1. CALCULUS & NUMERICAL
METHODS

2. Your lecturer:
 Name :
 Email:
 No Phone:

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. 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. 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. 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. FUNCTIONS AND GRAPHS

9. 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

10. 1.Relations and Functions
2.Representation of Functions

11. 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

12. 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)

13. 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)

14. Example 1:
 Let A = {1, 2, 3, 4} and B = {set of integers}. Illustrate
x 3.
the function f : x

15. 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

16. Example 2: Graph

17. Type of Function and Their Graph
Linear Function
f ( x)
 Where
are constant called the
coefficients of the linear
equation
x
;x
R

18. 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

19. Type of Function and Their Graph
Power Function
f ( x)
Where a is constant.
3
x ;x
R

20. ,
Type of Function and Their Graph
Exponential Function
f ( x)
Where a is a positive constant.
x
e ;x
R

21. ,
Type of Function and Their Graph
Logarithm Function
Where a is a positive constant.
f ( x)
ln x ; x
(0,
)

22. Example 10:
Consider for what value of x are the following
function defined?
1
f ( x)
x
2

23. 3. New Functions from Old
Function
1. TRANSFORMATIONS OF FUNCTIONS
2. COMBINATION OF FUNCTIONS
3. COMPOSITE FUNCTIONS

24. 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

25. TRANSFORMATIONS OF FUNCTIONS
 Vertical and horizontal shift

26. Example 3:
Use the graph of
f ( x)
x
g ( x)
x
to obtain the graph of
4

27. Example 4:
Use the graph of f ( x )
g ( x)
x
(x
2
to obtain the graph of
2)
2

28. TRANSFORMATIONS OF FUNCTIONS
 Vertical and horizontal shift

29. TRANSFORMATIONS OF FUNCTIONS
Vertical and Horizontal Reflecting and Stretching

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

31. 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

32. 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)

33. 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.

34. Example 6:
 If
f (x)=3x +1
of x
(a)
f ° g
(b)
g° f
and
g(x)=2-x , find as a function

35. COMPOSITE FUNCTIONS
 Determine the
Domain of the
Composite
Functions

36. 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

37. 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.

38. Example 8:
 Consider the graphs shown below and state whether
they represent functions:

39. 4. Inverse Function

40. 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

41. Find the inverse of
Example 11:
 Find the inverse of :
f (x)
x
3
2

42. 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.

44. Example 12:
 Find the inverse of :
1
f ( x)
1
2, x
x
 State the domain of the inverse
1.

45. FUNCTION WITH NO INVERSE
 An inverse function can only exist if the function is a
one-to-one function.

46. 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