Page 6 - Schaum's Outline of Theory and Problems of Advanced Calculus

NUMBERS
CHAPTER 1 For more information about this title, click here. 1
Sets. Real numbers. Decimal representation of real numbers. Geometric
representation of real numbers. Operations with real numbers. Inequal-
ities. Absolute value of real numbers. Exponents and roots. Logarithms.
Axiomatic foundations of the real number system. Point sets, intervals.
Countability. Neighborhoods. Limit points. Bounds. Bolzano-
Weierstrass theorem. Algebraic and transcendental numbers. The com-
plex number system. Polar form of complex numbers. Mathematical
induction.

CHAPTER 2 SEQUENCES 23
Definition of a sequence. Limit of a sequence. Theorems on limits of
sequences. Infinity. Bounded, monotonic sequences. Least upper bound
and greatest lower bound of a sequence. Limit superior, limit inferior.
Nested intervals. Cauchy’s convergence criterion. Infinite series.

CHAPTER 3 FUNCTIONS, LIMITS, AND CONTINUITY 39
Functions. Graph of a function. Bounded functions. Montonic func-
tions. Inverse functions. Principal values. Maxima and minima. Types
of functions. Transcendental functions. Limits of functions. Right- and
left-hand limits. Theorems on limits. Infinity. Special limits. Continuity.
Right- and left-hand continuity. Continuity in an interval. Theorems on
continuity. Piecewise continuity. Uniform continuity.
CHAPTER 4 DERIVATIVES 65
The concept and definition of a derivative. Right- and left-hand deriva-
tives. Differentiability in an interval. Piecewise differentiability. Differ-
entials. The differentiation of composite functions. Implicit
differentiation. Rules for differentiation. Derivatives of elementary func-
tions. Higher order derivatives. Mean value theorems. L’Hospital’s
rules. Applications.
v
Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.