Chapter0

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Chapter0

  1. 1. Int roduc t ory Mat hem at ic al Analysis for B u s i n e s s , Ec o n o m i c s a n d t h e L i f e a n d So c i a l Sc i e n c e s
  2. 2. Review of algebra More Algebra
  3. 3. 25 POINTS: 5 QUIZES 25 POINTS: FINAL 50 POINTS: CALULAS (after the mid-term)
  4. 4. Review of the Subsets of the Real Number System
  5. 5. N - counting numbers -
  6. 6. WHOLE NUMBERS (W (W )
  7. 7. INTEGERS (Z) ,
  8. 8. RATIONAL NUMBERS (Q (Q ) numbers that can be expressed as a quotient a/b, where a and b are integers. terminating or repeating decimals
  9. 9. c Any number that cannot be written as a 3 1.732050808 ratio (fraction) 3 2 1.492106248 Any number whose e1 2.718281828 decimal 3.141592654 representation neither repeats or stops.
  10. 10. Set of all rational and irrational numbers. c A Real Number is any number that can be graphed in the number line
  11. 11. . Graph the numbers 0.2, 7 , 1, 2 , and 4 on a number line. 10 7 4 1 0.2 2 10 4 3 2 1 0 1 2 3 4
  12. 12. I M c Q A G Z I N W A R Y N
  13. 13. Properties of Addition & Multiplication Properties of Equivalence or Equality Properties of Order or Inequality
  14. 14. CLOSURE PROPERTY Given real numbers a and b, Then, or
  15. 15. Is the set of integers CLOSED with respect to: 1. Addition 2. Subtraction 3. Multiplication 4. Division
  16. 16. Given real numbers a and b, Addition: Multiplication:
  17. 17. Given real numbers a, b and c, Addition: Multiplication:
  18. 18. Given real numbers a, b and c,
  19. 19. Given a real number a, Addition: Multiplication:
  20. 20. Given a real number a, Addition: Multiplication: -1 a 0
  21. 21. For any real numbers a, b, and c Addition Multiplication Commutative a b b a a b b a Associative (a b) c a (b c) (a *b)* c a *(b * c) Identity a+0=a=0+a a a a If a in not zero, then Inverse a ( a) 0 ( a) a 1 1 a *a 1 a *a
  22. 22. Pr o p e r t i e s o f Eq u i v a l e n c e o r Eq u a l i t y Addition and For any reals a, b, and c, if Subtraction a=b then a+c=b+c and a-c=b-c Multiplication For any reals a, b, and c, if and Division a=b then a*c=b*c, and, if c is not zero, a/c=b/c
  23. 23. Pr o p e r t i e s o f Or d e r o r Inequalit y Result: Between any two real numbers there is a rational number and an irrational number. 1- For any real numbers a and b either a < b , b < a or a=b 2- If a < b and b < c then a < c
  24. 24. 3- If a < b then a+c < b+c c R 4- If a < b and c > 0 ,then ac < bc and a b c c If a < b and c < 0, then ac > bc and a b c c
  25. 25. 5- If a < b and a = c , then c < b 6- If 0 < a < b or a < b < 0, then 1 1 a b
  26. 26. 7- If 0 < a < b and n > 0 , then an < bn and n a < n b
  27. 27. - is an element of ~ - not or negation of - Union - Intersection - is a subset of - and (conjunctive/ intersection of two sets) - or (disjunctive/ union of two sets) > - Greater than & Greater than or equal to < - Less than & Less than or equal to = - equal and not equal to
  28. 28. Tell w hic h of t he propert ies of real num bers just ifies eac h of t he follow ing st at em ent s. 1. (2 )(3 ) + (2 )(5 ) = 2 (3 + 5 ) 2. (1 0 + 5 ) + 3 = 1 0 + (5 + 3 ) 3. (2 )(1 0 ) + (3 )(1 0 ) = (2 + 3 )(1 0 ) 4. (1 0 )(4 )(1 0 ) = (4 )(1 0 )(1 0 ) 5. 1 0 + (4 + 1 0 ) = 1 0 + (1 0 + 4 ) 6. 1 0 [ (4 )(1 0 )] = [ (4 )(1 0 )] 1 0 7. [ (4 )(1 0 )] 1 0 = 4 [ (1 0 )(1 0 )] 8. 3 + 0 .3 3 i s a r e a l n u m b e r
  29. 29. n 1 n n
  30. 30. 4
  31. 31. n one +ve root no real one -ve root roots one real one +ve root no +ve roots root, 0 one -ve root no -ve roots
  32. 32. 1 3 3 3 1 4 4 4
  33. 33. 2 2 2 4 4 2 2 2
  34. 34. 3 2 3 2 3 6 3 3 3 3 3
  35. 35. Review basic laws of exponents and radicals on page 10
  36. 36. Simplify and express all answers in terms of positive exponents Answer:
  37. 37. Simplify the expressions Answer:
  38. 38. Write the expression in terms of positive exponents only. Avoid all radicals in the final form. Answer:
  39. 39. Write the exponential forms involving radicals Answer:
  40. 40. Simplify the expressions. Express all answers in terms of positive exponents. Rationalize the denominator where necessary to avoid fractional exponents of denominator. Answer:
  41. 41. Simplify the expressions. Express all answers in terms of positive exponents. Rationalize the denominator where necessary to avoid fractional exponents of denominator. Answer:
  42. 42. Algebraic expressions are numbers represented by symbols which are combined by any or all of the arithmetic operations such as addition, subtraction, multiplication and division as well as exponentiation and extraction of roots.
  43. 43. Algebraic expressions with exactly one term : monomials with exactly two terms: binomials with exactly three terms: trinomials with more than one term: multinomials
  44. 44. Adding Algebraic Expression Prob.7 Prob.7 (Sec. 0.4) Perform the indicated operations and simplify (6x2 + 10xy + 2) (2z xy + 4) 10xy Answer: (6x2 + 10xy + 2) (2z xy + 4) (6 10xy = 6x2 + 10xy + 2 2z + xy - 4 10xy = 6x2 + 10xy + xy - 2z + 2 4 10xy = 6x2 + 11xy - 2z + 2 4 11xy
  45. 45. Subtracting Algebraic Expression Prob.13 Prob.13 (Sec. 0.4) Perform the indicated operations and simplify 3(x2 + y2) x(y + 2x) + 2y(x + 3y) Answer: 3x2 + 3y2 xy - 2x2 + 2xy + 6y2 = 3x2 - 2x2 + 3y2 + 6y2 xy + 2xy = x2 + 9y2 + xy
  46. 46. Removing Grouping Symbols Prob.15 Prob.15 (Sec. 0.4) Perform the indicated operations and simplify 2[3[3(x2 + 2) 2(x2 5)]] )]] Answer: 2[3[3x2 + 6 2x2 + 10]]10] = 2[3[3x2 - 2x2 + 6 + 10]] (rearranging) 10] = 2[3[x2 + 16]] = 2[3x2 + 48] = 6x2 + 96 16] 48]
  47. 47. Special Products Refer to page 18 of textbook for list of rules for special products Prob.19 (Sec. 0.4) Perform the indicated Prob.19 operations and simplify (x + 4)(x + 5) Answer: (x +4)(x + 5) = x2 + 5x + 4x + 20 = x2 + 9x + 20
  48. 48. Multiplying Multinomials Prob.35 Prob.35 (Sec. 0.4) Perform the indicated operations and simplify (x2 - 4)(3x2 + 2x - 1) )(3 Answer: x2(3x2 + 2x 1) 4(3x2 + 2x 1) = 3x4 + 2x3 - x2 - 12x2 - 8x + 4 12x = 3x4 + 2x3 - 13x2 - 8x + 4 13x
  49. 49. Dividing a Multinomial by a Multinomial Prob.47 Prob.47 (Sec. 0.4) Perform the indicated operations and simplify Answer:
  50. 50. Long Division Prob.51 Prob.51 (Sec. 0.4) Perform the indicated operations and simplify (3x3-2x2+x 3)÷(x + 2)
  51. 51. Factoring is rewriting expression as a product of 2 or more factors E.g. If c = ab, then a and b are factors of c Refer to page 21 of textbook for list of rules for factoring
  52. 52. Common Factors Prob.5 Prob.5 (Sec. 0.5) Factor the following expressions completely 8a3bc - 12ab3cd + 4b4c2d2 12ab Answer: 4bc(2a3 - 3ab2d + b3cd2) bc(2
  53. 53. Factoring Trinomials Prob.9 Prob.9 (Sec. 0.5) Factor the following expressions completely p2 + 4p + 3 Answer: (p+1)(p+3 (p+1)(p+3)
  54. 54. Algebraic expressions which are fractions can be simplified multiplying and dividing both numerator and denominator of a fraction by the same non- non-zero quantity
  55. 55. Simplifying Fractions Prob.3 Prob.3 (Sec. 0.6) Simplify the expressions Answer:
  56. 56. Multiplying and Dividing Fractions Rule for multiplying with is Rule for dividing with where c 0 is
  57. 57. Multiplying and Dividing Fractions Prob.11 Prob.11 (Sec. 0.6) Simplify the expressions Answer:
  58. 58. Rationalizing the Denominator Prob.53 Prob.53 (Sec. 0.6) Simplify and express your answer in a form that is free of radicals in the denominator
  59. 59. Addition and Subtraction of Fractions Prob.29 Prob.29 (Sec. 0.6) Perform the operations and simplify as much as possible Answer:
  60. 60. Addition and Subtraction of Fractions Prob.39 (Sec. 0.6) Prob.39 Perform the operations and simplify as much as possible
  61. 61. Addition and Subtraction of Fractions Prob.47 Prob.47 (Sec. 0.6) Factor the following expressions completely Answer:
  62. 62. Sec. 0.1 4, 5, 7, 8 Sec. 0.2 3, 4, 10, 22 10, Sec. 0.3 55, 58, 65, 85, 55, 58, 65, 85, 90 Sec. 0.4 18, 37, 39, 18, 37, 39, 50 Sec. 0.5 38, 40, 46, 38, 40, 46, 50 Sec. 0.6 10, 34, 46, 10, 34, 46, 59 Sec. 0.7 30, 37, 46, 59, 71, 75, 85, 92, 30, 37, 46, 59, 71, 75, 85, 92, 105 Sec. 0.8 25, 32, 40, 47, 54, 74, 25, 32, 40, 47, 54, 74, 84
  63. 63. Mathematical Systems,groups and fields a set of elements One or more operations defined on this set Definitions and rules for applying the operations on the set. Theorems can be deduced from the given definitions and rules.
  64. 64. The set G is closed under the operation * The operation * is associative There is an identity element e of G for * There is an inverse element for every element of G
  65. 65. A Field is a mathematical system that consisting a set F and two operations that satisfy 11 properties:
  66. 66. Domain a set of all possible replacement values for a given variable. Ex. D= { x| x R} Quantifier a word or phrase that describes in general terms the part of the domain for which a sentence/ statement is true. Ex. - ,
  67. 67. Universal Quantifier a statement that has the same truth value for every element of the domain. X = > For all/ every x Existential Quantifier a phrase that describes a statement as being true for some or at least one element from the domain.: X : x + 3 = 7 = > There exist a value for x such that x + 3 = 7
  68. 68. Negating Quantified Statements The negation of a universally quantified statement p is an existentially quantified statement of the negation of p (~ p). The negation of x p is expressed as x ~p The negation of an existentially quantified statement p is a universally quantified statement of the negation of p (~ p). The negation of x p is expressed as x ~p
  69. 69. X :x= 2 X :x> 7 X ~ k Some animals can fly Some rectangles are squares
  70. 70. Set - a collection of objects or elements. 2 types - Finite & Infinite eg. { 1 , 2 , 3 } { 2 , 4 , 6 , 8 } { all w omen < 21 } 21} { } Subset - a set whose entire contents also belongs to another set
  71. 71. Define each of the following and show how they are represented? Empty Set Union of sets I ntersection of sets Universal set - Complement of a set
  72. 72. Empty Set a set with no members - { } or Union of sets A set containing the members of both/all given given sets Union of sets A set containing all all the members of both/all sets Intersection of sets a set containing only members that are elements of BOTH/ALL sets Universal set - The complete set or groups of elements from which solution variables/subsets can be chosen. Normally the Universal set is also the Domain. Complement of a set If a subset A of elements is identified within Universal set U, elements is identified within Universal set U, the complement A is all the elements that are NOT in the identified set, but are if the universal set. ex. if the universal set is the set of natural numbers, and the set of even numbers is identified, then the complement of that set is the set of odd numbers.

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