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calculus Chapter 1 notes

calculus Chapter 1 notes - MAT 132 CALCULUS WITH ANALYTIC...

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MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I Page 1 of 8 CHAPTER 1: PRELIMINARIES 1.1 Real Numbers and the Real Line Real Numbers Much of calculus is based on properties of the real number system. Real numbers are numbers that can be expressed as decimals. Rules for Inequalities If a, b and c are real numbers, then: 1. c b c a b a + < + < 2. c b c a b a - < - < 3. b a < and bc ac 0 c < 4. b a < and ac bc 0 c < < Special case: a b b a - < - < 5. 0 a 1 0 a 6. If a and b are both positive or both negative, then a 1 b 1 b a < < 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. Types of intervals Notation Set description Type Finite: ) b , a ( { } b x a x < < Open [ ] b , a { } b x a x Closed [ ) b , a { } b x a x < Half-open ] b , a ( { } b x a x < Half-open
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MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I Page 2 of 8 Infinite: ) , a ( { } a x x Open [ ) , a { } a x x Closed ) b , ( -∞ { } b x x < Open ] b , ( -∞ { } b x x Closed ) , ( -∞ (set of all real numbers) Both open & closed Example: Attend lecture. Absolute Value The absolute value of a number x, denoted by x , is defined by the formula < - = 0 x , x 0 x , x x Absolute Values and Intervals If a is any positive number, then 1. a x = if and only if a x ± = 2. a x < if and only if a x a < < - 3. a x if and only if a x or a x - < 4. a x if and only if a x a - 5. a x if and only if a x or a x - Example: Attend lecture.
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MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I Page 3 of 8 PROBLEM SET 1.1 Inequalities Solve the following inequalities. 1. 4 x 2 - 5. 6 7 x 7 2 1 x 2 + - 2. 5 x 3 8 - 6.
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