أكثر من 40 سؤال عن المعادلات التفاضلية الجزئية
Intervals
A subset of the real line is called an interval if it contains at least two numbers and contains
all the real numbers lying between any two of its elements. For example, the set of all
real numbers x such that is an interval, as is the set of all x such that
The set of all nonzero real numbers is not an interval; since 0 is absent, the set fails to contain
every real number between and 1 (for example).
Geometrically, intervals correspond to rays and line segments on the real line, along
with the real line itself. Intervals of numbers corresponding to line segments are finite intervals;
intervals corresponding to rays and the real line are infinite intervals.
A finite interval is said to be closed if it contains both of its endpoints, half-open if it
contains one endpoint but not the other, and open if it contains neither endpoint. The endpoints
are also called boundary points; they make up the interval’s boundary. The remaining
points of the interval are interior points and together comprise the interval’s interior.
Infinite intervals are closed if they contain a finite endpoint, and open otherwise.
The entire real line is an infinite interval that is both open and closed.
Solving Inequalities
The process of finding the interval or intervals of numbers that satisfy an inequality in x is
called solving the inequalit
OVERVIEW This chapter reviews the basic ideas you need to start calculus. The topics include
the real number system, Cartesian coordinates in the plane, straight lines, parabolas,
circles, functions, and trigonometry. We also discuss the use of graphing calculators and
...........
Real Numbers
Much of calculus is based on properties of the real number system. Real numbers are
numbers that can be expressed as decimals, such as
The dots in each case indicate that the sequence of decimal digits goes on forever.
Every conceivable decimal expansion represents a real number, although some numbers
have two representations. For instance, the infinite decimals and represent
the same real number 1. A similar statement holds for any number with an infinite tail
of 9’s.
The real numbers can be represented geometrically as points on a number line called
the real line.
The symbol denotes either the real number system or, equivalently, the real line.
The properties of the real number system fall into three categories: algebraic properties,
order properties, and completeness. The algebraic properties say that the real numbers
can be added, subtracted, multiplied, and divided (except by 0) to produce more real
numbers under the usual rules of arithmetic. You can never divide by 0.
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