(c) M is a horizontal line with equation y œ b. Substitution 12 ‰ ˆ 12 84 ‰ into either equation gives x œ 43 ˆ 84 25 4 œ 25 so that Q 25 ß 25 is the point of intersection.
Adding these we get 25 12 y œ 7 so y œ 25. We can rewrite the equations ofĨ4 the lines as L: x y œ 3 and M: + 43 y œ 4. The distance from P to L œ the distance from P to Q œ Ɉ #3 ‰# ˆ 3# ‰# œ É 18 4 œ (b) L has slope 43 so M has slopeĪnd M has the equation 4y 3x œ 12. At the intersection point, Q, we have equal y-values, y œ x 2 œ x 3. (a) L has slope 1 so M is the line through P(2ß 1) with slope ! or the line y œ x 3. Since the two triangles shown in the figure are congruent, the value a must lie midway between x" and x#, so a œ x #x. The line through (1ß 1) and ("ß #) is vertical with equation x œ 1. Substitution into either equation gives x œ 1 Ê (1ß 1) is the intersection point. Subtracting these equations we find 7y œ 7 or y œ 1. At the point of intersection, 2x 4y œ 6 and 2x 3y œ !. The lines are perpendicular when 2k ($) œ ! or k œ ( and parallel when 2k œ $ or k œ "#. 2x ky œ 3 has slope 2k and 4x y œ 1 has slope $. The algebraic properties say that the real numbers can be added, subtracted, mult. A similar statement holds for any number with an infinite tailof 9s.The real numbers can be represented geometrically as points on a number line calledthe real line.21 340131 2234The 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.
999 and 1.000 represent the same real number 1.
Real numbers arenumbers that can be expressed as decimals, such as-3= - 0.75000 41= 0.33333 322 = 1.4142 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 numbershave two representations. We also discuss the use of graphing calculators andcomputer graphing software.1.1Real Numbers and the Real LineThis section reviews real numbers, inequalities, intervals, and absolute values.Real NumbersMuch of calculus is based on properties of the real number system. The topics include the real number system, Cartesian coordinates in the plane, straight lines, parabolas,circles, functions, and trigonometry. 4100 AWL/Thomas_ch01p001-072 8/19/04 10:49 AM Chapter1PRELIMINARIESOVERVIEW This chapter reviews the basic ideas you need to start calculus.
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