- 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 DIFFERENTIALEQUATIONS (1 week) 5. Learning
Outcomes LO1: {C2}: Apply knowledge and fundamentalconcepts of
Calculus and Numerical Methods. LO3:{ C3,P3,CTPS}:Solve problems
particularly incomputer science with appropriate and high-level
programming language or tools. LO3:{C3, LL}:Solve real-life
application problems usingsuitable 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 2T1 (10%)T2 (10%) A2 (10%)MT1 (15%) F1 (15%) 40%
20%LO 3A1 (10%) MT2(15%) F3 (15%) 40% 7. FUNCTIONS AND GRAPHS 8.
Subtopics 1. Relations and Functions 2. Representation of Functions
3. New Function form Old Function 4. Inverse of Functions5.
Exponential Functions 6. Logarithm Functions, log x 9. 1.Relations
and Functions 2.Representation of Functions 10. Relations and
Functions Definition-A function is defined as a relation inwhich
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. Representation of Functions 1. Verbally ( by a
description in words) P(t) is the human population of the world of
time2. Numerically (by a table of values)
Year190019201940196019802000Population165018602300304044506080(millions)
12. Representation of Functions 3. Visually ( by a graph)
Population (millions)80006000 4000 20000 1900 1920 1940 1960 1980
2000 Year4. Algebraically ( by an explicit formula) 13. Example 1:
Let A = {1, 2, 3, 4} and B = {set of integers}. Illustrate x 3. the
function f : x 14. Example 2: Draw the graph of the function,f :x2x
,xRwhere 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 10 01 12 43 9 15. Example 2: Graph 16.
Type of Function and Their Graph Linear Function f ( x) Whereare
constant called the coefficients of the linear equationx;xR 17.
Type of Function and Their Graph Polynomial Where n is anonnegative
integer and the number are constant called the coefficients of the
polynomial. Quadraticf ( x)2x ;xR 18. Type of Function and Their
Graph Power Function f ( x)Where a is constant.3x ;xR 19. ,Type of
Function and Their Graph Exponential Function f ( x)Where a is a
positive constant.xe ;xR 20. ,Type of Function and Their Graph
Logarithm FunctionWhere a is a positive constant.f ( x)ln x ; x(0,)
21. Example 10: Consider for what value of x are the following
function defined?1f ( x) x2 22. 3. New Functions from Old Function
1. TRANSFORMATIONS OF FUNCTIONS 2. COMBINATION OF FUNCTIONS 3.
COMPOSITE FUNCTIONS 23. New Functions from Old Function
TRANSFORMATIONS OF FUNCTIONS The graph of one function can be
transform into the graph of adifferent function rely on a functions
equation. Vertical and horizontal shift 24. TRANSFORMATIONS OF
FUNCTIONS Vertical and horizontal shift 25. Example 3: Use the
graph off ( x)xg ( x)xto obtain the graph of 4 26. Example 4: Use
the graph of f ( x ) g ( x)x(x2to obtain the graph of 2)2 27.
TRANSFORMATIONS OF FUNCTIONS Vertical and horizontal shift 28.
TRANSFORMATIONS OF FUNCTIONS Vertical and Horizontal Reflecting and
Stretching 29. Example 5: Use the graph of f ( x ) g(x)h( x)xxxto
obtain the graph of 30. Example 5: Use the graph of f ( x ) g (
x)h(x)2x 1 22x2x2to obtain the graph of 31. COMBINATION OF
FUNCTIONS Functions can be added, subtracted, multiplied anddivided
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 andf ( x)x2and g(x)+f(x)
g(x)-f(x) g(x)/f(x) g(x).f(x) 32. COMPOSITE FUNCTIONS DefinitionWe
define f gConsider 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. Example 6: Iff (x)=3x +1of x
(a)f g(b)g fandg(x)=2-x , find as a function 34. COMPOSITE
FUNCTIONS Determine theDomain of the Composite Functions 35.
Example 7: Iff (x)=3x +1andg(x)=2-x , find as a functionof x
(a)Find f g and determine its domain and range(b)Find g f and
determine its domain and range 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. Example 8: Consider the graphs shown below and state
whetherthey represent functions: 38. 4. Inverse Function 39. The
Inverse of Functions If f is a function, the inverse is denoted by
Suppose y=f (x) then x y yy3211( y)f (x) 9 5 9x325 9f1( y)5(y32 )9
Since y could be any variable, we can rewritex5 xffas a function of
x as(y32 )f1(x)5 9(xf32 )1 40. Find the inverse ofExample 11: Find
the inverse of : f (x)x3 2 41. Graphical Illustration of an Inverse
Function Verify that the inverse of f (x)=2x-3 isf1(x)x3 2Figure
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. Example 12: Find the inverse of : 1f ( x) 12, x
x State the domain of the inverse1. 43. FUNCTION WITH NO INVERSE An
inverse function can only exist if the function is aone-to-one
function. 44. Subtopics 1. Relations and Functions 2.
Representation of Functions 3. New Function form Old Function 4.
Inverse of FunctionsNext week lecture: 1. Exponential Functions 2.
Logarithm Functions, log x
