MTH 170

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MTH 170

• Chapter 1
• Section 1

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Inductive Reasoning

• Is the process of reaching a general conclusion
  by examining specific examples.
• In other words, its looking for a trend to make
  a conclusion.

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Example of Induction

• 3, 6, 9, 12, 15, ?
• 1, 3, 6, 10, 15, ?
• 5, 10, 15, 20, 25, ?
• 2, 5, 10, 17, 26, ?

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Use Induction to Make a Conjecture

• Pick a number.
• Multiply the number by 8,
• add 6 to the product,
• divide the sum by 2,
• and then subtract that number by 3.
• Use inductive reasoning to make a conjecture
  about the relationship between the size of the
  resulting number and the original.

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Galileo and the Pendulum

• Used induction to discover the time for a
  pendulum to complete one full swing, called the
  Period of the pendulum, depends on the length of
  the pendulum.
• Used his heartbeat to time it, since he did not
  have a clock.

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Galileo
If the pendulum has a length of 25 units, what is
its period? If the length of the pendulum is
quadrupled, what happens to its period?
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Tsunamis
What happens to the height of the tsunami when
its velocity is doubled? What should be the
height of the tsunami if its velocity is 30
ft/sec?
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Counterexamples

• A statement is true if and only if it is true in
  all cases.
• If you can find one case for which a statement is
  not true (called a counterexample), then the
  statement is a false statement.

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Find a counterexample

• x gt 0
• x2 gt x

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Deductive Reasoning

• Is the process of reaching a conclusion by
  applying general assumptions, procedures, or
  principles

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Example

• Pick a number
• Multiply by 8,
• Add 6 to the product
• Divide sum by 2
• Subtract 3
• Let n be the number, so
• (8n6)/2 -3 4n3 -3 4n
• So the result is 4 times the original number

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Check your progress

• Use deductive reasoning to show the following
  procedure produces a number that is 3 times the
  original number
• Pick a number
• Multiply by 6
• Add 10 to the product
• Divide the sum by 2
• And subtract 5
• Hint let n be the original number

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Solve a logic puzzleFind occupations

• Maria gets home from work after the banker but
  before the dentist
• Sarah, who is the last to get home from work, is
  not the editor
• The dentist and sarah leave for work at the same
  time.
• The banker lives next door to Brian

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Solve a logic puzzle

• Maria gets home from work after the banker but
  before the dentist
• Sarah, who is the last to get home from work, is
  not the editor
• The dentist and sarah leave for work at the same
  time.
• The banker lives next door to Brian

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Solve a logic puzzle

• Maria gets home from work after the banker but
  before the dentist
• Sarah, who is the last to get home from work, is
  not the editor
• The dentist and sarah leave for work at the same
  time.
• The banker lives next door to Brian

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Solve a logic puzzle

• Maria gets home from work after the banker but
  before the dentist
• Sarah, who is the last to get home from work, is
  not the editor
• The dentist and sarah leave for work at the same
  time.
• The banker lives next door to Brian

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Solve a logic puzzle

• Maria gets home from work after the banker but
  before the dentist
• Sarah, who is the last to get home from work, is
  not the editor
• The dentist and sarah leave for work at the same
  time.
• The banker lives next door to Brian

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Check your progress

• Brianna, Ryan, Tyler and Ashley were recently
  elected as the new class officers (pres, v.p.,
  sec, and treasurer). From the clues determine
  which position each holds
• Ashley is younger than the president but older
  than the treasurer
• Brianna and the sec. are both the same age, and
  they are the youngest members of the group.
• Tyler and the sec. are next door neighbors

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Logic Puzzle
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HomeworkDue 1/23/08

• Exercise set 1.1
• 2, 4, 8, 12, 16, 28, 36, 40, 48
• All work must be shown and the answers must be
  complete sentences which have the question
  incorporated into the answer.
• Ex. Answer for 1 The next number in the
  sequence of 4,8,12,16,20,24 is 28.
• Quiz on Friday 1/25/08 over sec 1.1

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