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Determining extrema
Task number: 3046
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Variant 1
the maximum values of the function \(x^{1/x}\) over its domain of definition.
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Equations and inequalities
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Inequalities with fractions
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Two inequalities with absolute values
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Logic
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Mathematical induction
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coprime numbers; constructing a regular n-gon
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Measurement precision
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Second complex root
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Injective mapping
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Suprema and infima, properties of reals
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Suprema and infima of concrete sets
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Set operations
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Ordering by inclusion
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Ordering by divisibility
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infimum of set of real numbers bounded below
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The irrational numbers are dense
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Sequences and limits of sequences
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Monotonic sequences
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Limits by definition
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Statements about limits
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Alternating and constant sequences
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Arithmetic of limits
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Differences of roots
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Sequences going to e
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Monotonicity with a parameter
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Recursive sequences
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Approximating roots
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Simple series
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Simple series
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Series as limits
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Series
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Limits of products
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Sum of a series
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Convergence and absolute convergence I
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Divergent series
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Using the definition
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Comparison test
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Comparison test - difference of roots
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Ratio test
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Root test
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Various series
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Series with parameters
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An easy assertion about series
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Leibniz´s test
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Series with parameters - absolute convergence
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Limits of functions
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Operations preserving monotonicity
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Assertions about monotonicity and extremes
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Rational fractional functions
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Rational fractional functions
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Limit of the sum of a series around 1
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Limits with roots
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Applications to the convergence of series
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Limit of a composite function
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Exponential
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Various limits
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Limits requiring a bit of thought
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Removable discontinuities
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Derivatives of ratios, roots and logarithms
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Inverse functions
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Derivatives of trig. and inv trig. functions
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Derivatives of functions with exponents
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Functions with absolute values
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simple derivatives
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moderately complicated derivatives
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Applications of derivatives: extreme values
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Applications of derivatives: inequalities
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Limits of functions II
(4)
Limits of ratios
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Limits of products
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Limits of differences
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Limits of powers
(L1)
The shape of functions
(8)
Determining extrema
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Extrema
(L2)
Shape of a function (without convexity)
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Shape of a function (with convexity)
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Application - word problems
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Inequalities
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Increasing function
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Shape of a function
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Convergence and absolute convergence II
(1)
Series with a parameter in the exponent
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Polynomials, partial fraction decomposition
(3)
Factoring out a root
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Finding (guessing) roots
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Decomposition into partial fractions
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Taylor polynomials
(4)
Taylor polynomials of concrete functions
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Numerical approximations (no error estimates)
(L1)
Numerical approximation (with error estimate)
(L1)
Computing limits using Taylor polynomials
(L1)
Indefinite integrals
(12)
Table - basic integrals
(L1)
Simple integrals
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Integration by parts
(L1)
Recurrence relations
(L1)
Simple substitution
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Table - rational functions
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Integrals of rational functions
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Simple converions to rational functions
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Substituting for roots and Euler‘s substituti
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Integrals of goniometric functions
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Goniometric substitution
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Integration by parts 2
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The definite integral and its applications
(6)
The definite integral
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Area calculations
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The length of a curve
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Volumes of solids of revolution
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Surface area of solids of revolution
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Estimate of a factorial
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Metric spaces
(11)
Getting acquainted with metrics
(L1)
Line segment without a midpoint
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Getting acquainted with open and closed sets
(L1)
The Paris metric with an open ball
(L1)
The New York metric with an open ball
(L1)
The suprema metric and unit distances
(L2)
Continuity of the identity function
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A closed set generated by a formula
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Getting acquainted with compact sets
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Attaining a maximum
(L1)
A set for which all subsets are compact
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Functions of several variables
(9)
Limits
(L1)
A limit with parameters
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Domain, continuity, boundedness of functions
(L1)
Partial derivatives
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Equivalence of partial derivatives
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Extrema
(L1)
The total differential
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A function without a total differential
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The total differential for composite function
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Implicit functions
(5)
Derivative of a simple function
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Partial derivative of an implicit function
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Astroids
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An implicit function of two variables
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An increasing implicit function
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Bounded extrema
(1)
Bounded extrema
(L1)
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