## Index Index

aa@\("!\), Definition

Absolute Convergence Theorem, Theorem

absolute maximum, Definition

absolute minimum, Definition

Absolute Value Theorem, Theorem

accumulated error

using Euler's method, Paragraph

Alternating Series Test, Theorem

analytic function, Paragraph

angle of elevation, Paragraph

antiderivative, Definition

of vector-valued function, Definition

approximation

linear, Definition

tangent line, Definition

asymptote

horizontal, Item

vertical, Definition

average rate of change, Definition

average value of a function, Definition

average value of function, Definition

average velocity, Paragraph

bacterial growth, Example

Binomial Series, Paragraph

Bisection Method, Paragraphs

boundary point, Definition

bounded sequence, Definition

convergence, Theorem

bounded set, Definition

carrying capacity, Example

center of mass, Paragraph Theorem Theorem Theorem Definition

Chain Rule, Theorem

notation, Paragraphs

chain rule

as matrix multiplication, Theorem

change of variables, Paragraph

circle of curvature, Paragraph

circulation, Paragraph

closed, Definition

closed disk, Definition

concave down, Definition

concave up, Definition

concavity, Definition Paragraph

inflection point, Theorem

test for, Theorem

conic sections, Paragraph

degenerate, Paragraph

ellipse, Definition

hyperbola, Definition

parabola, Definition

connected, Paragraph

simply, Paragraph

conservative field, Definition Theorem Theorem

Constant Multiple Rule

of derivatives, Item

of integration, Item

of series, Item

constrained optimization, Paragraph

continuity

of exponential functions, Theorem

of logarithmic functions, Theorem

of polynomial functions, Theorem

of rational functions, Theorem

of trigonometric functions, Theorem

continuous

at a point, Item

everywhere, Item

on an interval, Item

continuous function, Definition Item

vector-valued, Definition Theorem

contour lines, Paragraph

convergence

Alternating Series Test, Theorem

conditional, Item

Direct Comparison Test, Theorem

Direct Comparison Test!for integration, Item

Integral Test, Theorem

interval of, Item

Limit Comparison Test, Theorem

Limit Comparison Test!for integration, Theorem

nth@\(n\)th-term test, Theorem

of geometric series, Item

of monotonic sequences, Item

of p@of \(p\)-series, Theorem

of power series, Theorem

of sequence, Definition Theorem

of series, Item

radius of, Item

Ratio Comparison Test, Theorem

Root Comparison Test, Theorem

coordinates

cylindrical, Paragraph

polar, Paragraph

spherical, Paragraph

critical number, Definition

critical point, Definition Definition Theorem Definition

critical value

of a function of two variables, Section

cross product

and derivatives, Theorem

applications, Paragraph

applications!area of parallelogram, Paragraphs

applications!torque, Paragraphs

applications!volume of parallelepiped, Paragraphs

definition, Definition

of conservative fields, Theorem

curvature, Definition

and motion, Paragraph

equations for, Theorem

radius of, Paragraph

curve

parametrically defined, Definition

rectangular equation, Paragraph

smooth, Definition

curve sketching, Key Idea

cusp, Definition

cycloid, Example

cylinder, Definition

cylindrical coordinates, Paragraph

decreasing function, Definition

finding intervals, Key Idea

definite integral, Definition

and substitution, Theorem

of vector-valued function, Definition

properties, Theorem

del operator, Paragraph

derivative

acceleration, Key Idea

as a function, Definition

at a point, Definition

basic rules, Theorem

Chain Rule, Theorem Paragraphs Theorem Item

Constant Multiple Rule, Item

Constant Rule, Item

differential, Definition

directional, Definition Theorem Key Idea Item Definition Theorem

exponential functions, Theorem

First Deriv. Test, Theorem

Generalized Power Rule, Theorem

higher order, Definition

higher order!interpretation, Paragraph

hyperbolic funct., Item

interpretation, Paragraph

inverse function, Paragraph

inverse hyper., Item

inverse trig., Theorem

logarithmic, Paragraph

Mean Value Theorem, Theorem

mixed partial, Definition

motion, Key Idea

multivariable differentiability, Item Definition

normal line, Definition

notation, Definition Definition

parametric equations, Key Idea

partial, Definition Definition

power series, Theorem

Product Rule, Theorem

Quotient Rule, Theorem

Second Deriv. Test, Theorem

Sum/Difference Rule, Item

tangent line, Definition

trigonometric functions, Theorem

vector-valued functions, Definition Item Theorem

velocity, Key Idea

derivative!general, Definition

difference quotient, Paragraph

differentiability

functions of several variables, Definition

differentiable, Definition Item Definition

general functions, Definition

on a closed interval, Paragraph

differential, Definition

notation, Key Idea

differential equation

definition, Definition

first order linear, Definition

general solution, Paragraph Definition

graphical solution, Paragraph

implicit soution, Example

integrating factor, Paragraph

modeling, Paragraph

numerical solution, Paragraph

order of, Definition

particular solution, Definition

separable, Definition

Direct Comparison Test

for integration, Item

for series, Theorem

direction field.

*See*slope fielddirectional derivative, Definition Theorem Key Idea Item Definition Theorem

directrix, Definition Definition

discontinuity

infinite, Item

jump, Item

removable, Item

Disk Method, Key Idea

distance

between lines, Item

between point and line, Item

between point and plane, Key Idea

between points in space, Definition

traveled, Theorem

divergence, Paragraph Definition

Alternating Series Test, Theorem

Direct Comparison Test, Theorem

Direct Comparison Test!for integration, Item

Integral Test, Theorem

Limit Comparison Test, Theorem

Limit Comparison Test!for integration, Theorem

nth@\(n\)th-term test, Theorem

of geometric series, Item

of p@of \(p\)-series, Theorem

of sequence, Definition

of series, Item

Ratio Comparison Test, Theorem

Root Comparison Test, Theorem

Divergence Theorem

in space, Theorem

in the plane, Theorem

dot product

and derivatives, Theorem

definition, Definition

double integral, Definition Theorem

in polar, Key Idea

properties, Theorem

eccentricity, Definition Definition

elementary function, Paragraph

ellipse

definition, Definition

eccentricity, Definition

parametric equations, Item

reflective property, Theorem

standard equation, Key Idea

Euler's Method, Key Idea

Euler's method

accumulated error, Paragraph

everywhere continuous, Item

exponential function

continuity of, Theorem

extrema

absolute, Definition Definition

and First Deriv. Test, Theorem

and Second Deriv. Test, Theorem

finding, Key Idea

relative, Item Item Definition Theorem

extreme values, Definition

factorial, Definition

First Derivative Test, Theorem

first octant, Paragraph

floor function, Example

flow, Paragraph Definition

fluid pressure/force, Definition Key Idea

focus, Definition Definition Definition

Fubini's Theorem, Theorem

function

continuous, Definition

floor, Example

of three variables, Definition

of two variables, Definition

vector-valued, Definition

and Chain Rule, Paragraph

Gabriel's Horn, Example

Gauss's Law, Example

general solution

of a differential equation, Paragraph

Generalized Power Rule, Theorem

geometric series, Definition Item

gradient, Definition Key Idea Item Definition Theorem Definition

and level curves, Item

and level surfaces, Theorem

half life, Exercise

Harmonic Series, Item

Head To Tail Rule, Paragraph

Hooke's Law, Paragraph

hyperbola

definition, Definition

eccentricity, Definition

parametric equations, Item

reflective property, Theorem

standard equation, Key Idea

hyperbolic function

definition, Item

derivatives, Item

identities, Item

integrals, Item

inverse, Paragraph

inverse!derivative, Item

inverse!integration, Key Idea

inverse!logarithmic def., Item

image

of a point, Definition

of a subset, Definition

improper integration, Item Definition

incompressible vector field, Paragraph

increasing function, Definition

finding intervals, Key Idea

indefinite integral, Definition

of vector-valued function, Definition

inflection point, Definition

initial condition, Definition

initial point, Definition

initial value problem, Paragraphs

for differential equations, Definition

Integral Test, Theorem

integration

arc length, Theorem

area, Definition Paragraph Item

average value, Definition

by parts, Theorem

by substitution, Theorem

definite, Definition

definite!and substitution, Theorem

definite!properties, Theorem

definite!Riemann Sums, Theorem

displacement, Paragraph

distance traveled, Theorem

double, Definition

fluid force, Definition Key Idea

general application technique, Key Idea

hyperbolic funct., Item

improper, Item Definition Item Theorem

indefinite, Definition

inverse hyperbolic, Key Idea

iterated, Definition

Mean Value Theorem, Theorem

multiple, Definition

notation, Paragraph Definition Paragraph Definition

numerical, Paragraph

of multivariable functions, Paragraph

of power series, Theorem

of trig. functions, Theorem

of vector-valued function, Definition

of vector-valued functions, Theorem

partial fraction decomp., Key Idea

Power Rule, Item

Sum/Difference Rule, Item

trig. subst., Item

triple, Definition Theorem Definition Definition Theorem

volume!cross-sectional area, Theorem

volume!Disk Method, Key Idea

with cylindrical coordinates, Theorem

with spherical coordinates, Theorem

work, Key Idea

interior point, Definition

Intermediate Value Theorem, Theorem

interval of convergence, Item

inverse

of a transformation, Paragraph

iterated integration, Definition Definition Theorem Definition Theorem Definition Definition Theorem

changing order, Paragraph

l'Hospital's Rule

infinity over infinity, Item

zero over zero, Theorem

Lagrange multipliers, Paragraph

lamina, Paragraph

Left/Right Hand Rule, Key Idea

limit

Absolute Value Theorem, Theorem

at infinity, Item

definition, Definition

difference quotient, Paragraph

left-handed, Item

of exponential functions, Theorem

of infinity, Definition

of logarithmic functions, Theorem

of multivariable function, Definition Theorem Item

of polynomial functions, Theorem

of rational functions, Theorem

of sequence, Definition

of trigonometric functions, Theorem

of vector-valued functions, Definition Item

pseudo-definition, Paragraph

right-handed, Item

Squeeze Theorem, Theorem

Limit Comparison Test

for integration, Theorem

for series, Theorem

line integral

over scalar field, Definition Item Paragraph

over vector field, Definition

path independent, Definition Theorem

properties over a scalar field, Item

properties over a vector field, Item

linear function, Definition

linearization, Definition Paragraph

functions of several variables, Definition

lines, Paragraph

distances between, Item

equations for, Definition

intersecting, Paragraph

parallel, Paragraph

skew, Paragraph

logarithmic differentiation, Paragraph

logarithmic function

continuity of, Theorem

Maclaurin Polynomial

definition, Definition

Maclaurin Polynomial|see{Taylor Polynomial}, Definition

Maclaurin Series

definition, Definition

Maclaurin Series|see{Taylor Series}, Definition

magnitude of vector, Definition

center of, Paragraph Definition

matrix

Jacobian, Definition

maximum

absolute, Definition Definition

and First Deriv. Test, Theorem

and Second Deriv. Test, Theorem

relative/local, Item Definition Theorem

Mean Value Theorem

of differentiation, Theorem

of integration, Theorem

minimum

absolute, Definition Definition

relative/local, Item Definition Theorem

moment, Definition Theorem Definition

monotonic sequence, Item

multi-index notation, Paragraph

multiple integration|see{iterated integration}, Definition

multivariable function, Definition Definition

differentiability, Item Theorem Theorem Definition Item

domain, Definition Definition

level curves, Paragraph

level surface, Paragraph

limit, Definition Theorem Item

range, Definition Definition

Möbius band, Paragraph

Newton's Law of Cooling, Paragraph

Newton's Method, Key Idea

norm, Definition

normal line, Definition Definition Definition

normal vector, Paragraph

nth@\(n\)th-term test, Theorem

numerical integration, Paragraph

Simpson's Rule!error bounds, Theorem

Trapezoidal Rule!error bounds, Theorem

octant

first, Paragraph

one to one, Subsection

one-to-one, Definition

onto, Definition

open, Definition

open ball, Item

open disk, Definition

optimization, Paragraph

constrained, Paragraph

with Lagrange multipliers, Paragraph

order

of a differential equation, Definition

orientable, Paragraph

orientation, Item

orthogonal, Definition Definition

decomposition, Key Idea

orthogonal decomposition of vectors, Key Idea

orthogonal projection, Definition

osculating circle, Paragraph

outer unit normal vector, Subsection

parabola

definition, Definition

general equation, Key Idea

reflective property, Item

parallel vectors, Item

Parallelogram Law, Paragraph

parametric equations

arc length, Theorem

concavity, Paragraph

definition, Definition

finding \(\frac{d^2y}{dx^2}\), Key Idea

finding \(\frac{dy}{dx}\), Key Idea

normal line, Definition

of a surface, Definition

surface area, Theorem

tangent line, Definition

parametrized surface, Definition

partial derivative, Definition Definition

high order, Paragraph

meaning, Paragraph

mixed, Definition

second derivative, Definition

total differential, Definition Definition

partition, Definition

size of, Definition

path independent, Definition Theorem

perpendicular|see{orthogonal}, Definition

piecewise smooth curve, Paragraph

planes

coordinate plane, Paragraph

distance between point and plane, Key Idea

equations of, Definition

introduction, Paragraph

normal vector, Paragraph

tangent, Definition

point of inflection, Definition

polar

coordinates, Paragraph

function!arc length, Paragraph

function!gallery of graphs, Paragraph

function!surface area, Theorem

functions, Paragraph

functions!area, Paragraph

functions!area between curves, Paragraphs

functions!finding \(\frac{dy}{dx}\), Key Idea

functions!graphing, Paragraph

polar coordinates, Paragraph

plotting points, Example

polynomial function

continuity of, Theorem

potential function, Paragraph Definition

Power Rule

integration, Item

power series, Definition

algebra of, Theorem

convergence, Theorem

derivatives and integrals, Theorem

quadric surface

definition, Definition

ellipsoid, Paragraph

elliptic cone, Paragraph

elliptic paraboloid, Paragraph

hyperbolic paraboloid, Paragraph

hyperboloid of one sheet, Paragraph

hyperboloid of two sheets, Paragraph

sphere, Paragraph

trace, Paragraph

Quotient Rule, Theorem

r@\(\mathbb{R}\), Paragraph

radius of convergence, Item

radius of curvature, Paragraph

Ratio Comparison Test

for series, Theorem

rational function

continuity of, Theorem

reduction formula

trigonometric integral, Paragraph

regular value, Paragraph

Related Rates, Key Idea

related rates, Paragraph

Riemann Sum, Paragraph Paragraph Definition

and definite integral, Theorem

right hand rule

of Cartesian coordinates, Paragraph

of the cross product, Paragraphs

Rolle's Theorem, Theorem

Root Comparison Test

for series, Theorem

saddle point, Definition Theorem

sensitivity analysis, Paragraph

separation of variables, Paragraph

sequence

Absolute Value Theorem, Theorem

positive, Subsection

sequences

boundedness, Definition

convergent, Definition Theorem Item

definition, Definition

divergent, Definition

limit, Definition

limit properties, Item

monotonic, Item

series

absolute convergence, Item

Absolute Convergence Theorem, Theorem

alternating, Definition

Alternating Series Test, Theorem

alternating!Approximation Theorem, Theorem

Binomial, Paragraph

conditional convergence, Item

convergent, Item

definition, Item

Direct Comparison Test, Theorem

divergent, Item

geometric, Definition Item

Integral Test, Theorem

interval of convergence, Item

Limit Comparison Test, Theorem

Maclaurin, Definition

nth@\(n\)th-term test, Theorem

partial sums, Item

power, Definition Theorem

power!derivatives and integrals, Theorem

properties, Item

radius of convergence, Item

Ratio Comparison Test, Theorem

Root Comparison Test, Theorem

Taylor, Definition

signed area, Definition

signed volume, Definition Theorem

simple curve, Paragraph

simply connected, Paragraph

error bounds, Theorem

slope field, Definition

smooth, Definition

curve, Definition

surface, Definition

smooth curve

piecewise, Paragraph

speed, Item

sphere, Paragraph

spherical coordinates, Paragraph

Squeeze Theorem, Theorem

Stokes' Theorem, Theorem

Sum/Difference Rule

of derivatives, Item

of integration, Item

of series, Item

summation

notation, Paragraph

properties, Theorem

surface, Definition

smooth, Definition

surface area, Definition

surface integral, Definition

tangent line, Definition Definition Key Idea Definition

directional, Definition

tangent plane, Definition Definition

to a graph, Definition

Taylor polynimial

of several variables, Paragraph

Taylor Polynomial

definition, Definition

Taylor's Theorem, Item

Taylor Series

common series, Key Idea

definition, Definition

equality with generating function, Theorem

Taylor's Theorem, Item

in several variables, Paragraph

terminal point, Definition

theorem

Intermediate Value, Theorem

torque, Paragraphs

total differential, Definition Definition

sensitivity analysis, Paragraph

total signed area, Definition

trace, Paragraph

transformation, Paragraph Definition

error bounds, Theorem

trigonometric function

continuity of, Theorem

triple integral, Definition Theorem Definition Definition Theorem

properties, Theorem

unbounded sequence, Definition

unbounded set, Definition

unit normal vector

an@\(a_\text{N}\), Paragraph

and curvature, Paragraph

definition, Definition

in \(\mathbb{R}^2\), Theorem

unit tangent vector

and curvature, Definition Paragraph

at@\(a_\text{T}\), Paragraph

definition, Definition

in \(\mathbb{R}^2\), Theorem

unit vector, Definition

properties, Item

standard unit vector, Item

unit normal vector, Definition

unit tangent vector, Definition

vector field, Item

conservative, Definition Theorem

curl of, Paragraph

divergence of, Paragraph Definition

over vector field, Key Idea

potential function of, Paragraph Definition

vector-valued function

algebra of, Definition

arc length, Theorem

average rate of change, Definition

continuity, Definition Theorem

definition, Definition

derivatives, Definition Item Theorem

describing motion, Item

displacement, Paragraph

distance traveled, Theorem

graphing, Paragraph

integration, Theorem

limits, Definition Item

smooth, Definition

tangent line, Definition

vectors, Definition

algebra of, Item

algebraic properties, Theorem

component form, Definition

cross product, Definition Theorem Theorem

definition, Definition

dot product, Definition Theorem Theorem

Head To Tail Rule, Paragraph

magnitude, Definition

norm, Definition

normal vector, Paragraph

orthogonal, Definition

orthogonal decomposition, Key Idea

orthogonal projection, Definition

parallel, Item

Parallelogram Law, Paragraph

resultant, Paragraph

standard unit vector, Item

unit vector, Definition Item

zero vector, Paragraph

average velocity, Paragraph

volume, Definition Theorem Theorem

work, Key Idea Definition