Glossary


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Glossary Entries
Word(s) Definition Image Caption Link Source
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 $$t$$.
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
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
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
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
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
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
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
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
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 $a_2(x)y''+a_1(x)y′+a_0(x)y=0 \nonumber$
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