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Math 203 Calculus II

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Title: Math 203 Calculus II


1
Math 203Calculus II Linear Algebra
  • Wafik Lotfallah

2
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
3
  • Course Assessment System
  • Homework (10)
  • Quizzes (25)
  • Midterm (25)
  • Final (40, comprehensive)

4
Prerequisites Math 103. However, apart from
being able to differentiate and integrate, we
will review anything you need to know for the
course.
5
Lecture 2
  • Sequences

6
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

7
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,

8
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

9
Example
  • Find the first 5 terms of the sequences defined
    by
  • an 1/n,
  • an (?1)n
  • an sin(n?)

10
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?

11
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,

12
Visualizing Sequences
13
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

14
Picture
15
Limit Rules
  • In other words, the limit operation distributes
    over the basic operation (, ?, ?, /).

16
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

17
Note Parts (b) and (d) can also be done using
LHoptials Rule.
18
Example Find the limit of the sequence
19
Example Find the limit of the sequence
Caution We can not use LHoptials Rule here, as
the limit is not 0/0 or ?/?.
20
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.
21
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22
Another useful Theorem
  • This theorem says limn?? f(an) f(limn?? an),
  • i.e. the limit operations can enter continuous
    functions.

23
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
24
Famous Limits
25
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

26
  • Thank you for listening.
  • Wafik
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