Age(John)=18, Age(Linda)=17…
BirthDay (Name) :
BirthDay(John)=02/04/1995, BirthDay(Linda)=16/07/1996, …
Sex (Name): Sex(John)=M, Sex(Linda)=F…
Graphs of Functions
Part Two
Functions Limites Derivatives Integration Differential Equations
Functions
What is a function? How does a function play in Calculus? What is any difference between the
Represent of Function
Numerically by tables Geometrically by Graphs Algebraically by formulas Verbally
Example of Function 1
x
0
1
2
3
f (x)
3
4
-1
3
f (0) 3, f (1) 4, f (2) 1, f (3) 3
Different from Social Science
Philosophy v.s Arts Logical v.s Image
Method of Studying Advanced Mathematics
Understanding deeply Deduce step by step Exercise
Advanced Mathematics D
Part One
Introduction
The Aim of this Course
Learn Advanced Mathematics
What is advanced mathematics What is advanced mathematics doing How is mathematical thinking
Example of Function 2
90
80
East(Season) 70
60
West(Season) 50
东部
North(Season)
40 30
西部 北部
20
10
0 第一季度 第二季度 第三季度 第四季度
East(1)=0, East(2)=28, East(3)=90, East(4)=20 West(1)=30, West(2)=38, West(3)=32, West(4)=31, North(1)=45, North(2)=46, North(3)=44, North(1)=43
F G m1m2 r2
Example of Function 6
The Age of the students in Class X The Birth Day of the students in Class X The Sex of the students in Class X
Students in Class X are: John, Linda, …. Age (Name) :
Example of Function 3
Example of Function 4
x 10-3 1.2
1
0.8
error
0.6
0.4
0.2
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1Biblioteka 0.11xExample of Function 5
Newton’s Law of Universal Gravitation:
Not all graph in xy-plane is a function
For every input variable, only one output corresponds
A curve in the xy-plane is the graph of a function, if and only if no vertical line intersects cure more than once
Learn English
What is saying in English about advanced mathematics
Calculus – Advanced Mathematics
Different from High School Mathematics
Calculate v.s Analysis Static v.s Dynamic
y
This is a function
y
This is
not a
function
x x
Domain & Range
Domain – the set of all allowable inputs
Range – the set of the all possible output when input over the domain
If a variable y depends on a variable x in a such way that each value of x determines exactly one value of y, then we say that y is a function of x.
A function f is a rule that associates a unique output with each input. If the input is denoted by x, then the output is denoted by f(x) (read “f of x”)
function in Calculus and the Mathematics in the Middle School?
Basic Concepts
Variables: x,y Value or image Relation Determination Region
Definition of Function
Denoted by Df , and Rf respectively