Chapter 1 takes the student on a direct route to the point where it is possible to study derivatives. Sections 1.1 through 1.3 are reviews of precalculus material and can be skipped in many calculus courses. Section 1.4 gives an intuitive explanation of the hyperreal numbers and how they can be used to find slopes of curves. This section has no problem set and is intended as the basis for an introductory lecture. The main content of Chapter 1 is in the last two sections, 1.5 and 1.6. In these sections, the student will learn how to work with the hyperreal numbers and in particular how to compute standard parts. Standard parts are used at the beginning of the next chapter to find derivatives of functions. Sections 1.5 and 1.6 take the place of the beginning chapter on limits found in traditional calculus texts.

 

For the benefit of the interested student, we have included an Epilogue at the end of the book that presents the theory underlying this chapter.

 

1.1 THE  REAL  LINE

Familiarity with the real number system is a prerequisite for this course. A review of the rules of algebra for the real numbers is given in the appendix. For convenience, these rules are also listed in a table inside the front cover. The letter R is used for the set of all real numbers. We think of the real numbers as arranged along a straight line with the integers (whole numbers) marked off at equal intervals, as shown in Figure 1.1.1. This line is called the real line.

 

 

In grade school and high school mathematics, the real number system is constructed gradually in several stages. Beginning with the positive integers, the systems of integers, rational numbers, and finally real numbers are built up. One way to construct the set of real number is as the set of all nonterminating decimals.

 

After constructing the real numbers, it is possible to prove the familiar rules for sums, differences, products, quotients, exponents, roots, and order. In this course, we take it for granted that these rules are familiar to the student, so that we can proceed as quickly as possible to the calculus.

 

Before going on, we pause to recall two special points that are important in the calculus. First, division by zero is never allowed. Expressions such as

 

 

are always considered to be undefined.

 

Second, a positive real number c always has two square roots, ______ and ______ , and _____always stands for the positive square root. Negative real numbers do not have real square roots. For each positive real number c,______ is positive and ___ is undefined.

 

On the other hand, every real number has one real cube root. If c >0, c has the positive cube root ____, and -c has the negative cube root_______ =_____.

 

In calculus, we often deal with sets of real numbers. By a set S of real numbers, we mean any collection of real numbers, called members of S, elements of S, or points in S.

 

A simple but important kind of set is an interval. Given two real numbers a and b with a< b, the closed interval [a, b] is defined as the set of all real numbers x such that a ≤ x and x≤ b, or more concisely, a≤x≤b.

 

The open interval (a, b) is defined as the set of all real numbers x such that a. Closed and open intervals are illustrated in Figure 1.1.2.

 

 

 

 

 

 

Figure 1.1.2

 

For both open and closed intervals, the number a is called the lower endpoint, and b the upper endpoint. The difference between the closed interval [a, b] and the open interval (a,b) is that the endpoints a and b are elements of [a, b] but are not elements of (a, b). When a≤x≤b, we say that x is between a and b; when a

 

Three other types of sets are also counted as open intervals: the set (a, ∞) of all real numbers x greater than a; the set ( -∞, b) of all real numbers x less than b, and the whole real line R. The real line R is sometimes denoted by ( -∞, ∞). The symbols ∞ and -∞, read “infinity” and “minus infinity,” do not stand for numbers; they are only used to indicate an interval with no upper endpoint, or no lower endpoint.

 

Besides the open and closed intervals, there is one other kind of interval, called a half-open interval. The set of all real numbers xsuch that a≤xis a half-open interval denoted by [a,b).the set of all real numbers x such that a ≤xis also a half-open interval and is written [a,∞). Here is a table showing the various kinds of intervals.

 

 

 

 

 

 

 

 

 

 

 

 

 

We list some other important examples of sets of real numbers.

(1)The empty set Ø, which has no elements.

(2)The finite set { a1, ……, an}, whose only elements are the numbers

    a1,a2,……an.

(3)the set of all x such that x ≠ 0.

(4)The set N = {1,2,3,4,…} of all positive integers.

(5)The set Z={…, -3,-2,-1,0,1,2,3,…} of all integers.

(6)The set Q of all rational numbers. A rational number is a quotient m/n where m and n are integers and n≠0.

 

While real numbers correspond to points on a line, ordered pairs of real numbers correspond to points on a plane. This correspondence gives us a way to draw pictures of calculus problems and to translate physical problems into the language of calculus. It is the starting point of the subject called analytic geometry.

 

An ordered pair of real numbers, (a, b), is given by the first number a and the second number b. For example, (1,3), (3,1), and (1,1) are three different ordered pairs. Following tradition, we use the same symbol for the open interval (a,b) and the ordered pair (a,b). However the open interval and ordered pair are completely different things. It will always be quite obvious from the context whether(a,b) stands for the open interval or the ordered pair.

 

We now explain how ordered pairs of real numbers correspond to points in a plane. A system of rectangular coordinates in a plane is given by a horizontal and a vertical copy of the real line crossing at zero. The horizontal line is called the horizontal axis, or x-axis, while the vertical line is called the vertical axis, or y-axis. The point where the two axes meet is called the origin and corresponds to the ordered pair (0,0). Now consider any point P in the plane. A vertical line through P will cross the x-axis at a real number x0, and a horizontal line through P will cross the y-axis at a real number y0. The ordered pair (x0, y0) obtained in this way corresponds to the point P. (See Figure 1.1.3) We sometimes call P the point(x0, y0) and sometimes write P(x0, y0). x0 is called the x-coordinate of P and y0 the y-coordinate of P.

 

Conversely, given an ordered pair (x0, y0) of real numbers there is a corresponding point

P(x0, y0) in the plane. P(x0, y0) is the point of intersection of the vertical line crossing the x-axis at x0 and the horizontal line crossing the y-axis at y0.

We have described a one-to-one correspondence between all points in the plane and all ordered pairs of real numbers.

 

From now on, we shall simplify things by identifying points in the plane with ordered pairs of real numbers, as shown in Figure 1.1.4.

 

 

 

 

 

 

 

 

 

DEFINITION

      The (x, y) plane is the set of all ordered pairs (x, y) of real numbers. The origin is the point (0,0). The x-axis is the set of all points of the form (x,0), and the y-axis is the set of all points of the form (0,y).

 

The x-and y-axes divide the rest of the plane into four parts called quadrants. The quadrants are numbered I through IV, as shown in Figure 1.1.5.

 

In Figure 1.1.6, P(x1, y1) and Q(x2,y2) are two different points in the (x,y) plane. As we move from P to Q, the coordinates x and y will change by amounts that we denote by Δx and Δy. Thus

                    change in x= Δx= x2-x1,

                    change in y= Δy= y2-y1,

The quantities Δx and Δy may be positive, negative, or zero. For example, when x2>x1, Δx is positive, and when x2<<i>x1, Δx is negative. Using Δx and Δy we define the basic notion of distance.

 

 

 

 

 

 

 

 

 

 

 

DEFINITION

       The distance between the points P(x1, y1) and Q(x2 , y2) is the quantity

                distance (P,Q) = ______________ =________________.

       When we square both sides of the distance formula, we obtain

                [ distance (P, Q)]² = (Δx)² +(Δy)²

 

One can also get this formula from the Theorem of Pythagoras in geometry: the square of the hypotenuse of a right triangle is the sum of the squares of the sides.

 

EXAMPLE 1  Find the distance between P(7,2) and Q(4, 6) (see Figure 1.1.7).

                     Δx= 4-7 = -3 ,          Δy= 6-2 = 4.

                  distance (P, Q) = _____=5.

 

We often deal with sets of points in the plane as well as on the line. One way to describe a set of points in the plane is by an equation or inequality in two variables, say x and y. A solution of an equation in x and y is a point (x0, y0) in the plane for which the equation is true. The set of all solutions is called the locus, or graph, of the equation. The circle is an important example of a set of points in the plane.

 

 

 

 

 

 

 

 

 

Figure 1.1.7

DEFINITION  OF  CIRCLE

        The set of all points in the plane at distance r from a point P is called the circle of

        radius r and center P.

   

        Using the distance formula, we see that the circle of radius r and center at the origin(Figure 1.1.8) is the locus of the equation

                  x² + y² = r²

The circle of radius r and center at P(h,k) (Figure 1.1.8) is the locus of the equation

                   (x-h) ² + (y-k) ² = r² .

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.1.8  

     

For example, the circle with radius 3 and center at P(2,-4) has the equation

                      (x-2)²+ ( y+4)²= 9.

 

Chapter 1 takes the student on a direct route to the point where it is possible to study derivatives. Sections 1.1 through 1.3 are reviews of precalculus material and can be skipped in many calculus courses. Section 1.4 gives an intuitive explanation of the hyperreal numbers and how they can be used to find slopes of curves. This section has no problem set and is intended as the basis for an introductory lecture. The main content of Chapter 1 is in the last two sections, 1.5 and 1.6. In these sections, the student will learn how to work with the hyperreal numbers and in particular how to compute standard parts. Standard parts are used at the beginning of the next chapter to find derivatives of functions. Sections 1.5 and 1.6 take the place of the beginning chapter on limits found in traditional calculus texts.

 

For the benefit of the interested student, we have included an Epilogue at the end of the book that presents the theory underlying this chapter.