Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition

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1 Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition

2 1 Appendix A : Numbers, Inequalities, and Absolute Values Sets A set is a collection of objects with an important property that we can tell whether any given object is or is not in the set. Each object in a set is called an element or a member of the set. Capital letters A, B, C, X, Y, are often used to represent sets and lowercase letters a, b, c, x, y, to represent elements of a set. A set is usually described in listing by listing the elements between braces { } or by enclosing within braces a rule that determines the elements. Example: If A is the set of numbers a such that a 2 = 9, then using the listing method we write A = { 3, 3} or using the rule method we write A = {a a 2 = 9}. Note that in the rule method, the vertical bar means "such that" and the entire symbolic form A = {a a 2 = 9} is read "the set of all numbers a such that a 2 = 9". Symbolically, a A means "a is an element of a set A" and a A means "a is not an element of a set A". Example: 2 {1, 2, 3} means 2 is an element of {1, 2, 3} and 4 {1, 2, 3} means 4 is not an element of {1, 2, 3}. A set is called finite if the number of elements in the set can be counted and infinite if there is no end in counting its elements. A set is called empty if it contains no elements. The empty set is also called the null set and is denoted by. Example: The set X of numbers such that x 2 = 1 is empty, that is X = {x x 2 = 1} =. A set A is called a subset of a set B if each element in A is an element of B and we write A B. Note that the definition of a subset allows that a set to be a subset of itself. Since the empty set has no elements, every element of is also an element of a given set. Thus, the empty set is a subset of every set. Example: {1,3} {1, 2, 3, 4} {1,3} {1, 3} {1, 3} Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Example: {1, 2, 3} = {3, 2, 1}. Notice that the order of listing elements in a set does not matter. If A and B are sets, the intersection of A with B denoted by A B, is the set consisting of elements that belongs to both A and B, that is A B = {x x A and x B}. The union of A with B denoted by A B, is the set consisting of elements that belongs to either A or B or both, that is A B = {x x A or x B}.

3 2 Appendix A : Numbers, Inequalities, and Absolute Values The set difference of a set A and a set B, is the set of all elements in A which are not in B, that is A B = {x x A and x B}. It is read as A difference B. Sometimes, we use A B in place of A B. Example: If A = {1, 2, 3, 6, 7} and B = {3, 6, 9}, then A B = A B = {1, 2, 7} B A = B A = {9}. Example: Let A = {1, 2, 3, 6, 7}, B = {3, 6, 9} and C = {2, 4, 6, 7}. Then A B = {3, 6} A B = {1, 2, 3, 6, 7, 9} (A B) C = {2, 6, 7} C (A B) = {2, 3, 4, 6, 7} The main sets of numbers: 1) Natural Numbers (N) = {1,2,3, } 2) Whole Numbers (W) = {0,1,2,3, } 3) Integer Numbers (Z) = {, 3, 2, 1,0,1,2,3, } 4) Rational Numbers (r) = { m : m, n Z and n 0} n If we can write any number on the form m, then this number will be a rational number. Every number has a decimal n representation. If the number is rational, then the corresponding decimal is one of the following three types Limited decimal numbers: 0.5 = 1 2, = 1 64, = Repeated decimal numbers: Terminated decimal numbers: = 0. 6 = 2 3, = 0.13 = = = 157, = = All these numbers can be written on the form m n. 5) Irrational Numbers (Ir): Any number can t be expressed as of the form m n For example: 6) Real Numbers (R) = r Ir 4 2, 5, 3, π, sin 1, log 2 5 is called irrational number. The set of real numbers is the union of the set of rational numbers and the set irrational numbers. Thus, R = r Ir = {x x is rational} {x x is irrational} Remarks: 1- N W Z r R 2- Ir R = {x x is rational or irrational}. = {x x r or x Ir}

4 3 Appendix A : Numbers, Inequalities, and Absolute Values The real numbers can be represented by points on a line as in the following figure. This line called a coordinate line, or a real number line or simply a real line. Intervals Example: Write each of the following in an inequality notion: [ 2,2) = [ 1, 5 2 ] = (, 2) = (0,3) = [ 5, ) = Example: Find ( 2,3] (1,5) = ( 2,3] (1,5) = ( 2,3] (1,5) = (1,5) ( 2,3] = Example: Find [0, ) {1,2} =

5 4 Appendix A : Numbers, Inequalities, and Absolute Values Equations An equation is an expression, which contains a variable (or more), constants and the equality symbol (=). Linear equations of one variable: The form: ax + b = 0, where a, b R and a 0. ax + b = 0 ax = b x = b a Example: Solve the equation 3x 4 = 8. Example: Solve the equation 3x + 1 = 8(7 x). Quadratic equations of one variable: The form: ax 2 + bx + c = 0, where a, b, c R and a Solving quadratic equations by the quadratic formula ax 2 + bx + c = 0 x = b ± b2 4ac 2a Let D = b 2 4ac is called the discriminant and it has three cases as follows: i. D < 0 There is no real root. ii. D = 0 There is only one repeated real root. iii. D > 0 There are two distinct real roots. Example: Solve the equation 2x 2 3x 10 = x 2 4x 4.

6 5 Appendix A : Numbers, Inequalities, and Absolute Values Zero Factor Theorem: u. v = 0 u = 0 or v = 0 2. Solving quadratic equations by factoring Example: Solve the equation x 2 = 5x. Example: Solve the equation 8x 4x 2 = 0. Example: Solve the equation x 2 + 9x + 20 = 0. Example: Solve the equation x 2 11x + 30 = 0. Example: Solve the equation x 2 + 4x 21 = 0. Example: Solve the equation x 2 2x 24 = 0. Example: Solve the equation 3x 2 + 5x + 2 = 0. Example: Solve the equation 3x 2 14x + 15 = 0. Example: Solve the equation 3x 2 + 5x 2 = 0. Example: Solve the equation 3x 2 5x 2 = 0.

7 6 Appendix A : Numbers, Inequalities, and Absolute Values 3. Solving quadratic equations by taking square roots Example: Solve the equation 2x 2 72 = 0. Example: Solve the equation (x 5) 2 49 = 0. Radical equations: A radical equation is an equation with a square root or cube root, etc. Example: Solve the equation 2x 3 = 5. 3 Example: Solve the equation 2x = 0. Example: Solve the equation x + 6 x = 0. Rational equations: Example: Solve the equation x x 3 = 2 Example: Solve the equation x 5 = 2 3 x

8 7 Appendix A : Numbers, Inequalities, and Absolute Values Inequalities An inequality is an expression, which contains a variable (or more), constants and one of the inequality symbols (<,, > or ). Rules for Inequalities 1. If a < b, then a + c < b + c. 2. If a < b and c < d, then a + c < b + d. 3. If a < b and c > 0, then ac < bc. 4. If a < b and c < 0, then ac > bc. 5. If 0 < a < b, then 1/a > 1/b. Linear inequality of one variable: Example 1: Solve the inequality 1 + x < 7x + 5 Example 2: Solve the inequality 4 3x 2 < 13 The solution set is The solution set is Second-degree inequality of one variable: Example 3: Solve the inequality x 2 5x The solution set is

9 8 Appendix A : Numbers, Inequalities, and Absolute Values Example 4: Solve x 3 + 3x 2 > 4x The solution set is Absolute Value The absolute value of a number a, denoted by a, is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have a 0 for every number a. x, x 0 x = { x, x < 0 Example: Rewrite the expression without using the absolute value symbol. 3 = 1 2 = 3 = 2 1 = 0 = π 6 = = 2 π = 2 1 = 5 23 = π = x = Remarks: a 2 = a Properties of Absolute Values Suppose a and b are any real numbers and n is an integer. Then 1. ab = a b 2. a b = a (b 0) b 3. a n = a n

10 9 Appendix A : Numbers, Inequalities, and Absolute Values For solving equations or inequalities involving absolute values, it s often very helpful to use the following statements. 1. x = a if and only if x = ±a 2. x < a if and only if a < x < a 3. x > a if and only if x > a or x < a Example 5: Express 3x 2 without using the absolutevalue symbol. Example 6: Solve 2x 5 = 3. Example 7: Solve x 5 < 2. Example 8: Solve 3x The distance between two real numbers x & y on the real line is given by d(x, y) = x y = y x Example: Find the distance between 2 & 5. Example: Find the distance between 1 4 & 3.

11 10 Appendix A : Numbers, Inequalities, and Absolute Values Appendix A. Exercises Page A9 Homework: Rewrite the expression without using the absolute value symbol x 2 if x < 2 9. x + 1 Solve the inequality in terms of intervals and illustrate the solution set on the real number line x x + 1 < 5x x < (2x + 3)(x 1) x 2 < 3 Solve the equation for x x = 3 Solve the inequality. 49. x 4 < x Tutorials: Rewrite the expression without using the absolute value symbol. 4. π 2 8. x 2 if x > x 1 Solve the inequality in terms of intervals and illustrate the solution set on the real number line x x > 5 3x x (x 1)(x 2) > x x < 4 Solve the equation for x x + 5 = 1 Solve the inequality. 52. x x 2 < 6