Advanced Mathematics?

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Advanced Mathematics?
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Instructor Dr. Ye Huajun T.A.
Miss Liu Kaihui Office E409
Tel [phone](office) Email
hjye_at_uic.edu.hk (Instructor)
Websitewww.uic.edu.hk/yehuajun
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They invented Calculus!
Sir Isaac Newton (1642-1727)
Gottfriend Wilhelm von Leibniz (1646-1716)
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Ancient Greek Wisdom
Socrates (????) 470-399 B.C.
Aristotle(?????) 384-322 B.C.
Plato(???) 428-347 B.C.
Archimedes(????) 287-212 B.C.
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Zenos Paradoxes(????)

• Achilles and the tortoise (????)

In a race, the quickest runner can never
overtake the slowest, since the pursuer must
first reach the point whence the pursued started,
so that the slower must always hold a lead.
-------- Aristotle
Zeno of Elea (??) 490-430 B.C.
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• The dichotomy paradox (????)

That which is in locomotion must arrive at the
half-way stage before it arrives at the
goal.-------- Aristotle
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• The arrow paradox (????)

If everything when it occupies an equal space is
at rest, and if that which is in locomotion is
always occupying such a space at any moment, the
flying arrow is therefore motionless.--------
Aristotle
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Calculus was continued.
Cauchy(??) (1789-1857)
Fermat(??) (1601-1665)
Euler(??) (1707-1783)
Gauss(??) (1777-1855)
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Calculus has practical applications, such as
understanding the true meaning of the
infinitesimals. (Image concept by Dr. Lachowska.)
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What is Calculus all about?

• Calculus is the study of change, or to be more
  precise, changing quantities.
• The two key areas of Calculus are Differential
  Calculus and Integral Calculus.
• The big surprise is that these two seemingly
  unrelated areas are actually connected via the
  Fundamental Theorem of Calculus.

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Why do I have to take Calculus?

• If you choose a career path that requires any
  level of mathematical sophistication (such as
  Engineering, Economics or any branch of Science),
  you will need Calculus.
• To succeed in Calculus (and Mathematics in
  general) you must be able to solve problems,
  reason logically and understand abstract
  concepts. The ability to master these skills is a
  key indicators of success in future academic
  endeavors. 

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What can I do to maximize my chances for success?

• Top five
• Understand, don't memorize.
• Ask why, not how.
• More practices and exercises.
• Learn techniques, not results.
• Make sure you understand each topic before going
  on to the next.

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Some Suggestions

• Bring your questions when you take lecture.
  Questions could be found by preview.
• Review what you have studied as soon as possible
  after lecture. Textbook should be read carefully.
  
• Try your best to attend all lectures and tutorial
  class.

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Score System

• Assessment grade system
• A and A- (Not more than 10)
• B (Not more than 15)
• A and B that include A, A-, B, B and B- (Not
  more than 65)
• Below C and not include C (No any limit ).

Letter Grade Academic Performance Grade Point Per Unit
A Excellent 4.00
A- Excellent 3.70
B Good 3.30
B Good 3.00
B- Good 2.70
C Satisfactory 2.30
C Satisfactory 2.00
D Marginal Pass 1.00
F Fail 0.00
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Some notices on this Course

• Assignments must be handed in before the
  deadline. After the deadline, we refuse to accept
  your assignments!
• For the mid-term test and final examination, you
  can
• not bring anything except some stationeries
  and water! Mobile are not allowed.
• For the final examination, we can not tell you
  the score before the AR inform the official
  results. If you have any question on the score,
  you can check the marked sheet via AR.

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General Information

• Textbook
• Calculus-Early Transcendental Functions
• 3rd Edition
• Smith and Minton
• 2007, McGraw Hill, International Edition
• Advantages
• textbook for two semesters
• More applications

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General Information

• References
• S. Salas, E. Hille and G.J. Etgen, Calculus, One
  and Several Variables, 8th edition, John Wiley
  Sons, 1999.
• L.D. Hoffmann, G.L. Bradley, Calculus, for
  Business, Economics and the Social and Life
  sciences, 9th edition, McGraw Hill, 2004.
• J. Stewart, Calculus, 4th edition, Books/Cole,
  1999.
• D. Hughes-Hallett, A.M. Gleason, W.G. McCallum et
  al., Calculus, Single and Multivariable, 2nd
  edition, Wiley, 1998.

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Chapter 0Preliminaries (????)

• In this Chapter, we will encounter some
  important concepts.
• Polynomials and Rational Functions
• Inverse Functions
• Trigonometric and Inverse Trigonometric Functions
  

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Section 0.1 Polynomials(???) and Rational
Functions (????)

• The Real Number (??) System
• Integers (??)0, 1,2,
• Rational numbers(???) A rational number is any
  number of the form p/q, where p and q are
  integers and q?0.
• Irrational numbers are those real numbers that
  cannot be written in the form p/q, where p and q
  are integers. Irrational numbers have decimal
  expansions that do not repeat or terminate.

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For instant, three familiar irrational numbers
and their decimal expansions are
We picture the real numbers arranged along the
number line displayed in Figure 0.2 (the real
line). The set of real numbers is denoted by the
symbol
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• Prove that is an irrational number
• ??????????????,?????????????,??????????????.??????
  ?????.
• Exercises
• Show that is an irrational number

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?????

• ?n?N, ?n???????,?? ???????
• ????????? , ?? p,q?N,
• ??n???????,??m?N?
• ???? ???? ??
• ???? ,?? ,???
• ?
• ? ,???

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???????,?????????????? ?? ??????,???p?q???????
??????????,????? ????????
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• The intervals
• The closed interval (???) a,b to be the set of
  numbers between a and b, including a and b (the
  endpoints), that is,
• The open interval (a,b) is the set of numbers
  between a and b, but not including the endpoints,
  that is

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• Addition and subtraction of real numbers
  (????????)
• II-1 abba (commutative law of addition ???)
• II-2 (ab)ca(bc) (associative law ???)
• II-3 a0a (property of zero number)
• II-4 For any number a, there exists a number a
  such that a(-a)0.
• II-5 From agtb, we have acgtbc.

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• Multiplication and division of real numbers
  (????????)
• III-1 abba (commutative law)
• III-2 (ab)ca(bc) (associative law)
• III-3 a1a (property of 1)
• III-4 For any number a?0, there exists its
• reciprocal number 1/a such that
• III-5 (ab)cacbc (distributive law ??? )
• III-6 From agtb and cgt0, we have acgtbc

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Triangle inequality (?????)
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• Equations of Lines (????)

Notice that a line is horizontal if and only if
its slope is zero.
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Functions(??)

• A function f is a rule that assigns to each
  object in a set A exactly one object in a set B.
  In this case, we write yf(x).
• The set A is called the domain (???) of the
  function, and the set of assigned objects in B is
  called the range (??). We refer to x as the
  independent variable (???)and to y as the
  dependent variable(???).

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Function, or not?
YES
NO
NO
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Rational Function (????)
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In general, there are two situations where a
number is not in the domain of a function 1)
division by 0 2) The even number root of a
negative number
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Intercepts

• x intercepts The points where a graph crosses
  the x axis.
• A y intercept A point where the graph crosses
  the y axis.
• How to find the x and y intercepts The only
  possible y intercept for a function is
  , to find any x intercept of yf(x), set
  y0 and solve for x.
• Note Sometimes finding x intercepts may be
  difficult.

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Unfortunately, factoring is not always so easy.
Of course, for the quadratic equation
(for a?0), the solutions are given by the
familiar quadratic formula
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Section 0.3 Inverse Functions (???)

• Composition of Functions(????) Given functions
  f(u) and g(x), the composition f(g(x)) is the
  function of x formed by substituting ug(x) for u
  in the formula for f(u).

Example
Find the composition function f(g(x)), where
and
Solution
Replace u by x1 in the formula for f(u) to get
Question How about g(f(x))? Note In general,
f(g(x)) and g(f(x)) will not be the same.
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Inverse Functions
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Basic Topology

• Finite, Countable and Uncountable Sets

Definition If there exists a 1-1 mapping of A
onto B, we say A and B are equivalent, and we
write
This relation clearly has the following
properties It is reflexive .
It is symmetric If , then
. It is transitive If
and , then
.
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• Definition For any positive integers n, let
  be the set
• whose elements are the integers
  let J be the
• set consisting of all positive integers. For any
  set A, we
• say
• A is finite if for some n.
  (How about the empty set?)
• A is infinite if A is not finite.
• A is countable if .
• A is uncountable if A is neither finite nor
  countable.
• A is at most countable if A is finite or
  countable.

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EXAMPLE
Let A be the set of all integers. Then A is
countable. Hints Considering the following
arrangement of the sets A and J
Remark A finite set cannot be equivalent to one
of its proper subsets. However, this is possible
for infinite sets.
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Definition (A Sequence) A function f defined on
the set J of all positive integers. If
, for , it is customary to
denote the sequence f by
. If A is a set and if for
all , then is said to be a
sequence in A, or a sequence of elements of A
Note that the terms of a
sequence need not be distinct.
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Theorem Every infinite subset of a countable
set A is countable. Remark What is story behind
this theorem?
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• Let A and be sets, and suppose that with
  each element of A there is associated a subset
  of which we denote by
• .
• The union of the set is defined to be the
  set S
• The intersection of the set is defined to
  be the set P

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EXAMPLE

1. Suppose consists of 1,2,3 and consists
   of 2,3,4. Then , ?
2. Let A be the set of real numbers x such that
   0ltx?1. For every , let be the set
   of real numbers y such that 0ltyltx. Then

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Theorem Let , n1,2,3,.., be a
sequence of countable sets, and put Then S is
countable.
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Theorem Let A be a countable set, and let
be the set of all n-tuples
, where , and the
elements need not be
distinct. Then is countable. Corollary The
set of all rational numbers is countable.
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Section 0.4 Trigonometric (????) and Inverse
Trigonometric Functions (?????)

• Many phenomena encountered in your daily life
  involve waves.
• Radar Signals
• Electromagnetic waves
• Electrocardiogram
• The mathematical description of such phenomena
  involves periodic functions, the most familiar of
  which are the trigonometric functions.

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Remark When we discuss the period of a function,
we most often focus on the fundamental period.
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The Inverse Trigonometric Functions
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Chapter 1Limits and Continuity (?????)

• In this Chapter, we will encounter some
  important concepts.
• Limits (??)
• Continuity (??)
• Limits Involving Infinity (????????)
• Formal Definition of the Limit

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Section 1.1 The Concept Of Limit

• To illustrate the limit process, consider a
  manager who determines that when x percent of her
  companys plant capacity is being used, the total
  cost is

hundred thousand dollars. The company has a
policy of rotating maintenance in such a way that
no more than 80 of capacity is ever in use at
any one time. What cost should the manager expect
when the plant is operating at full permissible
capacity?
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It may seem that we can answer this question by
simply evaluating C(80), but attempting this
evaluation results in the meaningless fraction
0/0. However, it is still possible to evaluate
C(x) for values of x that approach 80 from the
left (xlt80) and the right (xgt80), as indicated in
this table
x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right x approaches 80 from the left ? ?x approaches 80 from the right
x 79.8 79.99 79.999 80 80.0001 80.001 80.04
C(x) 6.99782 6.99989 6.99999 7.000001 7.00001 7.00043
The values of C(x) displayed on the lower line of
this table suggest that C(x) approaches the
number 7 as x gets closer and closer to 80. The
functional behavior in this example can be
describe by
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If f(x) approaches L as x tends toward c from the
left (xltc), we write
where L is called the limit from the left (or
left-hand limit)
Likewise if f(x) approaches M as x tends toward c
from the right (xgtc), then
M is called the limit from the right (or
right-hand limit.)
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• As a start, we consider the functions
• For first function f(x), we compute some values
  of
• the function for x close to 2, as in the
  following
• tables.

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Notice that the table and the graph both suggest
that as x approaches 2 from two sides (left side
and right side), f(x) gets closer and closer to
4, written
It is important to remember that limits describe
the behavior of a function near a particular
point, not necessarily at the point itself.
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For the second function g(x), we have following
tables and figures
Since neither limit exists, we say that the limit
of g(x) as x approaches 2 does not exist, written
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Existence of a Limit The two-sided limit
exists if and only if the two one-sided
limits and
exist and are equal, and then
For the function
evaluate the one-sided limits
and
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Nonexistent One-sided Limits
A simple example is provided by the function
As x approaches 0 from either the left or the
right, f(x) oscillates between -1 and 1
infinitely often. Thus neither one-sided limit at
0 exists.
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Since the function approaches the same value as
x?-3 both from the right and from the left, we
write
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Section 1.2 Computation Of Limits
For any constant k, That is, the limit of a
constant is the constant itself, and the limit of
f(x)x as x approaches c is c.
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THEOREM 3.2 Limits of Polynomials and Rational
Functions If p(x) and q(x) are polynomials,
then and
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Indeterminate Form
If and , then
is said to be indeterminate.
The term indeterminate is used since the limit
may or may not exist.
Example
Find the Limit by Factoring and Rationalizing
(a) Find (b) Find
Solution

b.
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Idea
Your first reaction might be to say that this is
a limit of a product and it must be the product
of the limits
Since does not exist, we can not
directly use Theorem 3.1.
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Squeeze Theorem (?????)
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The challenge in using the Squeeze Theorem is in
finding appropriate functions f and h that bound
a given function g from below and above,
respectively, and that have the same limit as
x?a.
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Solution
Recall that
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Section 1.3 Continuity and Its Consequences

• A continuous function is one whose graph can be
  drawn without the pen leaving the paper. (no
  holes or gaps )

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A hole at xc
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A gap at xc
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• Definition 4.1
• Continuity(??) A function f is continuous at c
  if all
• three of these conditions are satisfied
• If f(x) is not continuous at c, it is said to
  have a
• discontinuity there.

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• When we can remove a discontinuity by defining
  the function at that point, we call the
  discontinuity removable.
• Not all discontinuities are removable.

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Continuity Polynomials and Rational Functions

• If p(x) and q(x) are polynomials, then

A polynomial or a rational function is continuous
wherever it is defined
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Continuity on an Interval

• A function f(x) is said to be continuous on an
  open interval altxltb if it is continuous at each
  point xc in that interval.
• f is continuous on closed interval axb, if it
  continuous on the open interval altxltb and
• If f is continuous on all of (-8, 8) , we simply
  say that f is continuous everywhere.

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Example 4.8
Discuss the continuity of the function
on the open interval -2ltxlt3 and on the
closed interval -2x3
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Theorem 4.4 The Intermediate Value Property

• Suppose that f(x) is continuous on the interval
  axb and L is a number between f(a) and f(b),
  then there exists a number c between a and b,
  such that f(c)L.

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The condition f(x) is continuous on the closed
interval a,b, can not be relaxed. Can you find
an example?
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In following Corollary, we see an immediate and
useful application of the Intermediate Value
Theorem
Corollary 4.2
If f is continuous on the closed interval a,b,
and f(a) and f(b) have opposite signs i.e.
f(a)f(b)lt0, then there exists a number c in
(a,b) where f(c)0.
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Remark Notice that The Intermediate Value
Theorem and Corollary 4.2 are example of
existence theorems they tell you that there
exists a number c satisfying some condition, but
they do not tell you what c is and where c is.
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Example
Show that the equation has
a solution for 1ltxlt2
Solution
Let . Then
f(1)-3/2 and f(2)2/3. Since f(x) is continuous
for 1x2, it follows from the intermediate value
property that the graph must cross the x axis
somewhere between x1 and x2.
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The Method of Bisections (???)

• The method of Bisections can help us locate the
  zeroes of a function.