Title: Advanced Mathematics?
1Advanced Mathematics?
2 Instructor Dr. Ye Huajun T.A.
Miss Liu Kaihui Office E409
Tel 3620622(office) Email
hjye_at_uic.edu.hk (Instructor)
Websitewww.uic.edu.hk/yehuajun
3They invented Calculus!
Sir Isaac Newton (1642-1727)
Gottfriend Wilhelm von Leibniz (1646-1716)
4Ancient Greek Wisdom
Socrates (????) 470-399 B.C.
Aristotle(?????) 384-322 B.C.
Plato(???) 428-347 B.C.
Archimedes(????) 287-212 B.C.
5Zenos 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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7- The dichotomy paradox (????)
That which is in locomotion must arrive at the
half-way stage before it arrives at the
goal.-------- Aristotle
8If 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
9 Calculus was continued.
Cauchy(??) (1789-1857)
Fermat(??) (1601-1665)
Euler(??) (1707-1783)
Gauss(??) (1777-1855)
10Calculus has practical applications, such as
understanding the true meaning of the
infinitesimals. (Image concept by Dr. Lachowska.)
11What 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.
12Why 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.
13What 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.
14Some 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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16Score 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
17Some 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.
18General Information
- Textbook
- Calculus-Early Transcendental Functions
- 3rd Edition
- Smith and Minton
- 2007, McGraw Hill, International Edition
- Advantages
- textbook for two semesters
- More applications
19General 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.
20Chapter 0Preliminaries (????)
-
- In this Chapter, we will encounter some
important concepts. - Polynomials and Rational Functions
- Inverse Functions
- Trigonometric and Inverse Trigonometric Functions
-
21Section 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.
22For 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
23- Prove that is an irrational number
- ??????????????,?????????????,??????????????.??????
?????. - Exercises
- Show that is an irrational number
24?????
- ?n?N, ?n???????,?? ???????
- ????????? , ?? p,q?N,
- ??n???????,??m?N?
- ???? ???? ??
- ???? ,?? ,???
- ?
- ? ,???
-
25???????,?????????????? ?? ??????,???p?q???????
??????????,????? ????????
26- 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
27- 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.
28- 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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30Triangle inequality (?????)
31- Equations of Lines (????)
Notice that a line is horizontal if and only if
its slope is zero.
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34Functions(??)
- 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(???).
35Function, or not?
YES
NO
NO
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37Rational Function (????)
38In 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
39Intercepts
- 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.
40Unfortunately, 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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42Section 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.
43Inverse Functions
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47Basic 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
.
48- 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.
49EXAMPLE
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.
50Definition (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.
51Theorem Every infinite subset of a countable
set A is countable. Remark What is story behind
this theorem?
52- 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
53EXAMPLE
- Suppose consists of 1,2,3 and consists
of 2,3,4. Then , ? - 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
54Theorem Let , n1,2,3,.., be a
sequence of countable sets, and put Then S is
countable.
55Theorem 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.
56Section 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.
57Remark When we discuss the period of a function,
we most often focus on the fundamental period.
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60The Inverse Trigonometric Functions
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62Chapter 1Limits and Continuity (?????)
- In this Chapter, we will encounter some
important concepts. - Limits (??)
- Continuity (??)
- Limits Involving Infinity (????????)
- Formal Definition of the Limit
-
63Section 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?
64It 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
65If 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.)
66- 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.
67Notice 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.
68For 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
69Existence 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
70Nonexistent 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.
71Since the function approaches the same value as
x?-3 both from the right and from the left, we
write
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74Section 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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78THEOREM 3.2 Limits of Polynomials and Rational
Functions If p(x) and q(x) are polynomials,
then and
79Indeterminate 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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81 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.
82Squeeze Theorem (?????)
83The 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.
84 Solution
Recall that
85Section 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 )
86A hole at xc
87A gap at xc
88- 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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90- When we can remove a discontinuity by defining
the function at that point, we call the
discontinuity removable. - Not all discontinuities are removable.
91 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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94Continuity 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.
95 Example 4.8
Discuss the continuity of the function
on the open interval -2ltxlt3 and on the
closed interval -2x3
96Theorem 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.
97The condition f(x) is continuous on the closed
interval a,b, can not be relaxed. Can you find
an example?
98In 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.
99Remark 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.
100 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.
101The Method of Bisections (???)
- The method of Bisections can help us locate the
zeroes of a function.