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Introduction
to Functions |
Function definition, notation and
terminology |
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Domain, range and
codomain |
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Evaluating
a function |
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Composition
of functions (a function of a function) |
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Inverse
function
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The
graph of a function |
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Function's
behavior, properties
and characteristic points of the graph |
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Domain
and range |
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Roots or zero function
values, x-intercepts, y-intercepts |
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Increasing/decreasing
intervals |
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The
instantaneous
rate of change or the derivative |
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Continuity and discontinuity |
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Vertical,
horizontal and oblique or slant asymptotes |
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Stationary
points and/or critical points |
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Turning points (extremes, local or relative
maximums
or minimums) |
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Inflection points and intervals of concavity |
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Symmetry
of a function, parity - odd and even functions
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Transformations
of original or source function |
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How
some changes of a function's notation affect the
graph of the function |
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Translations
of the graph of a function
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Reflections
of the graph of a function |
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Types
of functions - basic classification |
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Algebraic functions
and Transcendental functions |
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Algebraic functions |
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The
polynomial function |
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Rational
functions |
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Reciprocal
function |
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Transcendental functions |
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Exponential
and logarithmic functions, inverse functions |
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Trigonometric
(cyclometric) functions and inverse trigonometric functions (arc-functions) |
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The
graphs of the elementary functions |
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The
graphs of algebraic and transcendental functions |
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The
graphs of algebraic functions |
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The
graphs of the polynomial functions |
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The
source
or original polynomial function |
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Translating
(parallel shifting) of the polynomial function |
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Coordinates of translations
and their role in the polynomial expression |
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The
graph of the linear function |
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The
graph of the quadratic function |
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The
graphs of the cubic function |
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The
graphs of the quartic function |
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Sigma
notation of the polynomial |
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Coefficients of the source
polynomial in the form of a recursive formula |
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Coefficients
of the source polynomial function are related to its derivative
at x0 |
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Translated
power function |
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Translated
sextic function example
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Graphs
of rational
functions |
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Basic
properties of rational functions |
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Vertical asymptotes of
rational functions |
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Horizontal asymptote
of rational functions |
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The oblique or slant
asymptote of rational functions |
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The
graph of the reciprocal
function, equilateral or rectangular hyperbola |
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Translation of the reciprocal function, linear rational
function |
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Graphs of rational
functions, examples |
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The graphs of transcendental functions |
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Exponential
and logarithmic functions are mutually inverse functions |
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The
graph of the exponential
function |
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The graph of
the
logarithmic
function |
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Trigonometric
(cyclometric) functions and inverse trigonometric functions (arc
functions) |
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The
graphs of the trigonometric
functions and inverse trigonometric functions or arc-functions |
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The graph of
the sine function |
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The graph of
the
cosine function |
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The graph of
the arc-sine function and the arc-cosine function |
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The
graph of the tangent function
and the
cotangent function |
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The
graph of the
arc-tangent function and the arc-cotangent function |
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The graph of
the
cosecant function |
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The graph of
the
secant function |
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The
graph of the
arc-cosecant and the
arc-secant function |
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Sequences
and limits
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Infinite
sequences |
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Sequences notation -
the
rule for the n-th
term of a sequence |
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Graphing the terms of a sequence on the number line |
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The limit of a sequence |
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The definition of the limit of a sequence |
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Convergence of a sequence |
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Verifying the convergence of a sequence from the definition, examples |
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Limits of sequences |
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Properties of
convergent sequences |
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The limit value is exclusively determined by the behavior of the terms
in its close neighborhood |
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Bounded
sequences |
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Every convergent sequence is bounded |
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Increasing,
decreasing, monotonic sequence |
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Every subsequence of
a convergent sequence
converges to the same limit |
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Every
bounded monotonic sequence is convergent, example |
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Sandwich
theorem (result) or squeeze rule |
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Least
upper bound (or supremum, abbrev. lub, sup) and greatest lower bound (infimum,
or glb, inf) |
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The
definition of the real number e |
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The limit of a sequence, theorems |
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The
cluster point or accumulation point |
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Divergent
sequences |
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Sufficient condition for convergence of a sequence |
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The Cauchy criterion (general principle of convergence) |
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Some
important limits |
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Operations
with limits |
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Operations
with limits examples |
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Series
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Infinite series |
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The sequence of partial
sums |
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The
sum of the series |
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Convergence
of infinite series |
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Divergence
of infinite series |
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Convergent
and divergent series
examples |
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Harmonic
series |
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The remainder
or tail
of the series |
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Necessary
and sufficient
condition for the convergence of a series |
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Necessary
condition for the convergence of a series |
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The
n-th
term test for divergence |
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Properties
of series |
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The
product of two series or the Cauchy product |
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Geometric
series |
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P-series |
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Alternating
series |
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Alternating series test
or Leibnitz's
alternating series test |
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Absolute
convergence |
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Conditional
convergence |
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Series
of positive terms |
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Tests
for convergence |
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Comparison
test |
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Limit
comparison test |
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Ratio
test |
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Root
test or Cauchy's root test |
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Power
series |
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Power
series or polynomial with infinitely many terms |
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Maclaurin and Taylor series |
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The power series expansion of the exponential
function |
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Properties
of the power series expansion of the exponential function |
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The radius of convergence or the interval of convergence |
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The power series expansion of the logarithmic function |
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Properties
of the power series expansion of the logarithmic function |
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The power series expansion of the sine function |
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Properties
of the power series expansion of the sine function |
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The power series expansion of the cosine function |
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Properties
of the power series expansion of the cosine function |
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The power series expansion of the hyperbolic sine and hyperbolic cosine function |
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Properties
of the power series expansion of the hyperbolic sine and hyperbolic cosine function |
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The
limit of a function |
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The
definition of the limit of a function |
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A limit
on the left (a left-hand limit) and a limit
on the right (a right-hand limit) |
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Continuous
function |
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Limits
at infinity (or limits of functions as x approaches
positive or negative infinity) |
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Infinite limits |
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The
limit of a function examples |
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Vertical, horizontal
and slant (or oblique) asymptotes |
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Monotone
functions - increasing or decreasing in value |
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Limit of a function
properties (theorems or laws) |
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Squeeze
rule
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Composition
rule
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Limits of
functions
properties use, examples |
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Limits of
rational functions |
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Evaluating
the limit of a rational function at infinity |
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Evaluating
the limit of a rational function at a point |
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The limit of a rational function that is defined at the given point |
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The limit of a rational function that is not defined at the
given point |
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The limit of a rational function at infinity
containing roots (irrational expressions) |
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The limit of a rational function at
a point
containing irrational expressions, use of substitution |
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Evaluating
the limit of a rational function
containing irrational expressions using rationalization |
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Limits of
trigonometric functions |
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Evaluating
trigonometric
limits, examples |
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Limits of functions
based on the definition of the natural number e |
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Evaluating
limits of functions
based on the definition of the natural number e |
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Use
of the composition rule to evaluate limits of functions |
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| Calculus
contents continue |
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