1. Advanced Mathematics D
Calculus
Advanced Mathematics D

2. Part One
Introduction

3. The Aim of this Course Learn Advanced Mathematics Learn English
What is advanced mathematics
What is advanced mathematics doing
How is mathematical thinking
Learn English
What is saying in English about advanced mathematics

4. Calculus – Advanced Mathematics
Different from High School Mathematics
Calculate v.s Analysis
Static v.s Dynamic
Different from Social Science
Philosophy v.s Arts
Logical v.s Image

5. Method of Studying Advanced Mathematics
Understanding deeply
Deduce step by step
Exercise

6. Part Two Functions Limites Derivatives Integration
Differential Equations

7. Functions What is a function? How does a function play in Calculus?
What is any difference between the function in Calculus and the Mathematics in the Middle School?

8. Basic Concepts Variables: x,y Value or image Relation Determination
Region

9. Definition of Function
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”)

10. Represent of Function Numerically by tables Geometrically by Graphs
Algebraically by formulas
Verbally

11. Example of Function 1 x 1 2 3
f (x) 4 -1

12. Example of Function 2

13. Example of Function 3

14. Example of Function 4

15. Example of Function 5
Newton’s Law of Universal Gravitation:

16. 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) :
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…

17. Graphs of Functions 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
This is not a function y y
This is a function x x

18. Domain & Range Domain – the set of all allowable inputs
Range – the set of the all possible output when input over the domain
Denoted by respectively
for Domain and Range of function
Range of f(x)
Domain of f(x)

19. Example of Domain & Range 1
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
Domain of Age(Name) , BirthDay(Name), Sex(Name) is {John, Linda, ……}
Range of Age(Name) is (0,200)
Range of BirthDay(Name) is {dates in the calendar }
Range of Sex(Name) is {M,F}

20. Example of Domain & Range 21 2 3 4 -1
Domain of is {0,1,2,3}
Range of is {-1,3,4}

21. Example of Domain & Range 3
Domain of
Error (Δx)
= ( 0.01,0.1 )
Range of
Error (Δx)
= ( , )

22. Example of Domain 4 x x

23. Example of Domain 5 x 1 3 x

24. Example of Domain 6 x x

25. Example of Domain 7 x x

26. Construct of a New Function
From given functions, we can construct a new function
e.x given f(x) and g(x), a new function can be created from them.

27. Arithmetic Operation on Functions
New domain is the intersection of the old domains, for the last one is also except the root of g(x)

28. Composition of Functions
The domain of the new function is consisting of all x in domain g for which g(x) is the domain of f

29. Simple Transfer from a Function
Translation
Reflection
Stretches and compressions

30. Even and Odd Functions
Even function
Odd function

31. Function Family
A function with a parameter
e.x.

32. Some Functions Inverse Proportions
Power functions with non-integer exponents
Polynomials
Rational functions
Algebraic functions
Trigonometric functions

33. Inverse Function If the function f and g satisfy then we say that
f is an inverse of g
g is an inverse of f
f and g are inverse functions

34. Domain and Range of Inverse Function

35. How to find an Inverse Function
Step 1: write down the equation y = f(x)
Step 2: If possible, solve this equation for x as a function of y
Step 3: The resulting will be , which provides a formula for with y as the independent variable
Step 4: If y is accepted, job is done, and determine the domain

36. Existence of Inverse Functions
A function has an inverse if and only if it is one to one
Increasing or decreasing function are invertible

37. Increasing/Decreasing Functions
The function is called increasing (decreasing) function if its graph always rising (falling)
They are invertible.

38. Mirror image of an inverse function
If f has an inverse, then the graph of and are reflections of one another about the line y=x. That is, each graph is the mirror image of the other with respect to that line.

39. Restriction Domain
If a function g is obtained from a function f by placing restrictions on the domain of f, then g is called the restriction of f
Sometime, it is possible to create an invertible function from a function that is not invertible by restricting the domain appropriately.

40. Inverse Trigonometric Function
By restricting skill, we can define inverse trigonometric function

41. Relations of Trigonometric Functions

42. Exponential & Logarithmic Functions
We know
Question: what is And
We can use a sequence to define it:
A function of the form , is called an exponential function with base b.

43. Properties of Exponential Function
The graphs are all pass (0,1)

44. Natural exponential function
The number of e comes from the limitation of

45. Logarithmic Function
A logarithmic function with base b (b>0,b≠1) is an inverse of a exponential function:
Logarithmic function with e is called natural logarithmic function, noted as ln(x)

46. Algebraic Properties of Logarithms

47. Change Base Formula

48. Parametric Equation
Graph can be explained also by parametric equation,
ex: a circle in Cartesian Coordinates is not a function
However, the circle in Polar Coordinates system : is a function
The circle can also expressed by parametric equation:

49. Parametric Equation-
Rose-rhodonea-curve

50. Parametric Equation --
Escher’s famous art work “Path of Life”

51. Reference
James Strewart, Calculus, Thomson Leaning