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APEX Calculus

Gregory Hartman, Ph.D., Sean Fitzpatrick, Ph.D. (Editor), Alex Jordan, Ph.D. (Editor), Carly Vollet, M.S. (Editor)

Index Index

aa@\("!\), Definition
Absolute Convergence Theorem, Theorem
absolute maximum, Definition
absolute minimum, Definition
Absolute Value Theorem, Theorem
acceleration, Paragraph Item
accumulated error
using Euler’s method, Paragraph
Alternating Harmonic Series, Item Paragraph Paragraph
Alternating Series Test, Theorem
an@\(a_\text{N}\), Paragraph Paragraph
analytic function, Paragraph
angle of elevation, Paragraph
antiderivative, Definition
of vector-valued function, Definition
approximation
linear, Definition
tangent line, Definition
arc length, Theorem Theorem Paragraph Theorem Paragraph
arc length parameter, Paragraph Theorem
asymptote
horizontal, Item
vertical, Definition
at@\(a_\text{T}\), Paragraph Paragraph
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
interval, Section
bounded sequence, Definition
convergence, Theorem
bounded set, Definition
carrying capacity, Example
center of mass, Paragraph Theorem Theorem Theorem Definition
Chain Rule, Theorem
multivariable, Theorem Item
notation, Paragraphs
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
properties, Theorem Theorem
vector-valued, Definition Theorem
continuously differentiable, Definition
contour lines, Paragraph
convergence
absolute, Item Theorem
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 improper int., Item Item Item Theorem
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
cross product
and derivatives, Theorem
applications, Paragraph
applications!area of parallelogram, Paragraphs
applications!torque, Paragraphs
applications!volume of parallelepiped, Paragraphs
definition, Definition
properties, Theorem Theorem
curl, Paragraph Item
of conservative fields, Theorem
curvature, Definition
and motion, Paragraph
equations for, Theorem
of circle, Example Paragraph
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
implicit, Paragraph Theorem
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 Rule, Item Theorem Theorem
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
difference quotient, Paragraph
differentiable, Definition Item 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
logistic, Example Example
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 field
directional derivative, Definition Theorem Key Idea Item Definition Theorem
directrix, Definition Definition
discontinuity
infinite, Item
jump, Item
removable, Item
Disk Method, Key Idea
displacement, Paragraph Paragraph Paragraph
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 improper int., Item Item Item Theorem
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
properties, Theorem Theorem
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 Value Theorem, Theorem 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
flux, Paragraph Definition Paragraph Definition
focus, Definition Definition Definition
Fubini’s Theorem, Theorem
function
continuous, Definition
floor, Example
of three variables, Definition
of two variables, Definition
vector-valued, Definition
Fundamental Theorem of Calculus, Theorem Theorem
and Chain Rule, Paragraph
Fundamental Theorem of Line Integrals, Paragraph Theorem
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
Green’s Theorem, Paragraph 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
implicit differentiation, Paragraph Theorem
improper integration, Item Definition
incompressible vector field, Paragraph
increasing function, Definition
finding intervals, Key Idea
indefinite integral, Definition
of vector-valued function, Definition
indeterminate form, Paragraph Paragraph Paragraph Key Idea
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
area between curves, Paragraph Theorem
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
Fun. Thm. of Calc., Theorem Theorem
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
numerical!Left/Right Hand Rule, Paragraph Key Idea
numerical!Simpson’s Rule, Paragraph Key Idea Theorem
numerical!Trapezoidal Rule, Paragraph Key Idea Theorem
of multivariable functions, Paragraph
of power series, Theorem
of trig. functions, Theorem
of trig. powers, Key Idea Key Idea
of vector-valued function, Definition
of vector-valued functions, Theorem
partial fraction decomp., Key Idea
Power Rule, Item
Sum/Difference Rule, Item
surface area, Theorem Theorem Theorem
trig. subst., Item
triple, Definition Theorem Definition Definition Theorem
volume!cross-sectional area, Theorem
volume!Disk Method, Key Idea
volume!Shell Method, Key Idea Key Idea
volume!Washer Method, Key Idea Key Idea
with cylindrical coordinates, Theorem
with spherical coordinates, Theorem
work, Key Idea
interior point, Definition
Intermediate Value Theorem, Theorem
interval of convergence, Item
iterated integration, Definition Definition Theorem Definition Theorem Definition Definition Theorem
changing order, Paragraph
properties, Theorem Theorem
lamina, Paragraph
Left Hand Rule, Paragraph Paragraph Paragraph
Left/Right Hand Rule, Key Idea
level curves, Paragraph Item
level surface, Paragraph Theorem
limit
Absolute Value Theorem, Theorem
at infinity, Item
definition, Definition
difference quotient, Paragraph
does not exist, Paragraph Theorem
indeterminate form, Paragraph Paragraph Paragraph Paragraph Key Idea
left-handed, Item
l’Hospital’s Rule, Theorem 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
one-sided, Item Item
properties, Theorem Theorem
pseudo-definition, Paragraph
right-handed, Item
Squeeze Theorem, Theorem
Limit Comparison Test
for integration, Theorem
for series, Theorem
line integral
Fundamental Theorem, Paragraph Theorem
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
linearization, Definition
lines, Paragraph
distances between, Item
equations for, Definition
intersecting, Paragraph
parallel, Paragraph
skew, Paragraph
logarithmic differentiation, Paragraph
logarithmic function
continuity of, Theorem
l’Hospital’s Rule
infinity over infinity, Item
zero over zero, 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
mass, Paragraph Definition Definition Definition
center of, Paragraph 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
Midpoint Rule, Paragraph Paragraph
minimum
absolute, Definition Definition
and First Deriv. Test, Theorem Theorem
relative/local, Item Definition Theorem
moment, Definition Theorem Definition
monotonic sequence, Item
multiple integration|see{iterated integration}, Definition
multivariable function, Definition Definition
continuity, Item Theorem Item Theorem Item
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
Left/Right Hand Rule, Paragraph Key Idea
Simpson’s Rule, Paragraph Key Idea
Simpson’s Rule!error bounds, Theorem
Trapezoidal Rule, Paragraph Key Idea
Trapezoidal Rule!error bounds, Theorem
octant
first, Paragraph
one to one, Subsection
open, Definition
open ball, Item
open disk, Definition
optimization, Paragraph
constrained, Paragraph
order
of a differential equation, Definition
orientable, Paragraph
orthogonal, Definition Definition
decomposition, Key Idea
orthogonal decomposition of vectors, Key Idea
orthogonal projection, Definition
osculating circle, Paragraph
outer unit normal vector, Subsection
p@\(p\)-series, Item Item Theorem
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
differentiation, Item Theorem Theorem Theorem
integration, Item
power series, Definition
algebra of, Theorem
convergence, Theorem
derivatives and integrals, Theorem
projectile motion, Paragraph Key Idea Example
quadric surface
definition, Definition
ellipsoid, Paragraph
elliptic cone, Paragraph
elliptic paraboloid, Paragraph
gallery, Paragraph 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
rearrangements of series, Paragraph Theorem
reduction formula
trigonometric integral, Paragraph
Related Rates, Key Idea
related rates, Paragraph
Riemann Sum, Paragraph Paragraph Definition
and definite integral, Theorem
Right Hand Rule, Paragraph Paragraph Paragraph
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
Second Derivative Test, Theorem 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
p@\(p\)-series, Item Theorem
partial sums, Item
power, Definition Theorem
power!derivatives and integrals, Theorem
properties, Item
radius of convergence, Item
Ratio Comparison Test, Theorem
rearrangements, Paragraph Theorem
Root Comparison Test, Theorem
Taylor, Definition
telescoping, Example Paragraph
Shell Method, Key Idea Key Idea
signed area, Definition
signed volume, Definition Theorem
simple curve, Paragraph
simply connected, Paragraph
Simpson’s Rule, Paragraph Key Idea
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
of parametrized surface, Paragraph Theorem
solid of revolution, Theorem Theorem Theorem
surface integral, Definition
surface of revolution, Key Idea Key Idea
tangent line, Definition Definition Key Idea Definition
directional, Definition
tangent plane, Definition Definition
to a graph, Definition
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
telescoping series, Example Paragraph
terminal point, Definition
theorem
Intermediate Value, Theorem
torque, Paragraphs
total differential, Definition Definition
sensitivity analysis, Paragraph
total signed area, Definition
trace, Paragraph
Trapezoidal Rule, Paragraph Key Idea
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 acceleration, Paragraph Paragraph
and curvature, Paragraph
definition, Definition
in \(\mathbb{R}^2\), Theorem
unit tangent vector
and acceleration, Paragraph Paragraph
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
of constant length, Theorem Paragraph Key Idea Paragraph
projectile motion, Paragraph Key Idea
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
velocity, Paragraph Item
average velocity, Paragraph
volume, Definition Theorem Theorem
Washer Method, Key Idea Key Idea
work, Key Idea Definition
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