs r=f(\vecs r(b))−f(\vecs r(a))\) | ||||

function of two variables | a function \(z=f(x,y)\) that maps each ordered pair \((x,y)\) in a subset \(D\) of \(R^2\) to a unique real number \(z\) | ||||

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

Fubini’s theorem | if \(f(x,y)\) is a function of two variables that is continuous over a rectangular region \(R = \big\{(x,y) \in \mathbb{R}^2 \,|\,a \leq x \leq b, \, c \leq y \leq d\big\}\), then the double integral of \(f\) over the region equals an iterated integral, \[\displaystyle\iint_R f(x,y) \, dA = \int_a^b \int_c^d f(x,y) \,dx \, dy = \int_c^d \int_a^b f(x,y) \,dx \, dy \nonumber \] | ||||

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

Frenet frame of reference | (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector | ||||

formal definition of an infinite limit | \(\displaystyle \lim_{x→a}f(x)=\infty\) if for every \(M>0\), there exists a \(δ>0\) such that if \(0<|x−a|<δ\), then \(f(x)>M\) \(\displaystyle \lim_{x→a}f(x)=-\infty\) if for every \(M>0\), there exists a \(δ>0\) such that if \(0<|x−a|<δ\), then \(f(x)<-M\) | ||||

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 | ||||

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

flux integral | another name for a surface integral of a vector field; the preferred term in physics and engineering | ||||

flux | the rate of a fluid flowing across a curve in a vector field; the flux of vector field \(\vecs F\) across plane curve \(C\) is line integral \(∫_C \vecs F·\frac{\vecs n(t)}{‖\vecs n(t)‖} \,ds\) | ||||

first derivative test | let \(f\) be a continuous function over an interval \(I\) containing a critical point \(c\) such that \(f\) is differentiable over \(I\) except possibly at \(c\); if \(f'\) changes sign from positive to negative as \(x\) increases through \(c\), then \(f\) has a local maximum at \(c\); if \(f'\) changes sign from negative to positive as \(x\) increases through \(c\), then \(f\) has a local minimum at \(c\); if \(f'\) does not change sign as \(x\) increases through \(c\), then \(f\) does not have a local extremum at \(c\) | ||||

Fermat’s theorem | if \(f\) has a local extremum at \(c\), then \(c\) is a critical point of \(f\) | ||||

extreme value theorem | if \(f\) is a continuous function over a finite, closed interval, then \(f\) has an absolute maximum and an absolute minimum | ||||

exponential growth | systems that exhibit exponential growth follow a model of the form \(y=y_0e^{kt}\) | ||||

exponential decay | systems that exhibit exponential decay follow a model of the form \(y=y_0e^{−kt}\) | ||||

exponent | the value \(x\) in the expression \(b^x\) | ||||

explicit formula | a sequence may be defined by an explicit formula such that \(\displaystyle a_n=f(n)\) | ||||

even function | a function is even if \(f(−x)=f(x)\) for all \(x\) in the domain of \(f\) | ||||

Euler’s Method | a numerical technique used to approximate solutions to an initial-value problem | ||||

equivalent vectors | vectors that have the same magnitude and the same direction | ||||

equilibrium solution | any solution to the differential equation of the form \( y=c,\) where \( c\) is a constant | ||||

epsilon-delta definition of the limit | \(\displaystyle \lim_{x→a}f(x)=L\) if for every \(ε>0\), there exists a \(δ>0\) such that if \(0<|x−a|<δ\), then \(|f(x)−L|<ε\) | ||||

end behavior | the behavior of a function as \(x→∞\) and \(x→−∞\) | ||||

elliptic paraboloid | a three-dimensional surface described by an equation of the form \( z=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}\); traces of this surface include ellipses and parabolas | ||||

elliptic cone | a three-dimensional surface described by an equation of the form \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=0\); traces of this surface include ellipses and intersecting lines | ||||

ellipsoid | a three-dimensional surface described by an equation of the form \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\); all traces of this surface are ellipses | ||||

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 | ||||

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 \((\ln 2)/k\) | ||||

double Riemann sum | of the function \(f(x,y)\) over a rectangular region \(R\) is \[\sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \,\Delta A, \nonumber \] where \(R\) is divided into smaller subrectangles \(R_{ij}\) and \((x_{ij}^*, y_{ij}^*)\) is an arbitrary point in \(R_{ij}\) | ||||

double integral | of the function \(f(x,y)\) over the region \(R\) in the \(xy\)-plane is defined as the limit of a double Riemann sum, \[ \iint_R f(x,y) \,dA = \lim_{m,n\rightarrow \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \,\Delta A. \nonumber \] | ||||

dot product or scalar product | \(\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2+u_3v_3\) where \(\vecs{ u}=⟨u_1,u_2,u_3⟩\) and \(\vecs{ v}=⟨v_1,v_2,v_3⟩\) | ||||

domain | the set of inputs for a function | ||||

divergent sequence | a sequence that is not convergent is divergent | ||||

divergence test | if \(\displaystyle \lim_{n→∞}a_n≠0,\) then the series \(\displaystyle \sum^∞_{n=1}a_n\) diverges | ||||

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

divergence | the divergence of a vector field \(\vecs{F}=⟨P,Q,R⟩\), denoted \(\vecs ∇× \vecs{F}\), is \(P_x+Q_y+R_z\); it measures the “outflowing-ness” of a vector field | ||||

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

discriminant | the value \(4AC−B^2\), which is used to identify a conic when the equation contains a term involving \(xy\), is called a discriminant | ||||

discriminant | the discriminant of the function \(f(x,y)\) is given by the formula \(D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)−(f_{xy}(x_0,y_0))^2\) | ||||

discontinuity at a point | A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point | ||||

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 | ||||

directional derivative | the derivative of a function in the direction of a given unit vector | ||||

gradient | the gradient of the function \(f(x,y)\) is defined to be \(\vecs ∇f(x,y)=(∂f/∂x)\,\hat{\mathbf i}+(∂f/∂y)\,\hat{\mathbf j},\) which can be generalized to a function of any number of independent variables | ||||

direction vector | a vector parallel to a line that is used to describe the direction, or orientation, of the line in space | ||||

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 | ||||

direction cosines | the cosines of the angles formed by a nonzero vector and the coordinate axes | ||||

direction angles | the angles formed by a nonzero vector and the coordinate axes | ||||

differentiation | the process of taking a derivative | ||||

differential form | given a differentiable function \(y=f'(x),\) the equation \(dy=f'(x)\,dx\) is the differential form of the derivative of \(y\) with respect to \(x\) | ||||

differential equation | an equation involving a function \(y=y(x)\) and one or more of its derivatives | ||||

differential calculus | the field of calculus concerned with the study of derivatives and their applications | ||||

differential | the differential \(dx\) is an independent variable that can be assigned any nonzero real number; the differential \(dy\) is defined to be \(dy=f'(x)\,dx\) | ||||

differentiable on \(S\) | a function for which \(f'(x)\) exists for each \(x\) in the open set \(S\) is differentiable on \(S\) | ||||

differentiable function | a function for which \(f'(x)\) exists is a differentiable function | ||||

differentiable at \(a\) | a function for which \(f'(a)\) exists is differentiable at \(a\) | ||||

differentiable | a function \( f(x,y)\) is differentiable at \( (x_0,y_0)\) if \( f(x,y)\) can be expressed in the form \( f(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x−x_0)+f_y(x_0,y_0)(y−y_0)+E(x,y),\) where the error term \( E(x,y)\) satisfies \( \lim_{(x,y)→(x_0,y_0)}\dfrac{E(x,y)}{\sqrt{(x−x_0)^2+(y−y_0)^2}}=0\) | ||||

difference rule | the derivative of the difference of a function \(f\) and a function \(g\) is the same as the difference of the derivative of \(f\) and the derivative of \(g\): \(\dfrac{d}{dx}\big(f(x)−g(x)\big)=f′(x)−g′(x)\) | ||||

difference quotient | of a function \(f(x)\) at \(a\) is given by \(\dfrac{f(a+h)−f(a)}{h}\) or \(\dfrac{f(x)−f(a)}{x−a}\) | ||||

difference law for limits | the limit law \[\lim_{x→a}(f(x)−g(x))=\lim_{x→a}f(x)−\lim_{x→a}g(x)=L−M \nonumber \] | ||||

derivative of a vector-valued function | the derivative of a vector-valued function \(\vecs{r}(t)\) is \(\vecs{r}′(t) = \lim \limits_{\Delta t \to 0} \frac{\vecs r(t+\Delta t)−\vecs r(t)}{ \Delta t}\), provided the limit exists | ||||

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

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

dependent variable | the output variable for a function | ||||

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 | ||||

degree | for a polynomial function, the value of the largest exponent of any term | ||||

definite integral of a vector-valued function | the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function | ||||

definite integral | a primary operation of calculus; the area between the curve and the \(x\)-axis over a given interval is a definite integral | ||||

decreasing on the interval \(I\) | a function decreasing on the interval \(I\) if, for all \(x_1,\,x_2∈I,\;f(x_1)≥f(x_2)\) if \(x_1<x_2\) | ||||

cylindrical coordinate system | a way to describe a location in space with an ordered triple \((r,θ,z),\) where \((r,θ)\) represents the polar coordinates of the point’s projection in the \(xy\)-plane, and z represents the point’s projection onto the \(z\)-axis |
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cylinder | a set of lines parallel to a given line passing through a given curve | ||||

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

cusp | a pointed end or part where two curves meet | ||||

curvature | the derivative of the unit tangent vector with respect to the arc-length parameter | ||||

curl | the curl of vector field \(\vecs{F}=⟨P,Q,R⟩\), denoted \(\vecs ∇× \vecs{F}\) is the “determinant” of the matrix \[\begin{vmatrix} \mathbf{\hat i} & \mathbf{\hat j} & \mathbf{\hat k} \ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \ P & Q & R \end{vmatrix}. \nonumber \] and is given by the expression \((R_y−Q_z)\,\mathbf{\hat i} +(P_z−R_x)\,\mathbf{\hat j} +(Q_x−P_y)\,\mathbf{\hat k} \); it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point | ||||

cubic function | a polynomial of degree 3; that is, a function of the form \(f(x)=ax^3+bx^2+cx+d\), where \(a≠0\) | ||||

cross-section | the intersection of a plane and a solid object | ||||

cross product | \(\vecs u×\vecs v=(u_2v_3−u_3v_2)\mathbf{\hat i}−(u_1v_3−u_3v_1)\mathbf{\hat j}+(u_1v_2−u_2v_1)\mathbf{\hat k},\) where \(\vecs u=⟨u_1,u_2,u_3⟩\) and \(\vecs v=⟨v_1,v_2,v_3⟩\) determinant a real number associated with a square matrix parallelepiped a three-dimensional prism with six faces that are parallelograms torque the effect of a force that causes an object to rotate triple scalar product the dot product of a vector with the cross product of two other vectors: \(\vecs u⋅(\vecs v×\vecs w)\) vector product the cross product of two vectors. | ||||

critical point of a function of two variables | the point \((x_0,y_0)\) is called a critical point of \(f(x,y)\) if one of the two following conditions holds: 1. \(f_x(x_0,y_0)=f_y(x_0,y_0)=0\) 2. At least one of \(f_x(x_0,y_0)\) and \(f_y(x_0,y_0)\) do not exist | ||||

critical point | if \(f'(c)=0\) or \(f'(c)\) is undefined, we say that c is a critical point of \(f\) | ||||

coordinate plane | a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the \(xy\)-plane, \(xz\)-plane, or the \(yz\)-plane | ||||

convergent sequence | a convergent sequence is a sequence \(\displaystyle {a_n}\) for which there exists a real number \(\displaystyle L\) such that \(\displaystyle a_n\) is arbitrarily close to \(\displaystyle L\) as long as \(\displaystyle n\) is sufficiently large | ||||

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

contour map | a plot of the various level curves of a given function \(f(x,y)\) | ||||

continuity over an interval | a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function \(f(x)\) is continuous over a closed interval of the form [\(a,b\)] if it is continuous at every point in (\(a,b\)), and it is continuous from the right at \(a\) and from the left at \(b\) | ||||

continuity from the right | A function is continuous from the right at a if \(\displaystyle \lim_{x→a^+}f(x)=f(a)\) | ||||

continuity from the left | A function is continuous from the left at b if \(\displaystyle \lim_{x→b^−}f(x)=f(b)\) | ||||

continuity at a point | A function \(f(x)\) is continuous at a point a if and only if the following three conditions are satisfied: (1) \(f(a)\) is defined, (2) \(\displaystyle \lim_{x→a}f(x)\) exists, and (3) \(\displaystyle \lim{x→a}f(x)=f(a)\) | ||||

constraint | an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem | ||||

constant rule | the derivative of a constant function is zero: \(\dfrac{d}{dx}(c)=0\), where \(c\) is a constant | ||||

constant multiple rule | the derivative of a constant \(c\) multiplied by a function \(f\) is the same as the constant multiplied by the derivative: \(\dfrac{d}{dx}\big(cf(x)\big)=cf′(x)\) | ||||

constant multiple law for limits | the limit law \[\lim_{x→a}cf(x)=c⋅\lim_{x→a}f(x)=cL \nonumber \] | ||||

conservative field | a vector field for which there exists a scalar function \(f\) such that \(\vecs ∇f=\vecs{F}\) | ||||

connected set | an open set \(S\) that cannot be represented as the union of two or more disjoint, nonempty open subsets | ||||

connected region | a region in which any two points can be connected by a path with a trace contained entirely inside the region | ||||

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

conditional convergence | if the series \(\displaystyle \sum^∞_{n=1}a_n\) converges, but the series \(\displaystyle \sum^∞_{n=1}|a_n|\) diverges, the series \(\displaystyle \sum^∞_{n=1}a_n\) is said to converge conditionally | ||||

concavity test | suppose \(f\) is twice differentiable over an interval \(I\); if \(f''>0\) over \(I\), then \(f\) is concave up over \(I\); if \(f''<\) over \(I\), then \(f\) is concave down over \(I\) | ||||

concavity | the upward or downward curve of the graph of a function | ||||

concave up | if \(f\) is differentiable over an interval \(I\) and \(f'\) is increasing over \(I\), then \(f\) is concave up over \(I\) | ||||

concave down | if \(f\) is differentiable over an interval \(I\) and \(f'\) is decreasing over \(I\), then \(f\) is concave down over \(I\) | ||||

computer algebra system (CAS) | technology used to perform many mathematical tasks, including integration | ||||

composite function | given two functions \(f\) and \(g\), a new function, denoted \(g∘f\), such that \((g∘f)(x)=g(f(x))\) | ||||

component functions | the component functions of the vector-valued function \(\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}\) are \(f(t)\) and \(g(t)\), and the component functions of the vector-valued function \(\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}+h(t)\hat{\mathbf{k}}\) are \(f(t)\), \(g(t)\) and \(h(t)\) | ||||

component | a scalar that describes either the vertical or horizontal direction of a vector | ||||

complementary equation | for the nonhomogeneous linear differential equation \[a+2(x)y″+a_1(x)y′+a_0(x)y=r(x), \nonumber \] the associated homogeneous equation, called the complementary equation, is \[a_2(x)y''+a_1(x)y′+a_0(x)y=0 \nonumber \] |
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comparison test | If \(0≤a_n≤b_n\) for all \(n≥N\) and \(\displaystyle \sum^∞_{n=1}b_n\) converges, then \(\displaystyle \sum^∞_{n=1}a_n\) converges; if \(a_n≥b_n≥0\) for all \(n≥N\) and \(\displaystyle \sum^∞_{n=1}b_n\) diverges, then \(\displaystyle \sum^∞_{n=1}a_n\) diverges. | ||||

closed set | a set \(S\) that contains all its boundary points | ||||

closed curve | a curve for which there exists a parameterization \(\vecs r(t), a≤t≤b\), such that \(\vecs r(a)=\vecs r(b)\), and the curve is traversed exactly once | ||||

closed curve | a curve that begins and ends at the same point | ||||

circulation | the tendency of a fluid to move in the direction of curve \(C\). If \(C\) is a closed curve, then the circulation of \(\vecs F\) along \(C\) is line integral \(∫_C \vecs F·\vecs T \,ds\), which we also denote \(∮_C\vecs F·\vecs T \,ds\). | ||||

characteristic equation | the equation \(aλ^2+bλ+c=0\) for the differential equation \(ay″+by′+cy=0\) | ||||

change of variables | the substitution of a variable, such as \(u\), for an expression in the integrand | ||||

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

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 | ||||

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

catenary | a curve in the shape of the function \(y=a\cdot\cosh(x/a)\) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary | ||||

carrying capacity | the maximum population of an organism that the environment can sustain indefinitely | ||||

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 \(r=a(1+\sin θ)\) or \(r=a(1+\cos θ)\) | ||||

bounded sequence | a sequence \(\displaystyle {a_n}\) is bounded if there exists a constant \(\displaystyle M\) such that \(\displaystyle |a_n|≤M\) for all positive integers \(\displaystyle n\) | ||||

bounded below | a sequence \(\displaystyle {a_n}\) is bounded below if there exists a constant \(\displaystyle M\) such that \(\displaystyle M≤a_n\) for all positive integers \(\displaystyle n\) | ||||

bounded above | a sequence \(\displaystyle {a_n}\) is bounded above if there exists a constant \(\displaystyle M\) such that \(\displaystyle a_n≤M\) for all positive integers \(\displaystyle n\) | ||||

boundary-value problem | a differential equation with associated boundary conditions | ||||

boundary point | a point \(P_0\) of \(R\) is a boundary point if every \(δ\) disk centered around \(P_0\) contains points both inside and outside \(R\) | ||||

boundary conditions | the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times | ||||

binormal vector | a unit vector orthogonal to the unit tangent vector and the unit normal vector | ||||

binomial series | the Maclaurin series for \( f(x)=(1+x)^r\); it is given by \( (1+x)^r=\sum_{n=0}^∞(^r_n)x^n=1+rx+\dfrac{r(r−1)}{2!}x^2+⋯+\dfrac{r(r−1)⋯(r−n+1)}{n!}x^n+⋯\) for \( |x|<1\) | ||||

base | the number \(b\) in the exponential function \(f(x)=b^x\) and the logarithmic function \(f(x)=\log_bx\) | ||||

average velocity | the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval [\(t,a\)] (if \(t<a\) or [\(a,t\)] if \(t>a\)), with a position given by \(s(t)\), that is \(v_{ave}=\dfrac{s(t)−s(a)}{t−a}\) | ||||

average value of a function | (or \(f_{ave})\) 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 | ||||

average rate of change | is a function \(f(x)\) over an interval \([x,x+h]\) is \(\frac{f(x+h)−f(a)}{b−a}\) | ||||

autonomous differential equation | an equation in which the right-hand side is a function of \(y\) alone | ||||

asymptotically unstable solution | \( y=k\) if there exists \( ε>0\) such that for any value \( c∈(k−ε,k+ε)\) the solution to the initial-value problem \( y′=f(x,y),y(x_0)=c\) never approaches \( k\) as \( x\)approaches infinity | ||||

asymptotically stable solution | \( y=k\) if there exists \( ε>0\) such that for any value \( c∈(k−ε,k+ε)\) the solution to the initial-value problem \( y′=f(x,y),y(x_0)=c\) approaches \( k\) as \( x\) approaches infinity | ||||

asymptotically semi-stable solution | \( y=k\) if it is neither asymptotically stable nor asymptotically unstable | ||||

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

arc-length parameterization | a reparameterization of a vector-valued function in which the parameter is equal to the arc length | ||||

arc-length function | a function \(s(t)\) that describes the arc length of curve \(C\) as a function of \(t\) | ||||

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 | ||||

antiderivative | a function \(F\) such that \(F′(x)=f(x)\) for all \(x\) in the domain of \(f\) is an antiderivative of \(f\) | ||||

angular coordinate | \(θ\) 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 | ||||

amount of change | the amount of a function \(f(x)\) over an interval \([x,x+h] is f(x+h)−f(x)\) | ||||

alternating series test | for an alternating series of either form, if \( b_{n+1}≤b_n\) for all integers \( n≥1\) and \( b_n→0\), then an alternating series converges | ||||

alternating series | a series of the form \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}b_n\) or \(\displaystyle \sum^∞_{n=1}(−1)^nb_n\), where \( b_n≥0\), is called an alternating series | ||||

algebraic function | a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable \(x\) | ||||

acceleration vector | the second derivative of the position vector | ||||

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

absolute value function | \(f(x)=\begin{cases}−x, & \text{if } x<0\x, & \text{if } x≥0\end{cases}\) | ||||

absolute minimum | if \(f(c)≤f(x)\) for all \(x\) in the domain of \(f\), we say \(f\) has an absolute minimum at \(c\) | ||||

absolute maximum | if \(f(c)≥f(x)\) for all \(x\) in the domain of \(f\), we say \(f\) has an absolute maximum at \(c\) | ||||

absolute extremum | if \(f\) has an absolute maximum or absolute minimum at \(c\), we say \(f\) has an absolute extremum at \(c\) | ||||

absolute error | if \(B\) is an estimate of some quantity having an actual value of \(A\), then the absolute error is given by \( |A−B|\) | ||||

absolute convergence | if the series \(\displaystyle \sum^∞_{n=1}|a_n|\) converges, the series \(\displaystyle \sum^∞_{n=1}a_n\) is said to converge absolutely | ||||

\(δ\) disk | an open disk of radius \(δ\) centered at point \((a,b)\) | ||||

\(δ\) ball | all points in \(\mathbb{R}^3\) lying at a distance of less than \(δ\) from \((x_0,y_0,z_0)\) | ||||

steady-state solution | a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution |