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Terrell Maria Shea - Calculus With Applications

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Terrell Maria Shea Calculus With Applications

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Peter D. Lax and Maria Shea Terrell Undergraduate Texts in Mathematics Calculus With Applications 2nd ed. 2014 10.1007/978-1-4614-7946-8_1 Springer Science+Business Media New York 2014
1. Numbers and Limits
Peter D. Lax 1 and Maria Shea Terrell 2
(1)
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
(2)
Department of Mathematics, Cornell University, Ithaca, NY, USA
Abstract
This chapter introduces basic concepts and properties of numbers that are necessary prerequisites for defining the calculus concepts of limit, derivative, and integral.
1.1 Inequalities
One cannot exaggerate the importance in calculus of inequalities between numbers. Inequalities are at the heart of the basic notion of convergence, an idea central to calculus. Inequalities can be used to prove the equality of two numbers by showing that one is neither less than nor greater than the other. For example, Archimedes showed that the area of a circle was neither less than nor greater than the area of a triangle with base the circumference and height the radius of the circle.
A different use of inequalities is descriptive. Sets of numbers described by inequalities can be visualized on the number line.
Fig 11 The number line To say that a is less than b denoted by a lt b - photo 1
Fig. 1.1
The number line
To say that a is less than b , denoted by a < b , means that b a is positive. On the number line in Fig., a would lie to the left of b . Inequalities are often used to describe intervals of numbers. The numbers that satisfy a < x < b are the numbers between a and b , not including the endpoints a and b . This is an example of an open interval, which is indicated by round brackets, ( a , b ).
To say that a is less than or equal to b , denoted by a b , means that b a is not negative. The numbers that satisfy a x b are the numbers between a and b , including the endpoints a and b . This is an example of a closed interval, which is indicated by square brackets, [ a , b ]. Intervals that include one endpoint but not the other are called half-open or half-closed . For example, the interval a < x b is denoted by ( a , b ] (Fig.).
Fig 12 Left the open interval a b Center the half open interval - photo 2
Fig. 1.2
Left : the open interval ( a , b ). Center : the half open interval ( a , b ]. Right : the closed interval [ a , b ]
The absolute value| a |of a number a is the distance of a from 0; for a positive, then,| a |= a , while for a negative, The absolute value of a difference a b can be interpreted as the - photo 3 . The absolute value of a difference,| a b |, can be interpreted as the distance between a and b on the number line, or as the length of the interval between a and b (Fig. ).
Calculus With Applications - image 4
Fig. 1.3
Distances are measured using absolute value
The inequality
Calculus With Applications - image 5
can be interpreted as stating that the distance between a and b on the number line is less than . It also means that the difference between a and b is no more than and no less than:
Calculus With Applications - image 6
(1.1)
In Problem 1.9, we ask you to use some of the properties of inequalities stated in Sect.).
Example 1.1.
The inequality Calculus With Applications - image 7 describes the numbers x whose distance from 5 is less than Picture 8 . This is the open interval (4.5,5.5). It also tells us that the difference x 5 is between Calculus With Applications - image 9 and Calculus With Applications - image 10 . See Fig.. The inequality Calculus With Applications - image 11 describes the closed interval [4.5,5.5].
Calculus With Applications - image 12
Fig. 1.4
Left : the numbers specified by the inequality Calculus With Applications - image 13 in Example 1.1. Right : the difference x 5 is between Calculus With Applications - image 14 and Calculus With Applications - image 15
The inequality Calculus With Applications - image 16 can be interpreted as a statement about the precision of 3.141 as an approximation of . It tells us that 3.141 is within of and that is in an interval centered at 3141 of length Fig 15 - photo 17 of , and that is in an interval centered at 3.141 of length Fig 15 Approximations to We can imagine smaller intervals contained - photo 18 .
Fig 15 Approximations to We can imagine smaller intervals contained inside - photo 19
Fig. 1.5
Approximations to
We can imagine smaller intervals contained inside the larger one in Fig..
We use ( a ,) to denote the set of numbers that are greater than a , and [ a ,) to denote the set of numbers that are greater than or equal to a . Similarly, (, a ) denotes the set of numbers less than a , and (, a ] denotes those less than or equal to a . See Fig..
Calculus With Applications - image 20
Fig. 1.6
The intervals (, a ), (, a ], [ a ,), and ( a ,) are shown from left to right
Example 1.2.
The inequality Calculus With Applications - image 21 describes the numbers whose distance from 5 is greater than or equal to These are the numbers that are in 45 or in 55 See Fig Fig - photo 22 . These are the numbers that are in (,4.5] or in [5.5,). See Fig..
Fig 17 The numbers specified by the inequality in Example 12 11a Rules - photo 23
Fig. 1.7
The numbers specified by the inequality in Example 1.2
1.1a Rules for Inequalities
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