Math 203 Calculus II

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Math 203Calculus II Linear Algebra

• Wafik Lotfallah

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Course Material Textbooks T1) Thomas' Calculus,
11th edition, by Weir, Hass, and Giordano, 05 T2)
Linear Algebra A Modern Introduction, 2nd
edition, by Poole, 06 Supplementary Material on
the inter/intranet Intranet \Faculties\Math\2006
_Spring\M203 Engineering Spring 2005 Internet
http//math.guc.edu.eg/ss06math203/ References T
3) Advanced Engineering Mathematics, 8th edition,
by E. Kreyszig, J. Wiley Sons, 1999 T4) Shaums
Outline of Advanced Mathematics for Engineers and
Scientists, 1st edition, by M.R. Spiegel,
McGraw-Hill, 1971
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• Course Assessment System
• Homework (10)
• Quizzes (25)
• Midterm (25)
• Final (40, comprehensive)

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Prerequisites Math 103. However, apart from
being able to differentiate and integrate, we
will review anything you need to know for the
course.
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Lecture 2

• Sequences

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Lecture 2 Objectives

• Find the first few terms of a sequence defined
• Explicitly (by its nth term)
• Recursively
• Find a formula for the nth term of a sequence
  given by its first few terms
• Find the limit of a sequence using
• Algebraic Manipulation (Division Up and Down, and
  Taking Logarithms)
• L'Hopital's Rule
• The Sandwich Theorem

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Example

• Consider the following infinite sequence (list,
  array) of real numbers 1, 4, 9, 16,
  25,
• Question Can you guess the next term?
• Answer 36
• Question Whats the pattern?
• Answer The nth term is just n2.
• We can thus represent this sequence by the
  formula an n2, where n runs over the natural
  numbers 1, 2, 3, 4,

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Note

• an n2 is a function similar to the function
  f(x) x2, except that n must be a natural
  number, while x is usually allowed to take any
  real number.
• We thus define

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Example

• Find the first 5 terms of the sequences defined
  by
• an 1/n,
• an (?1)n
• an sin(n?)

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Example (Recursive Definitions)

• Find the first 5 terms of the sequences defined
  by
• a1 1, an1 2an
• a1 1, a2 1, an2 an an1 (This is
  the recursive definition of the so-called
  Fibonacci Sequence.)
• Can you find the general nth term of these
  sequences?

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Example

• Find the nth term of the sequences
• 2, 4, 6, 8, 10,
• 1, ?3, 5, ?7, 9,
• 2, 5, 10, 17, 26,
• 1, 0, ?1, 0, 1, 0, ?1, 0,

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Visualizing Sequences
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Limit of a sequence

• We sometimes need to know if the terms of the
  sequence an approach a real number L as n goes
  to ?.
• We say that L is the limit of the sequence an,
  or that an converges to L and write limn?? an
  L
• For a more precise definition, we have

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Picture
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Limit Rules

• In other words, the limit operation distributes
  over the basic operation (, ?, ?, /).

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Example Use the limit rules to find the limits
of the following sequences

• an ?1/n
• an (n ? 1)/n
• an 5/n2
• an (4 ? 7n6)/(n6 3)
• Remember 1/? 0, while 0/0, 0??, ?/?, 00, ?0,
  and 1? are not defined

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Note Parts (b) and (d) can also be done using
LHoptials Rule.
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Example Find the limit of the sequence
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Example Find the limit of the sequence
Caution We can not use LHoptials Rule here, as
the limit is not 0/0 or ?/?.
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The previous example can be solved using the
following Sandwich Theorem
I.e If a sequence bn is sandwiched between two
sequences an and cn having a common limit L, then
bn has that same limit.
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Another useful Theorem

• This theorem says limn?? f(an) f(limn?? an),
• i.e. the limit operations can enter continuous
  functions.

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Example Find the limit of the sequence
Hint When n occurs in both the base and the
power, rewrite the expression using e and ln.
Thus
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Famous Limits
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Lecture 2 Objectives (revisited)

• Find the first few terms of a sequence defined
• Explicitly (by its nth term)
• Recursively
• Find a formula for the nth term of a sequence
  given by its first few terms
• Find the limit of a sequence using
• Algebraic Manipulation (Division Up and Down, and
  Taking Logarithms)
• L'Hopital's Rule
• The Sandwich Theorem

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• Thank you for listening.
• Wafik