CALCULUS & NUMERICAL
METHODS
Your lecturer:
 Name :
 Email:
 No Phone:
PART ONE: CALCULUS
 FUNCTIONS AND GRAPHS (2 weeks)
 LIMITS AND CONTINUITY (1 week)
 DIFFERENTIATION (1 week)
 INTEGRAT...
PART TWO: NUMERICAL METHODS
 ERRORS (1 week)
 ROOT FINDING (1 week)
 INTERPOLATION (1 week)
 NUMERICAL DIFFERENTIATION...
Learning Outcomes
 LO1: {C2}: Apply knowledge and fundamental

concepts of Calculus and Numerical Methods.
 LO3:{ C3,P3,...
Assessment Methods
LO 1
Assessment Methods
Test(2)
= 20%
Assignments(2) = 20%
Mid Term (1) = 30%
Final (1)
= 30%
Total
= 1...
FUNCTIONS AND GRAPHS
Subtopics
1. Relations and Functions
2. Representation of Functions
3. New Function form Old Function
4. Inverse of Functi...
1.Relations and Functions
2.Representation of Functions
Relations and Functions
 Definition-A function is defined as a relation in

which every element in the domain has a uniqu...
Representation of Functions
1. Verbally ( by a description in words)
P(t) is the human population of the world of time

2....
Representation of Functions
3. Visually ( by a graph)
Population (millions)

8000

6000
4000
2000

0
1900 1920 1940 1960 1...
Example 1:
 Let A = {1, 2, 3, 4} and B = {set of integers}. Illustrate
x 3.
the function f : x
Example 2:
 Draw the graph of the function

,

f :x

2

x ,x

R

where R is the set of real numbers.

Solution
Assume the...
Example 2: Graph
Type of Function and Their Graph
Linear Function
f ( x)

 Where

are constant called the
coefficients of the linear
equat...
Type of Function and Their Graph
Polynomial

 Where n is a

nonnegative integer
and the number are
constant
called the co...
Type of Function and Their Graph
Power Function
f ( x)

Where a is constant.

3

x ;x

R
,

Type of Function and Their Graph
Exponential Function
f ( x)

Where a is a positive constant.

x

e ;x

R
,

Type of Function and Their Graph
Logarithm Function

Where a is a positive constant.

f ( x)

ln x ; x

(0,

)
Example 10:
Consider for what value of x are the following
function defined?

1

f ( x)
x

2
3. New Functions from Old
Function
1. TRANSFORMATIONS OF FUNCTIONS
2. COMBINATION OF FUNCTIONS
3. COMPOSITE FUNCTIONS
New Functions from Old Function
 TRANSFORMATIONS OF FUNCTIONS
 The graph of one function can be transform into the graph...
TRANSFORMATIONS OF FUNCTIONS
 Vertical and horizontal shift
Example 3:
Use the graph of

f ( x)

x

g ( x)

x

to obtain the graph of
4
Example 4:
Use the graph of f ( x )
g ( x)

x

(x

2

to obtain the graph of
2)

2
TRANSFORMATIONS OF FUNCTIONS
 Vertical and horizontal shift
TRANSFORMATIONS OF FUNCTIONS
Vertical and Horizontal Reflecting and Stretching
Example 5:
Use the graph of f ( x )
g(x)

h( x)

x

x

x

to obtain the graph of
Example 5:
Use the graph of f ( x )
g ( x)

h(x)

2x
1
2

2

x

2

x

2

to obtain the graph of
COMBINATION OF FUNCTIONS

 Functions can be added, subtracted, multiplied and

divided in a many ways.
For example consid...
COMPOSITE FUNCTIONS

 DefinitionWe define f  g

Consider two functions f(x) and g(x).
fg ( x ) f [ g ( x )] meaning that...
Example 6:
 If

f (x)=3x +1

of x
(a)

f ° g

(b)

g° f

and

g(x)=2-x , find as a function
COMPOSITE FUNCTIONS
 Determine the

Domain of the
Composite
Functions
Example 7:
 If

f (x)=3x +1

and

g(x)=2-x , find as a function

of x
(a)

Find f ° g and determine its domain and range
...
Properties for Graph of Functions
 All forms of relations can be represented on
coordinates
 To test if a graph displaye...
Example 8:
 Consider the graphs shown below and state whether

they represent functions:
4. Inverse Function
The Inverse of Functions
 If f is a function, the inverse is denoted by
 Suppose y=f (x) then x
y
y

y

32

1

1

( y)

...
Find the inverse of

Example 11:
 Find the inverse of :
f (x)

x

3
2
Graphical Illustration of an Inverse Function
Verify that the inverse of f (x)=2x-3 is

f

1

(x)

x

3
2

Figure above sh...
Example 12:
 Find the inverse of :
1

f ( x)
1

2, x
x

 State the domain of the inverse

1.
FUNCTION WITH NO INVERSE

 An inverse function can only exist if the function is a

one-to-one function.
Subtopics
1. Relations and Functions
2. Representation of Functions
3. New Function form Old Function
4. Inverse of Functi...
Calculus and Numerical Method =_=
Calculus and Numerical Method =_=
Calculus and Numerical Method =_=
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Calculus and Numerical Method =_=

  1. 1. CALCULUS & NUMERICAL METHODS
  2. 2. Your lecturer:  Name :  Email:  No Phone:
  3. 3. PART ONE: CALCULUS  FUNCTIONS AND GRAPHS (2 weeks)  LIMITS AND CONTINUITY (1 week)  DIFFERENTIATION (1 week)  INTEGRATION (2 weeks)  DIFFERENTIAL EQUATIONS (1 week)  Total : 7 weeks
  4. 4. PART TWO: NUMERICAL METHODS  ERRORS (1 week)  ROOT FINDING (1 week)  INTERPOLATION (1 week)  NUMERICAL DIFFERENTIATION (1 week)  NUMERICAL INTEGRATION (1 week)  SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (1 week)
  5. 5. Learning Outcomes  LO1: {C2}: Apply knowledge and fundamental concepts of Calculus and Numerical Methods.  LO3:{ C3,P3,CTPS}:Solve problems particularly in computer science with appropriate and high-level programming language or tools.  LO3:{C3, LL}:Solve real-life application problems using suitable techniques in Calculus or Numerical Methods
  6. 6. Assessment Methods LO 1 Assessment Methods Test(2) = 20% Assignments(2) = 20% Mid Term (1) = 30% Final (1) = 30% Total = 100% LO 2 T1 (10%) T2 (10%) A2 (10%) MT1 (15%) F1 (15%) 40% 20% LO 3 A1 (10%) MT2(15%) F3 (15%) 40%
  7. 7. FUNCTIONS AND GRAPHS
  8. 8. Subtopics 1. Relations and Functions 2. Representation of Functions 3. New Function form Old Function 4. Inverse of Functions 5. Exponential Functions 6. Logarithm Functions, log x
  9. 9. 1.Relations and Functions 2.Representation of Functions
  10. 10. Relations and Functions  Definition-A function is defined as a relation in which every element in the domain has a unique image in the range. In other words, a function is one to one relation and many to one relation
  11. 11. Representation of Functions 1. Verbally ( by a description in words) P(t) is the human population of the world of time 2. Numerically (by a table of values) Year 1900 1920 1940 1960 1980 2000 Population 1650 1860 2300 3040 4450 6080 (millions)
  12. 12. Representation of Functions 3. Visually ( by a graph) Population (millions) 8000 6000 4000 2000 0 1900 1920 1940 1960 1980 2000 Year 4. Algebraically ( by an explicit formula)
  13. 13. Example 1:  Let A = {1, 2, 3, 4} and B = {set of integers}. Illustrate x 3. the function f : x
  14. 14. Example 2:  Draw the graph of the function , f :x 2 x ,x R where R is the set of real numbers. Solution Assume the domain is x = -3, -2, -1, 0, 1, 2, 3. A table of values is constructed as follows: x f(x) -3 9 -2 4 -1 1 0 0 1 1 2 4 3 9
  15. 15. Example 2: Graph
  16. 16. Type of Function and Their Graph Linear Function f ( x)  Where are constant called the coefficients of the linear equation x ;x R
  17. 17. Type of Function and Their Graph Polynomial  Where n is a nonnegative integer and the number are constant called the coefficients of the polynomial.  Quadratic f ( x) 2 x ;x R
  18. 18. Type of Function and Their Graph Power Function f ( x) Where a is constant. 3 x ;x R
  19. 19. , Type of Function and Their Graph Exponential Function f ( x) Where a is a positive constant. x e ;x R
  20. 20. , Type of Function and Their Graph Logarithm Function Where a is a positive constant. f ( x) ln x ; x (0, )
  21. 21. Example 10: Consider for what value of x are the following function defined? 1 f ( x) x 2
  22. 22. 3. New Functions from Old Function 1. TRANSFORMATIONS OF FUNCTIONS 2. COMBINATION OF FUNCTIONS 3. COMPOSITE FUNCTIONS
  23. 23. New Functions from Old Function  TRANSFORMATIONS OF FUNCTIONS  The graph of one function can be transform into the graph of a different function rely on a function’s equation. Vertical and horizontal shift
  24. 24. TRANSFORMATIONS OF FUNCTIONS  Vertical and horizontal shift
  25. 25. Example 3: Use the graph of f ( x) x g ( x) x to obtain the graph of 4
  26. 26. Example 4: Use the graph of f ( x ) g ( x) x (x 2 to obtain the graph of 2) 2
  27. 27. TRANSFORMATIONS OF FUNCTIONS  Vertical and horizontal shift
  28. 28. TRANSFORMATIONS OF FUNCTIONS Vertical and Horizontal Reflecting and Stretching
  29. 29. Example 5: Use the graph of f ( x ) g(x) h( x) x x x to obtain the graph of
  30. 30. Example 5: Use the graph of f ( x ) g ( x) h(x) 2x 1 2 2 x 2 x 2 to obtain the graph of
  31. 31. COMBINATION OF FUNCTIONS  Functions can be added, subtracted, multiplied and divided in a many ways. For example consider a) f(x)+g(x) b) f(x)-g(x) c) f(x)/g(x) d) f(x).g(x) and and and and f ( x) x 2 and g(x)+f(x) g(x)-f(x) g(x)/f(x) g(x).f(x)
  32. 32. COMPOSITE FUNCTIONS  DefinitionWe define f  g Consider two functions f(x) and g(x). fg ( x ) f [ g ( x )] meaning that the output values of the function g are used as the input values for the function f.
  33. 33. Example 6:  If f (x)=3x +1 of x (a) f ° g (b) g° f and g(x)=2-x , find as a function
  34. 34. COMPOSITE FUNCTIONS  Determine the Domain of the Composite Functions
  35. 35. Example 7:  If f (x)=3x +1 and g(x)=2-x , find as a function of x (a) Find f ° g and determine its domain and range (b) Find g ° f and determine its domain and range
  36. 36. Properties for Graph of Functions  All forms of relations can be represented on coordinates  To test if a graph displayed is a function, vertical lines are drawn parallel to the y – axis.  The graph is a function if each vertical line drawn through the domain cuts the graph at only one point.
  37. 37. Example 8:  Consider the graphs shown below and state whether they represent functions:
  38. 38. 4. Inverse Function
  39. 39. The Inverse of Functions  If f is a function, the inverse is denoted by  Suppose y=f (x) then x y y y 32 1 1 ( y) f (x) 9 5 9 x 32 5 9 f 1 ( y) 5 (y 32 ) 9 Since y could be any variable, we can rewrite x 5 x f f as a function of x as (y 32 ) f 1 (x) 5 9 (x f 32 ) 1
  40. 40. Find the inverse of Example 11:  Find the inverse of : f (x) x 3 2
  41. 41. Graphical Illustration of an Inverse Function Verify that the inverse of f (x)=2x-3 is f 1 (x) x 3 2 Figure above shows the graph of these two functions on the same pair axes. The dotted line is the graph y=x. These graphs illustrate a general relationship between the graph of a function and that of its inverse, namely that one graph is the reflection of the other in the line y = x.
  42. 42. Example 12:  Find the inverse of : 1 f ( x) 1 2, x x  State the domain of the inverse 1.
  43. 43. FUNCTION WITH NO INVERSE  An inverse function can only exist if the function is a one-to-one function.
  44. 44. Subtopics 1. Relations and Functions 2. Representation of Functions 3. New Function form Old Function 4. Inverse of Functions Next week lecture: 1. Exponential Functions 2. Logarithm Functions, log x

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