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MATH 100- General Mathematics Reviewed 97-2003 051051

Education Arts (Chuka University)

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APPENDIX 2: CHUKA UNIVERSITY MODULE STRUCTURE

CHUKA UNIVERSITY

DEPARTMENT OF DISTANCE LEARNING

IN COLLABORATION WITH
FACULTY OF SCIENCE, ENGINEERING AND TECHNOLOGY.
DEPARTMENT: PHYSICAL SCIENCES
UNIT CODE: MATH 100
UNIT NAME: GENERAL MATHEMATICS

WRITTEN BY: MARK OKONGO EDITED BY: DR. SAMMY MUSUNDI

COPYRIGHT©CHUKA UNIVERSITY

ALL RIGHTS RESERVED

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INTRODUCTION
This is a common course intended for all students undertaking their bachelor`s degree
programme and not specializing in Mathematics. The prerequisite for this course is high
school secondary mathematics. You are expected to complete the course in 45 hours within
a period of one semester.

COURSE OBJECTIVES
By the end of the course, the learner should be able to:
 Classify real numbers
 State and explain the elementary properties of real numbers
 State and apply the laws of indices and logarithms
 State and apply the laws of logarithmic and exponential functions.
 Solve problems involving algebraic expansions and factorizations including
quadratic expressions and equations
 Define relations and functions and solve problems involving composite and inverse
functions.
 Differentiate various functions and determine the gradient function of any given
function
 Calculate the measures of central tendency i.e. mean ,mode and median of any
given data
 Calculate the measures of dispersion e.g. variance and standard deviation of a
given data

TABLE OF CONTENTS
1. ELEMENTARY PROPERTIES OF REAL NUMBERS
2. LAWS OF INDICES
3. EXPONENTIAL AND LOGARITHMIC FUNCTIONS
4. EXPANSION AND FACTORIZATION OF ALGEBRAIC AND QUADRATIC
EXPRESSIONS
5. SOLUTIONS OF QUADRATIC FORMULAE
6. RELATIONS AND FUNCTIONS; DEPENDENT AND INDEPENDENT
VARIABLES
7. COMPOSITE AND INVERSE FUNCTIONS
8. POLYNOMIAL FUNCTIONS, FACTOR AND REMAINDER THEOREM
9. DIFFERENTIATION OF SIMPLE FUNCTIONS AND GRADIENT TO A
CURVE
10. TURNING POINTS AND CURVE SKETCHING
11. RAW DATA ORGANIZATION, MEASURES OF CENTRAL TENDENCY;
MEAN, MEDIAN AND MODE
12. MEASURES OF DISPERSION; STANDARD DEVIATION
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TOPIC 1 TITLE: ELEMENTARY PROPERTIES OF REAL NUMBERS

LECTURE ONE
1.1 INTRODUCTION
This unit starts with the concept of number classification. Numbers are classified as
follows; Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers,
Real numbers and Complex numbers. It then introduces the concept of the properties of
each of the number classification which allows for the manipulation of expressions and
equations and obtaining the values of a variable.

1.2 LECTURE OBJECTIVES

 Classify number systems
 State the elementary properties of real numbers

1.3.1SUBTOPIC 1: NUMBER CLASSIFICATION

1. Natural numbers: These are the counting numbers. They include 1, 2, 3,…
2. Whole Numbers: These are the natural numbers and zero and include 0, 1, 2, 3…
3. Integers: These are whole numbers, their opposites/negatives and zero. They include the
following . . . -3, -2, -1, 0, 1, 2, 3…
p
4. Rational numbers: These are numbers that can be expressed in the form of , where,
q
both p and q are integers and q  0 . They can be written as decimal fractions which may
1 4 5
be recurring or terminating. Examples include , , .
3 7 3
p
5. Irrational numbers: These are numbers which cannot be written in the form of ,
q
where, both p and q are integers and q  0 . There decimal fractions are neither recurring
nor terminating. They are sometimes referred to as surds and include the following
numbers. 2, , 7 etc.

6. Real numbers: All the numbers above from 1 to 5 form the real numbers or are the sub
sets of real numbers.
7. Complex numbers: These are numbers written as a  bi , where a and b are real
numbers and i  1 . All real numbers are complex numbers. Examples include 1+3i,
1
5+5i, 9i, 3, etc.
3
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1.3.2SUBTOPIC 2: PROPERTIES OF REAL NUMBERS

1. Commutative property
These deals with the order of addition or multiplication as shown below.
Order of addition
or multiplication does not matter.
2. Associative property
Deals with grouping. i.e. (a + b) + c = a + (b + c), (nm)p = n(mp), where a, b, c, n, m, p are
all real numbers.
3. Additive Identity property
Zero added to any real number equals to the real number
a+ 0 = a, 4 + 0 = 4.
4. Inverse property of Addition
Any real number added to its negative is that real number.
a + -a = 0, 7 + -7 = 0
5. Multiplicative identity property
1 x a = a, 1 x 7 = 7.
6. Inverse property of multiplication
1
a a
a
7. Distributive property
a(b+c)=ab + ac, 4(3  7)  4  3  4  7

8. Reflexive property
(a + b) = (a + b), (2 - 7) = (2 - 7).
The same expression is written on both sides of the equation.
9. Symmetric property
If a  b , then b  a , if 4  5  9 , then 9  4  5
10. Transitive property
If a  b and b  c , then a  c , If 3(3) = 9, and 9 = 4 + 5, then 3(3) = 4 +5.

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11. Substitution property

If a = b, then a can be replaced by b. i.e. (3 + 2) = 5.

1.4 TOPIC SUMMARY

Number can be classified into Natural, Whole, Integers, Rational, Irrational, Real and
Complex. Properties of real numbers help us manipulate various expressions and equations
and in solving the values of a variable.

1.5 NOTE
All Natural numbers are whole numbers, all whole numbers are integers, all integers are
rational numbers. Rational numbers are not irrational numbers and all real numbers are
complex numbers therefore all numbers are complex numbers.
1.6 TOPIC ACTIVITIES
Draw figure showing the various number subsets.
1.8 FURTHER READING
References:
 Secondary mathematics students book one to four, 3rd edition by Kenya
literature bureau 2015.

 E. H. Connell, (1999) Elements of Abstract and Linear Algebra, University of

Miami.

 Chandler S. and Bostock L. (1990), Core Mathematics for A- level. Stanley

Thornes Publishers Ltd.
.

 Wolfram MathWorld
http://mathworld.wolfram.com/. A complete and comprehensive guide to all topics
in mathematics. The students are expected to become familiar with this web site and
to follow up key words and module topics at the site.

 Wolfram MathWorld
http://mathworld.wolfram.com/. Find demonstrations of numbers and number
systems and use the mathematica player programme from the Wofram Mathworld
website to practice the demonstrations.

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 Wikipedia
http://www.wikipedia.org/ Wikipedia provides encyclopaedic coverage of all
mathematical topics. Students should follow up key words by searching at wikipedia.

1.9 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.
Assessments
1.9.1 Classify each of the following numbers
a) -1
b) 6
c) 0
d) -2.222
1
e)
2
f) 5

1.9.2 State the properties of real numbers in the equations below

a) 5=5+0
b) 5(2x + 7) = 10x + 35
c) 24(2) = 2(24)
d) (7 + 8) + 2 = 2 + (7 + 8)
e) 5(4+6)=5(10)
f) 5(4+6) = 5(4+6)
g) If 5(4+6)=5(10), then 5(10) = 5(4+6)
h) If 5(10)=5(4+6), and 5(4+6)=20+30, then 5(10)=20+30

Solutions to Assessments 1.9.1

a) -1: real, rational, integer
b) 6: real, rational, integer, whole, natural.
c) 0: real, rational, integer, whole, natural
d) -2.222: real, rational
1
e) : real, rational
2
f) 5: real, irrational
g) 4 : complex

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Solutions to Assessments 1.9.2

h) 5 = 5 + 0: Additive identity property

i) 5(2x + 7) = 10x + 35: Distributive
j) 24(2) = 2(24): Commutative
k) (7 + 8) + 2 = (2 + (7) + 2): Associative
l) 5( 4 + 6) = 5(10): Substitution
m) 5(4 + 6) = 5(4 + 6): Reflexive
n) If 5(4 + 6) = 5(10), then 5(10) = 5(4 + 6): Symmetric
o) If 5(10) = 5(4 + 6), and 5(4 + 6) = 20 + 30, then 5(10) = 20 + 30:Transitive

Further Assessments
1. Identify the property of real numbers being applied in each of the following
(i) 11 + 0 = 11
(ii) (3 + 6) + 4 = 4 + (3 + 6)
(iii) (5 + 9) + 7 = 5 + (9 + 7)

2. Using examples justify the following statements

(i). All Natural numbers are Integers but all integers are not Natural numbers

(ii). An integer is a rational number

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TOPIC 2 TITLE: LAWS OF INDICES

LECTURE ONE
2.1 INTRODUCTION
The lowest factors of 2000 are 2×2×2×2×5×5×5. These factors are written as 24×53, where
2 and 5 are called bases and the numbers 4 and 3 are called indices or exponents. When an
index is an integer it is called a power. Thus, 24 is called ‗two to the power of four, and has
a base of 2 and an index of 4. Similarly, 53 is called ‗five to the power of 3 and has a base
of 5 and an index of 3.Special names may be used when the indices are 2 and 3, these being
called ‗squared‘ and ‗cubed, respectively. Thus 72 is called ‗seven squared‘ and 93 is called
‗nine cubed‘. When no index is shown, the power is 1, i.e. 2 means 21.

Reciprocal
The reciprocal of a number is when the index is (−1) and its value is given by 1 divided
1
by the base. Thus the reciprocal of 2 is2−1 and its value is or 0.5. Similarly, the reciprocal
2
1
of 5 is 5−1which is or 0.2
5

2.2 LECTURE OBJECTIVES

 State and explain the laws of indices
 Solve problems involving indices

2.3.1 SUBTOPIC 1: LAWS OF INDICES

I f m and n are integers, then the following laws of indices apply.
1) am x an = am + n

am
2) n
 a mn
a

3) (am )n = amn

4) (ab)m = am x bm

n
a an
5)    n
b b

6) a0 = 1

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7) a1 = a
1
8) a  n 
an
1
9) a n  n a

 a
m
m
10) a n  n

Useful results
1
1) If a x  b , then a  b x
y x

2) If a  b , then a  b
x y x
and b  a y

1
3) If a  b , then a  b x
x

4) If a x  a y , then x  y

2.3.2 SUBTOPIC 2: SOLVING PROBLEMS INVOLVING INDICES

1) Simplify the following
4
64 x  8

3

Solution
4
 26 x  23 
4 
64 x  8

3 3  264  4
2) Solve the equation 4x2  22 x3  96

Solution
   22 x 3  96  22 x  24   22 x  23   96
x2
4x2  22 x3  96  22
 22 x  24  23   96 ,  24  22 x   96 ,  x  1

2.4 TOPIC SUMMARY

 When simplifying calculations involving indices, certain basic rules or laws can be
applied, called the laws of indices.
 When multiplying two or more numbers having the same base, the indices are
added.

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 When a number is divided by a number having the same base, the indices are
subtracted. Thus
 When a number which is raised to a power is raised to a further power, the indices
are multiplied.
 When a number has an index of 0, its value is 1.
 A number raised to a negative power is the reciprocal of that number raised to a
positive power. Similarly, when a number is raised to a fractional power the
denominator of the fraction is the root of the number and the numerator is the
power.

2.5 NOTE
The laws of indices only apply to terms having the same base. Group the terms having the
same base, and then applying the laws of indices to each of the groups independently.

2.6 TOPIC ACTIVITIES

Solve problems in the self assessment exercises.
2.7 FURTHER READING
References:
 Secondary mathematics students book one to four, 3rd edition by Kenya
literature bureau 2015.
 A Textbook for High School Students Studying Maths by the Free High School
Science Texts authors, 2005.

 E. H. Connell, (1999) Elements of Abstract and Linear Algebra, University of

Miami.

 Chandler S. and Bostock L. (1990), Core Mathematics for A- level. Stanley

Thornes Publishers Ltd.
.
 Wolfram MathWorld
http://mathworld.wolfram.com/. A complete and comprehensive guide to all topics
in mathematics. The student is expected to become familiar with this web site and
to follow up key words and module topics at the site.

 Wikipedia
http://www.wikipedia.org/ Wikipedia provides encyclopaedic coverage of all
mathematical topics. Students should follow up key words by searching at wikipedia.

 Wolfram MathWorld
http://mathworld.wolfram.com/. Find demonstrations on indices and use the
mathematica player programme from the Wofram Mathworld website to practice the
demonstrations.

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2.8 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.
2.8.1 Assessments and Solutions
Classify each of the following numbers
1) Show that:

2) Show that =1
1 1

3) If a  2 3  2 3
, show that 2a3  6a  3  0
4) Simplify x 1 y  y 1 y  z 1 y
1
5) Solve for x given that 9  81x 
27 x2
6) Solve for x 4x  3  2x2  25  0

Solutions to Assessments 2.8.1

4. Simplify to 1
4
5. x  .
7
6. x = 1 or 2
Further Assessments
1
1. Solve for x in the 9  81x 
27 x2

( x a b ) 2  ( xb  c ) 2  ( x c  a ) ) 2
2. Simplify
x a  xb  x c )4

3. Without using a calculator find the value of

3. Find the simplest form of following

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TOPIC 3 TITLE: LOGARITHMS

LECTURE ONE
3.1 INTRODUCTION
With the use of calculators firmly established, logarithmic tables are now rarely used for
calculation. However, the theory of logarithms is important, for there are several scientific
and engineering laws that involve the rules of logarithms. If a number y can be written in
the form ax, then the index x is called the ‗logarithm of y to the base of a‘,
i.e. y  a x ,  x  log a y  3  log10 1000
Check this using the ‗log‘ button on your calculator.

(a) Logarithms having a base of 10 are called common logarithms

The following values may be checked by using a calculator: log 17.9 = 1.2528 . . . ,
log 462.7 = 2.6652 . . .and log 0.0173=−1.7619…

(b) Logarithms having a base of e (where ‗e‘ is a mathematical constant approximately

equal to 2.7183) are called hyperbolic, Napierian or natural logarithms, and log e is
usually abbreviated to . The following values may be checked by using a calculator:

3.2 LECTURE OBJECTIVES

- State and explain the laws and properties of logarithms
- Solve problems involving logarithms

3.3.1 SUBTOPIC 1: LAWS OF LOGARITHMS

There are three laws of logarithms, which apply to any base
1) log  A  B   log A  log B
 A
2) log    log A  log B
B
3) log An  n log A

Change of base formulae

logb x
The log a x 
logb a
log10 7 ln 7
Example 1. Evaluate log5 7   1.20906 to 5 dp. Or log 7   1.20906
log10 5 ln 5

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3.3.2 SUBTOPIC 2: SOLVING PROBLEMS INVOLVING LOGARITHMS

1). Evaluate each of the following

a). log 4 16

Soln. let log 4 16  y ,  4 y  16 ,  4 y  44 ,  y  4

1
b). log 5 .
125
1 1
Soln. let log5  y ,  5y   53 ,  y  3
125 125
1
c). ln .
e
1 1
Soln. let ln e  y ,  e y   e1 ,  y  1 .
e e
2). Solve for x given that log x  1  log( x  3)

Soln.

 10 
log x  log10  log( x  3) ,  log x  log   ,  x( x  3)  10 . Solving the quadratic
 x 3
equation yields x  2 or x  5

3.4 TOPIC SUMMARY

1. The three laws of logarithms only apply if the bases are the same, though the bases
can be changed to be similar.
2. logb 1  0
3. logb b  1
4. logb b f ( x )  f ( x)

3.5 NOTE
The logarithm of zero or a negative number does not exist.

3.6 TOPIC ACTIVITIES

Solve various logarithmic problems of base 10 and base e and confirm the answers using a
calculator.

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3.7 FURTHER READING

References:

 John Bird, (2007). Engineering Mathematics, 5th edition, Elsevier.

 A Textbook for High School Students Studying Maths by the Free High School
Science Texts authors, 2005.

 E. H. Connell, (1999) Elements of Abstract and Linear Algebra, University of

Miami.

 Chandler S. and Bostock L. (1990), Core Mathematics for A- level. Stanley

Thornes Publishers Ltd.
.
 Wolfram MathWorld
http://mathworld.wolfram.com/. A complete and comprehensive guide to all topics
in mathematics. The student is expected to become familiar with this web site and
to follow up key words and module topics at the site.

 Wikipedia
http://www.wikipedia.org/ Wikipedia provides encyclopaedic coverage of all
mathematical topics. Students should follow up key words by searching at wikipedia.

 Wolfram MathWorld
http://mathworld.wolfram.com/. Find demonstrations On logarithms and indices and
use the mathematica player programme from the Wofram Mathworld website to
practice the demonstrations.

3.8 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.

Assessments and Solutions

3.9.1 Solve for x in each of the following
a. 7 x  9
b. 104 x1  100x  0
c. 105 x  8
d. log x  log( x  1)  log(3x  12)
e. log 2 ( x 2  6 x)  3  log 2 (1  x)

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Solutions to Assessments 3.8.1

a. 1.1292
b. - 0.5
c. 4.0969
d. X = 6 or -2
e. X = - 4, or 2.

Further Assessments
1. solve
1 2
2. Solve for x given (Log 7 x ) 2 - log 7 x =
3 3
3. Given , and , express the following in terms of

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TOPIC 4 TITLE: EXPANSION AND FACTORIZATION OF ALGEBRAIC AND

QUADRATIC EXPRESSIONS AND SOLUTIONS OF QUADRATIC FORMULAE

LECTURE ONE
4.1 INTRODUCTION
Algebra is that part of mathematics in which the relations and properties of numbers are
investigated by means of general symbols. For example, the area of a rectangle is found by
multiplying the length by the breadth; this is expressed algebraically as A = l ×b, where A
represents the area, l the length and b the breadth. The basic laws introduced in arithmetic
are generalized in algebra. Let a, b, c and d represents any four numbers. Then:

(i) a +(b+c)=(a+b)+c

(ii) a(bc)=(ab)c

(iii)a+b=b+a

(iv) ab=ba

(v) a(b+c)=ab+ac

ab a b
(vi)  
c b c

(vii) (a+b)(c+d)=ac+ad+bc+bd

4.2 LECTURE OBJECTIVES

 Expand and factorize algebraic and quadratic expressions
 Solve quadratic equations using various methods.

4.3.1 SUBTOPIC 1: EXPAND AND FACTORIZE ALGEBRAIC AND QUADRATIC

EXPRESSIONS.
Expansion involves the opening/removal of brackets.

Example 1
Expand the following (x-7y)(2x - 3y)

soln.
x(2x – 3y)-7y(2x – 3y) = 2x2 - 3yx-14xy+21y2 = 2x2-17xy+21y2

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Example 2

Show that (x + y)2 + (y + z)2 + (x - z)2 = 2(x + y)(x - z)

Soln
(x + y)2 + (y + z)2 + (x - z)2 = x2+2xy+y2+y2+2yz+z2+x2-2xz-z2

= 2x2+2xy+2y2+2yz-2xz=2(x2+xy+y2+yz-xz) = 2(x+y)(x-z)

Brackets and factorization

When two or more terms in an algebraic expression contain a common factor, then this
factor can be shown outside of a bracket.

For example
ab + ac = a(b + c)
which is simply the reverse of law (v) of algebra, and 6px + 2py − 4pz = 2p(3x + y − 2z)
This process is called factorization.

Factorizing quadratic expressions

Quadratic expressions are given in the form ax2 + bx + c. look for two numbers m and n
whose product is (ac) and sum is (b) then rewrite the expression then factorize.

Example 1
Factorize x2+13x+42,

Soln. here a = 1, b=13 and c = 42. Let m = 7 and n =6.

x2+13x+42 = x2+7x+6x+42 = x(x+7)+6(x+7) then factor out (x+7)to get
(x+7)(x+6)

Example 2
Factorize 2x2+7x-15

Soln.ac = -30 and b = 7. let m = 10 and n = -3  2x2+7x-15 = 2x2+10x-3x-15

= 2x (x+5)-3(x+5) = (x+5)(2x-3)

4.3.2 SUBTOPIC 2: SOLVING QUADRATIC EQUATIONS

Quadratic equations are equations of the form ax2 + bx + c = 0. It is a second degree
polynomial.
Methods of solving quadratic equations
(1). Factor method
The quadratic equation can be factorized using the methods discussed above for quadratic
expressions then each factor equated to zero to obtain the two values of x.

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Example.
Solve x2+3x+2=0
Soln.
x2+3x+2=0 = (x+2)(x+1)=0  x = -2 or x = -1

(2). Completing square method

Solve the equation 2x2+3x+1= 0. First ensure that the co efficient of x2 is 1
2
3 1 3  3 3
 x  x   0 . Represent x 2  x , by  x   , where , is obtained by dividing
2

2 2 2  4 4
2
 3
the co efficient of x by 2. Make sure the two are equal by expanding x  =
 4
3 9
x2  x  , therefore
2 16
2 2
3  3 9 3 1  3 9
x  x = x  -
2
, hence the equation x 2  x   0 becomes (  x   - )+
2  4  16 2 2  4  16
1
=0
2
2 2
 3 1  3 1
Which simplify to  x     0 ,   x    . Find the square root on both
 4  16  4  16
sides

 3 1 1
 x     ,  x   or x = -1
 4 4 2

(3). The quadratic formulae

The quadratic formulae is obtained by solving the quadratic equation ax2+bx+c = 0 using
b  b 2  4ac
the completing square method. The formulae is given by x  .
2a
Example
Solve the equation x2-5x+4=0, using the quadratic formulae

5  52  4(1)(4) 5  25  16
Soln. x  , x ,  x  2.5  1.5
2(1) 2

 x  4 or -1
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4.4 TOPIC SUMMARY

Quadratic equations can be solve using four methods namely the graphical method, the
factor method, the completing square method and the formulae method.

4.5 NOTE
Given the general quadratic equation ax2+bx+c =0,
(i). When b2>4ac then the roots are real and different.
(ii) When b2=4ac then the roots are real and identical
(iii). When b2<4ac then the roots are imaginary.

4.6 TOPIC ACTIVITIES

Derive the formulae method from the completing square method.
4.7 FURTHER READING
References:
References:
 John Bird, (2007). Engineering Mathematics, 5th edition, Elsevier.

 Secondary mathematics students book one to four, 3rd edition by Kenya

literature bureau 2015.

 A Textbook for High School Students Studying Maths by the Free High School
Science Texts authors, 2005.

 E. H. Connell, (1999) Elements of Abstract and Linear Algebra, University of

Miami.

 Chandler S. and Bostock L. (1990), Core Mathematics for A- level. Stanley

Thornes Publishers Ltd.
.
 Wolfram MathWorld
http://mathworld.wolfram.com/. A complete and comprehensive guide to all topics
in mathematics. The students is expected to become familiar with this web site and
to follow up key words and module topics at the site.

 Wikipedia
http://www.wikipedia.org/ Wikipedia provides encyclopaedic coverage of all
mathematical topics. Students should follow up key words by searching at wikipedia.

 Wolfram MathWorld
http://mathworld.wolfram.com/. Find demonstrations on quadratic equations and use
the mathematica player programme from the Wofram Mathworld website to practice
the demonstrations.

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4.8 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.

4.9.1 Assessments and Solutions

1. Factorize 3x2 + 5x + 0.75

2. Form quadratic equations from the following roots

a. (3, 5)
b. (-2,1)
c. (-3,4)
 1
d.  3, 
 2
3. Solve the equations below using the factor method, completing square method and
the formulae method and compare your answers.
a. x2 + 3x + 2 = 0
b. 7x2 + 9x + 2 = 0

Solutions to Assessments 4.9.2

(2 x  3)(6 x  1)
1.
4
2.
a. (3, 5)  x2-8x+15=0
b. (_2, 1)  x2+x – 2=0
c. (-3,4)  x2+7x+12=0
 1 1 1
d.  3,   x 2  3 x  1  0
 2 2 2

3.
a. x = -2, or -1
7
b. x  1 or 
2

Further Assessments
1. Find two numbers whose sum is 8 and whose product is 11
2. The roots of the quadratic equation 2x2– px+4=0 differ by 1. Find the value of p.
3. Given that the roots of the quadratic equation 4x2=8x+5, are  and  . find a
quadratic equation whose roots are    and   
4. The height s metres of a mass projected vertically upwards at time t seconds is
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S = u t – 12gt2. Determine how long the mass will take after being projected to
reach
a height of 16m (a) on the ascent and (b) on the descent, when u = 30 m/s and g =
9.81 m/s2.

Answer: The mass will reach a height of 16m after 0.59 s on the ascent and
after 5.53 s on the descent.

5. Calculate the diameter of a solid cylinder which has a height of 82.0 cm and a total
surface area of 2.0m2.

Answer. Diameter of the cylinder = 2 × 0.2874 = 0.5748m or

57.5 cm correct to 3 significant figures

NB: The icons will be put later in consultation with ICT.

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TOPIC 5 TITLE: POLYNOMIAL FUNCTIONS, FACTOR AND REMAINDER

THEOREM

LECTURE ONE
5.1 INTRODUCTION
Polynomials ➤38 74➤
A monomial is an algebraic expression consisting of a single term, such as 3x, while a
binomial consists of a sum of two terms, such as x + 3y. A polynomial is an algebraic
expression consisting of a sum of terms each of which is a product of a constant and one or
more variables raised to a non-negative integer power. An example with a single variable,
x, isx3− 2x2+ x + 4 and the general form of a polynomial in x is written as
pn x  an x n  an1 x n1  an2 x n2  . . .  a1 x  a0 , where the ai, i = 0, 1, . . . , n are given
numbers called the coefficients of the polynomial. We use pn(x) to denote a polynomial in
x of degree n. The notation ai is often used when we have a list of quantities to describe and
– i is called the subscript. If n is the highest power that occurs, as in the above, and if
an  0, then we say that the polynomial is of nth degree. An important property of a
polynomial in x is that it exists (i.e. has a definite value) for every possible value of x.

5.2 LECTURE OBJECTIVES

-Apply the long division method to obtain the remainder and factors of polynomials.
- Apply the factor and remainder theorems in expansion and factorization of polynomials.

5.3.1 SUBTOPIC 1: THE LONG DIVISION

The long division method is used to obtain the quotient when 2 x3  x2  6 x  9 is divided
by x+2
as shown below.

2 x 2  3x
x  2 2 x3  x 2  6 x  9

2 x3  4 x 2
0 3x 2  6 x  9

3x 2  6 x
009

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2 x3  x 2  6 x  9 9
Therefore  x2
x2 x2

5.3.2 SUBTOPIC 2: REMAINDER THEOREM

The remainder when a polynomial p(x) is divided by x − a is given by
p( x) r
 q( x)  , where r = p(a) and q(x) is a polynomial of degree less than that of
x2 xa
p(x).
We can see this by re-writing the above result asp(x) = (x − a)q(x) + r and putting x = a to
give p(a) = r

For example, the remainder when x2+ 2 is divided by x − 1 is 12+ 2 = 3,

5.3.3 SUBTOPIC 3: THE FACTOR THEOREM

If x − a is a factor of the polynomial p(x) then p(a) = 0.

For example, x – 1 is a factor of x2– 1, a = 1 and p(x) = x2 - 1  12 – 1 = 0 =p(1)

5.4 TOPIC SUMMARY

The remainder when a polynomial p(x) is divided by x − a is r = p(a). If x –a is a factor of
the polynomial p(x) then p(a) = 0.

5.5 NOTE
If x – a is a factor of the polynomial p(x) then the remainder when the polynomial p(x)
is divided by x − a is 0.

5.6 TOPIC ACTIVITIES

Confirm that the remainder when the polynomial p(x) = x4 + 2x3 – x - 2is divided by (x –
1), the remainder is zero. Confirm using the long division method.

5.7 FURTHER READING

References:

 Bill Cox, (2001). Understanding Engineering mathematics, Elsevier.

 John Bird, (2007). Engineering Mathematics, 5th edition, Elsevier.
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 E. H. Connell, (1999) Elements of Abstract and Linear Algebra, University of

Miami.

 Chandler S. and Bostock L. (1990), Core Mathematics for A- level. Stanley

Thornes Publishers Ltd.
.
 Wolfram MathWorld
http://mathworld.wolfram.com/. A complete and comprehensive guide to all topics
in mathematics. The students is expected to become familiar with this web site and
to follow up key words and module topics at the site.

 Wikipedia
http://www.wikipedia.org/ Wikipedia provides encyclopaedic coverage of all
mathematical topics. Students should follow up key words by searching at wikipedia.

 Wolfram MathWorld
http://mathworld.wolfram.com/. Find demonstrations on quadratic equations and use
the mathematica player programme from the Wofram Mathworld website to practice
the demonstrations.

5.8 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.

5.8.1 Assessments and Solutions

1. Factorize the polynomial x3− x2− x + 1 = (x − 1), (x -1), (x + 1), hence solve the
equation x3− x2− x + 1 = 0
2. Obtain the remainder when x3- 3x2+6x+5 is divided by (x-2) using the remainder
theorem. Confirm your answer using the long division method.
3. Confirm whether (x + 1) is a factor of x3 + 2x _ 5x - 6

Solutions to Assessments 5.8.1

1. x3− x2− x + 1 = (x − 1), (x -1), (x + 1),x =1, 1,-1
2. Remainder, r = 13.
3. p(1) = 0, hence a factor.

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Further Assessments
a. The expression is divisible fully by .
(i) Find the value of
ii. Use the remainder theorem to find the remainder when the expression is
divided by Hence confirm your answer of the remainder using the
long division method.
b. If and if , evaluate and find the values
of for which
c. The expression is divisible exactly by . Find the value
of and the remainder when the expression is divided by
d. Use long division method to find the remainder when 5 9 is divided by the
factor
e. Find the value of ( a), given that when f(x) = x 5 + 4x4 - 6x2 + ax + 2, is divided by
x+2, the remainder is 6.
f. . If f(x) = x4 + hx3 + gx2 – 10x – 12 has factors (x + 1) and (x - 2). Find the
constants h and g and the remaining factors
g. If f(x) = ax2 + bx + c leaves remainder 1, 25 and 1 on division by (x - 1) and (x +
1) and (x – 2) respectively. Show that f(x) is a perfect square.
h. The expression is divisible exactly by . Find the value
of and the remainder when the expression is divided by

NB: The icons will be put later in consultation with ICT.

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TOPIC 6 TITLE: RELATIONS AND FUNCTIONS; COMPOSITE AND INVERSE

FUNCTIONS.

LECTURE ONE
6.1 INTRODUCTION
The notions of a set and a function are the most fundamental concepts which together
constitute the foundations of Mathematics. Indeed, different branches of Mathematics start
with these two fundamental concepts. In this activity, we are simply demonstrating how
sets of objects are easily extracted from our surroundings.

Key Concepts
Function: This is a special type of mapping where an object is mapped to a unique image.
Mapping: This is simply a relationship between any two given sets
Set: This is a collection of objects or items with same properties

Consider the function y = 20x2 - 5t2, for 0  x  4 shown in table 1.

Table 1

x 0 1 2 3 4
y 0 15 20 15 0

The set of x variables are the independent variables while the y variables are the dependent
variables i.e. they depend on the given values of x. the set of values of the independent
variables are called the domain. The rule which is applied to the independent variables to
produce the dependent variables is called the function. The set of values of the dependent
variable is called the range or image.

6.2. LECTURE OBJECTIVES

 Define a function and evaluate a combination of functions
 Determine the inverse of a function
 Define composite functions and compose various functions
 Determine the inverse of a composite function.

6.3.1 SUBTOPIC 1: COMBINATION AND INVERSE FUNCTIONS

Definition of a function ➤88 107➤
A function is a relation which expresses how the value of one quantity, the dependent
variable, depends on the value of another, the independent variable.

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a) Story of Maize Grinding Machine

Jane walks in a village to a nearby market carrying a basket of maize to be ground into
flour. She puts the maize into a container in the grinding machine (figure 1) and starts
rotating the handle. The maize is then ground into flour which comes out of the machine
for her to take home.

Figure 1
Question
What relation can you make among the maize, the grinding machine and the flour?

b) Story of children born on the Christmas day in the year 2005

It was reported on the 25th of December 2005 in Pumwani Maternity Hospital which is in
Nairobi the Capital City of Kenya that mothers who gave birth to single babies were a total
of 52. This was the highest tally on that occasion. As it is always the case each baby was
given a tag to identify him or her with the mother.

Questions
1. In the situation above given the mother how do we trace the baby?
2. Given the baby how do we trace the mother?

Activity
Note that we can now represent the story of the maize grinding machine diagrammatically
as shown in figure 2.

Figure 2

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A = Set of some content (in this case maize) to be put in the grinding machine.

f = The mapping or function representing the process in the grinding machine

B = Set of the product content (in this case flour) to be obtained

Example 1
In this example we define two sets and a relation between them as follows:
Let A = {2, 3, 4} and B = {2, 4, 6, 8}

f is a relationship which says ―is a factor of‖ e.g. 3 is a factor of 6. In this case we have the
following mapping shown in figure 3.

Figure 3

Example 2
Think of a number of such situations and represent them with a mapping diagram as shown
above.

In our second story of each mother giving birth to only one child can be represented in a
mapping diagram as shown in figure 3.

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Figure 3

A = Set of babies
B = Set of mothers
f = Relation which says ―baby to‖

Remarks 3
(i). Notice that in this mapping each object is mapped onto a unique image. In this case it is
a function. We write f: A→ B

(ii). Note also that in the mapping above even if we interchanged the roles of sets A and B
we still have that each object has a unique image. Thus we have the mapping shown in
figure 4.

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Figure 4
In this case we have
B = Set of mothers
A = Set of babies
g = Relation which says ―is mother of ‖
In this case we say that the function f has an inverse g. We normally denote this inverse
e.g. as f 1

Thus for f: A → B we have f 1 : B → A

Example
Let A = {1, 2, 3, 4, 5}
B = {2, 3, 5, 7, 9, 11, 12}
f : x → 2x + 1
Then we have the mapping as follows: f : x → 2x+ 1

Domain and codomain

Given the general mapping f(x) = 2x + 1, we can write: f(1) = 3, f(2) = 5 etc as shown in
figure 5, The set A is called the domain off and the set B is called the co domain of f. The
set { 3, 5, 7, 9, 11} within which all elements of A are mapped is called the range off.

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A B
Figure 5
Evaluating functions
From f(x) = 5x – 2x +4, and g(x) =x + 6, f(1)= 5(1)- 2(1)+4 = 7 and
g(3) = 3+6 = 9.
Combining functions
Given two functions f(x) and g(x), the following holds true
1. (f + g)(x) = f(x) + g(x)
2. (fg)(x) = f(x)g(x)
3. (f - g)(x) = f(x) - g(x)
4.
f ( x) f ( x)

g ( x) g ( x)

Piecewise Functions
Is a function whose equation is broken into pieces.

 3x 2  4, if x  4 
 
Example f ( x)  10, if  4  x  15
1  x if x  15 
 
f(-5) = 79, f(2) = 10 and f(20) = -19.

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Inverse of a function
To find the inverse of a function f : x → 2x+ 3. Let y = 2x+3 and make x the subject of the
y 3
formulae to obtain x  , but f-1(y) = x
2
y 3 x3
 f 1 ( y )  x  , hence f 1 ( x)  . Confirm that f(1) = 5 and f 1 (5)  1
2 2

6.4.1. SUBTOPIC 2: COMPOSITE FUNCTIONS

Composite functions are about combinations of different simple mappings in order to yield
one function. The process of combining even two simple statements in real life situations in
order to yield one compound statement is important. We are set to verify that two
elementary functions whose formulae are known if combined in a certain order will yield
one composite formula and if the order in which they are combined is reversed then this
may yield a different formula. It is equally important to be able to represent a composite
function pictorially by drawing its graph and examine the shape. Indeed, the learner will be
able to draw these graphs starting with linear functions and quadratic functions.

Composite Function: This is a function obtained by combing two or more other simple
functions in a given order.
a) A Story of Nursery School Children
Two Children brother and sister called John and Jane go to a Nursery school called
Little Friends. One morning they woke up late and found themselves in a hurry to
put on clothes and run to school, Jane fist put on socks then shoes. But her brother
John fist put on shoes then socks. Jane looked at him and burst into laughter as she
run to school to be followed by her brother.

Question
Why did Jane burst into laughter?
b) Story of a visit to a beer brewing factory
A science Club in a secondary school called Nabumali High School in Uganda, one
Saturday made a trip to Jinja town to observe different stages of brewing beer called
Nile Beer. It was noted that of special interest was the way some equipment used in
the process would enter some chamber and emerge transformed. For example an
empty bottle would enter a chamber and emerge transformed full of Nile Beer but
without the bottle top. Then it would enter the next chamber and emerge with the
bottle top on (figure 6).

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Figure 6

Question
Can you try to explain what happens in each chamber of the brewing factory?

Activity
We note that in our story of the Nursery school Children what is at stake is the
order in which we should take instruction in real life situations. Jane laughed at
her brother because she saw the socks on top of the shoes. In other words her
brother had ended up with composite instruction or function which was untenable.
We can also look at other such cases through the following example.

Example 1
I think of a number, square it then add 3 or I think of a number add 3 then square
it. If we let the number to be x, then we will end up with two different results
namely x2 + 3 and (x + 3)2respectively.

Example 2
Can you now come up with a number of examples similar to the one above?
If we now consider our story on the brewing of Uganda Warangi we note that
each Chamber has a specific instruction on the job to perform. This is why whatever
item passes through the chamber must emerge transformed in some way.
We can also look at an example where instructions are given in functional form
with explicit formulae as shown below.

Example 3
Consider the composition of the functions.
f : x → 2 x and g : x → x + 5
Here if we are operating f followed by g then we double x before we add 5. But
if operate g followed by f then we add 5 to x before we double the result.
For notation purposes
( f  g )( x)  f ( g (( x)) means g then f. While ( g  f )( x)  g ( f (( x)) means
f then g. Thus we have:

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Representing the composite function f ( g ( x)) = 2(x+ 5

While:

Exercise 4
Given f : x → 3 x + 1 g : x → x − 2
Determine the following functions:
(a) ( g  f )( x)
(b) ( f  g )( x)
(c) ( f o g)−1
(d) ( g o f )−1

Taking x = 3 draw a diagram for each of the composite functions above as is the
case in example 3 above.

Exercise
Sketch the graph for each of the following function: assuming the domain for
each one of them is the whole set ℜ of real numbers.
a). f ( x) = 2 x − 3
b. ) g(x) = 4x2 − 12x
c). h(x) = x3 − 3 x + 1
d). k( x) = 2 sin x

6.5. TOPIC SUMMARY

The important point about a function is that it must have a single unique value, y, for every
value, x, for which the function is defined. X is also called the argument of the function
f (x). The set X of all values for which the function f is defined is called its domain. The
set of all corresponding values of y = f (x),= Y, is called the range of f (x). We sometimes
express a function as a mapping between the sets X, Y denoted f : X → Y . The value of a
particular function for a particular value of x, say x = a, is called the image of (a) under f,
denoted f (a).

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6.6. NOTE
Composite functions are about combinations of different simple mappings in order
to yield one function.
6.7 TOPIC ACTIVITIES
Given f : x → 3 x + 1, g : x → x − 2
Determine the following functions:
(a) ( g  f )( x)
(b) ( f  g )( x)
(c) ( f o g)−1
(d) ( g o f )−1

Taking x = 3 draw a diagram for each of the composite functions above as is the
case in example 3 above.

6.8 FURTHER READING

References:

 Bill Cox, (2001). Understanding Engineering mathematics, Elsevier.

 John Bird, (2007). Engineering Mathematics, 5th edition, Elsevier.

 Chandler S. and Bostock L. (1990), Core Mathematics for A- level. Stanley

Thornes Publishers Ltd.

 Wolfram MathWorld
http://mathworld.wolfram.com/. Find demonstrations on functions and use the
mathematica player programme from the Wofram Mathworld website to practice the
demonstrations.

 Wolfram MathWorld
http://mathworld.wolfram.com/SetTheory.html
• Read this entry for Set Theory.
• Follow links to explain specific concepts as you need to.

 Wikipedia
http://www.wikipedia.org/
• Type ‗Set Theory‘ into the search box and press ENTER.
• Follow links to explain specific concepts as you need to.

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6.9 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.

6.9.1 Assessments and Solutions

x5
1. Given the function defined by f ( x)  , obtain f 1 ( x) and f 1 (3)
x6
1
2. Given that f ( x)  x 2  1, and g ( x)  , obtain ( f  g )( x) and ( g  f )( x)
x

Solutions to Assessments 6.9.1

6x  3 23
1. f 1 ( x)  , f 1 (3) 
x 1 2
1 1
2. ( f  g )( x)  2  1 , and ( g  f )( x)  2
x x 1

Further Assessments
x
1) . Given that f ( x)  sin x and g ( x)  
2
Calculate: (i). (g o f) (ii). (f o g)
2). Given the function defined by f ( x)  25  x 2 and g ( x)  x , evaluate

(i). f ( g ( x)) (ii). f ( g (3))

3). Given that

(a) Evaluate
(i)

(ii)

(b) Show that

3
4). The function f is defined by f(x) = x  . Evaluate f (-3)
x
x
5). Functions f and g are defined by f: x  7x+1 and g: x   1.
3

Find: (i) f 1 (x) (ii) g 1 (x)

(iii)Verify that

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6). Starting with the set

A = {2, 4, 7, 9, 11, 12} as the domain find the range for each of the following
functions.
a) f(x) = 3x – 2
b) g(x) = 2x2 + 1

7). State the inverse of the following functions:

a) f (x) = 3 − 2x
b) g(x) = 1/1 − x
c) h(x) = 3x/2 − 2

8). Using as many different sets of real numbers as domains give examples of the
following:
a) A mapping which is not a function
b) A mapping which is a function
c) A function whose inverse is not a function
d) A function whose inverse is also a function
Demonstrate each example on a mapping diagram. If you are in a group each
member should come up with an example of his or her own for each of the cases
above.
NB: The icons will be put later in consultation with ICT.

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TOPIC 7 TITLE: DIFFERENTIATION OF SIMPLE FUNCTIONS: GRADIENT

TO A CURVE, TURNING POINTS AND CURVE KETCHING.

LECTURE ONE
7.1 INTRODUCTION
Calculus is a branch of mathematics involving or leading to calculations dealing with
continuously varying functions. Calculus is a subject that falls into two parts:
(i) differential calculus (or differentiation) and (ii) integral calculus (or integration).

7.2 LECTURE OBJECTIVES

 geometrically interpret differentiation
 Perform differentiation from first principles
 Obtain derivatives of various functions using the rules of differentiation
 Determine the maximum and minimum values of functions
 Sketch the curves of various functions

7.3.1 SUBTOPIC 1: THE GRADIENT OF A CURVE

If a tangent is drawn at a point P on a curve, then the gradient of this tangent is said to be
the gradient of the curve at P. In Figure 7, the gradient of the curve at P is equal to the
gradient of the tangent PQ

Figure 7
For the curve shown in Figure 8, let the points A and B have co-ordinates (x1, y1) and
(x2, y2), respectively. In functional notation, y1= f(x1) and y2 = f(x2) as shown

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The gradient of the chord AB

Figure 8
The gradient of the chord AB is given by

BC BD  CA f ( x2 )  f ( x1 )
 
AC ED x2  x1

Differentiation from first principles

In Figure 9, A and B are two points very close together on a curve, δx (delta x) and δy delta
y) representing small increments in the x and y directions, respectively

Figure 9
y
Gradient of chord AB  , however  y  f ( x   x)  f ( x) , hence
x

 y f ( x   x)  f ( x)

x x

dy
As δx approaches zero, , approaches a limiting value and the gradient of the chord
dx
approaches the gradient of the tangent at A.

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When determining the gradient of a tangent to a curve there are two notations used. The
 f ( x   x)  f ( x) 
gradient of the curve at A in figure 9 can either be written as: lim  
 x 0
 x 
dy  f ( x   x)  f ( x) 
In functional notation  lim  
dx  x0  x 

dy
is the same as f ( x) and is called the differential coefficient or the derivative. The
dx
process of finding the differential coefficient is called differentiation

dy y  f ( x   x)  f ( x) 
Summarizing, the differential coefficient,  f ( x)  lim  lim  
dx  x  0 x  x  0
 x 

Example. Differentiate from first principles f(x) = 2x and determine the value of the
gradient of the curve at x = 2

Soln.
dy y  f ( x   x)  f ( x) 
 f ( x)  lim  lim  
 x 0  x x
dx  x 0
 

dy y  2( x  x)  2 x 
  f ( x)  lim  lim  
 x 0  x x
dx  x 0
 

dy y  2x  dy
 f ( x)  lim  lim     2 . The gradient is constant.
 x 0  x  x 0  x
dx   dx

1
Example. Differentiate from first principles f ( x)  and determine the value of the
x
gradient of the curve at x = 2

Soln.
dy y  f ( x   x)  f ( x) 
 f ( x)  lim  lim  ,
dx  x  0 x  x  0
 x 

dy y  1 1 1
  f ( x)  lim  lim    ,
 x 0  x  x 0 x   x x x
dx 

dy y  x  x  x  1
 f ( x)  lim  lim  
dx  x  0 x  x  0
 x( x   x)   x

dy  x  1 dy  1  x dy  1 
  lim   ,  lim   ,  
dx  x0  x( x   x)   x dx  x0  x( x   x)   x dx  x 2 
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7.3.2. SUBTOPIC 2: RULES OF DIFFERENTIATION

1. The constant rule

dy
If y  c ,   0.
dx
Example
dy
Given that f ( x)  y  5 ,  0
dx
2. The power rule
dy
If y  ax n , then  anx n 1
dx
Example
dy
Given that y  3x 2 , then  3  2 x 21  6 x1
dx
3. The product rule

dy
If y  f ( x)  g ( x) , then  f ( x) g ( x)  g ( x) f ( x)
dx

Example : Given that y  3x 2 ( x3  1) , then

dy
 6 x( x3  1)  3x 2 (3x 2 )  6 x 4  6 x  9 x 4  15 x 4  6 x
dx

f ( x) dy f ( x) g ( x)  g ( x) f ( x)
4. If y  , then 
g ( x) dx ( g ( x))2
Example
3x  5 dy 3( x)  1(3x  5) 5
Given that y  , then   2
x dx x2 x

5. Chain rule
If y  g ( f ( x)) , let u  f ( x) ,  u  g (u) ,

dy dy du
 
dx du dx

Example

Given that y  (2 x 4  7)6 , let u  2 x4  7  y  u 6

du dy dy du
 8 x3     6u 5  8 x3  48 x3 (2 x 4  7)5
dx dx du dx

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7.3.3. SUBTOPIC 3: APPLICATION OF DIFFERENTIATION:

TURNING POINTS AND CURVE KETCHING
In general terms, the usefulness of calculus lies in the application of differentiation as a
rate of change or slope of a curve and integration as an area under a curve. These are
essentially applications to very useful mathematical methods. The applications to
mathematical methods are the main subject of this sub topic.
Tangent and normal to a curve
The derivative at a point (a, b) on a curve y = f (x) will give us the slope, m, of the
tangent to the curve at that point. This tangent is a straight line with gradient m passing
through the point (a, b) and therefore has an equation of y − b = m(x − a)

The normal to the curve y = f (x) at the point (a, b) is the line through (a, b)
perpendicular to the tangent – see Figure 10.

Figure 10

In mechanics, the normal is important because it defines the direction of the reaction to
a force applied to a smooth surface represented by the curve. If the gradient of y = f (x)
1
at (a, b) is m then the gradient of the normal through (a, b) will be
m
1
So the equation of the normal is y  b  ( x  a)
m
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dy
For the curve y = x2 − x − 1 we have = 2x − 1. So at (2, 1), the gradient is m = 2 ×
dx
2 − 1 = 3.
The equation of the tangent at (2, 1) is therefore y − 1 = 3(x − 2) or y = 3x – 5

1 1
The gradient of the normal at (2, 1) is  , so the equation of the normal is
m 3
1
y−1= ( x  1) , or x + 3 y − 5 = 0
3
Stationary points and points of inflection ➤ 291 310 ➤
Since the derivative describes rate of change, or the slope of a curve, it can tell us a
great deal about the shape of a curve, and the corresponding behaviour of the function.
Figure 11 shows the range of possibilities one can meet.

Figure 11
In all cases we assume that the function is continuous and smooth (the graph has no
dy
breaks and no sharp points). At points A, C, E, where is zero, y is not actually
dx
changing at all as x varies – the tangent to the curve is then parallel to the x-axis at
such points as shown. These are called stationary points. So, at a stationary point of
dy
the function y = f (x) we have  f ( x)  0
dx
The value of f (x) at a stationary point is called a stationary value of f (x). There are a
number of different types of stationary points illustrated in Figure 11, displaying the
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different possibilities – C is a minimum point, and A is a maximum. E is a stationary

point where the tangent crosses the curve – this is an example of an important point
called a point of inflection. B and D are also points of inflection of a different kind –
the tangent crosses the curve, but is not horizontal. The maximum and minimum points
are called turning points, since the curve turns at such points and changes direction. A
minimum point identified by differentiation is only a local minimum, not necessarily an
overall global minimum of the function. Similarly for a maximum. Indeed, for a
1
function such as y  x  the local minimum actually has a higher value than the
x
local maximum.
While the graphical representation of the behaviour of functions is very suggestive,
we need a means of distinguishing stationary points and points of inflection that
depends only on the derivatives of the function in question. The graphical form
provides a hint as to how this might work. As an example, consider a minimum point at
say x = x0, see figure 12

Figure 12
To the left of x0, i.e. for x < x0, y = f (x) is decreasing as x increases, and f ( x) <0.
For x > x0, y increases as x increases and so f ( x) >0. And of course at x = x0 we have
f ( x) = f(x0) = 0. So, near to a minimum point we can summarize the situation as in
table 2.

Table 2

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This characterization no longer relies on the graphical representation – it depends only

on the values of the derivative at different points. Rather than use this table as a means
of verifying a minimum point, we look at the implications it has for the second
derivative of the function. In this case we see that as x passes through x0,f(x) is steadily
d  dy  d 2 y
increasing. That is    2  f ( x)  0 , So a minimum point x = x0 on the
dx  dx  dx
curve y = f (x) is characterized by f ( x0 ) = 0 and f ( x0 )  0 , Similarly, for a
maximum point we obtain table 3.

Table 3
and so in this case the derivative is decreasing as we increase x through x = x0 and thus
a maximum point x = x0 is characterized by f ( x0 ) = 0 and f ( x0 )  0

Now what happens if f ( x0 ) = 0?. It is not clear whether or not the gradient, f ( x0 ) is
changing and a more careful examination is necessary. An example of one
possibility occurs at the point E in Figure 13, at which point f ( x0 )  f ( x0 )  0

This is a particular example of a point of inflection. B and D are other examples, but
these are clearly not stationary points – the gradient is not zero. In these cases f ( x0 ) =
0 but f ( x0 )  0 . So what precisely characterizes a point of inflection?
Look more closely at the point B in Figure 13. To emphasize the point we will bend
the curve somewhat near to B. Take x = x0 as the coordinate of B here.

Figure 13

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On either side of B the gradient is negative. However, on the left of B, x < x0, the curve is
below the tangent. We say the curve here is concave down. To the right of B, x > x0, the
curve is above the tangent and we say it is concave up. Looking at points D and E in
figure 11we see a similar change:

left of D – concave up
right of D – concave down
left of E – concave down
right of E – concave up

In general, any point where there is such a change of sense of concavity is called a point
of inflection, meaning a change in the direction of bending of the curve. At such a point
you will see that it is the rate of change of the gradient that is changing sign. For
example: left of B, gradient steepens as x increases – curve concave downright of B,
gradient decreases as x increases – curve concave up. So even though the gradient of f (x)
may not be zero at a point of inflection – i.e. we may not have a stationary value – the rate
of change of the gradient, f ( x0 ) , at such a point must be zero. However, note that the
condition f ( x0 ) = 0 does not itself guarantee a point of inflection.

Example

(i) For the function f (x ) = 16 x − 3 x3 we have f(x) = 16 − 9x2. Solving

4
f(x) = 16 − 9x2 = 0 gives stationary values at x =  . To classify these consider
3
4 4
the second derivative f  (x)= −18x. So, for x =  , f  (  )= −24 <0
3 3
and we have a maximum.

4 4
For x = − we have f  (- )= 24 >0 and we have a minimum.
3 3

We also note that f  (x) = 0 at x = 0, so there is a possibility of a point of

inflection at this point. To investigate this we have to consider how the gradient
is changing on either side of x = 0. This is indicated by the sign of f  (x):

x <0, f  (x) = −18x >0

x >0, f  (x)= −18x <0

So for x <0, f  (x) = f ( x ) is positive and therefore the gradient of f(x) is

increasing as x increases and the curve is concave upwards. For x >0, f  (x) =
f ( x ) is negative, so the gradient f ( x ) is decreasing and x increases and the

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curve is concave downwards. Thus x = 0 is a point where the curve changes

from concave up to concave down as x increases – it is a point of inflection.

The curve is sketched in Figure 14

Figure 14

Curve sketching in Cartesian coordinates ➤291 310 ➤

Curve sketching is exactly that – it does not mean plotting the curve, although one might
actually plot a few special points of the curve, such as where it intercepts the axes. A sketch
of a curve shows its general shape and main features, it is not necessarily an accurate
drawing. In sketching a curve we deduce what we can about it from general observations,
such as where it increases, decreases, or remains bounded as x becomes large or small.
Similarly we might look for stationary values, intercepts on the axes and so on. The main
benefit of the skills of curve sketching is that it gives you an appreciation of how functions
behave, and what the properties of the derivative can tell you about this.
Of course, we already know a large number of different graphs of the various ‗elementary
functions‘ we have considered, and the first step in sketching a curve is to see whether it
is easily converted to one with which we are already familiar. The kinds of transformation
are listed below.
1. y = f (x) + c translates the graph of y = f (x) by c units in the direction 0y.
2. y = f (x + c) translates the graph of y = f (x) by -c units in the 0x direction.
3. y = −f (x) reflects the graph of y = f (x) in the x-axis.
4. y = f (-x) reflects the graph of y = f (x) in the y-axis.
5. y = af (x) stretches y = f (x) parallel to the y-axis by a factor a.
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6. y = f (ax) stretches y = f (x) parallel to the x-axis by a factor 1/a.

However, once these transformations are taken advantage of, we might still have quite
complicated graphs to sketch. We can follow a systematic procedure for sketching graphs
which can be nicely summarized in the acronym S(ketch) GRAPH:
• Symmetry – does the curve have symmetry about the x- or y-axes, is the function odd or
even?
• Gateways – where does the curve cross the axes?
• Restrictions – are there any limits on the variable or function values?
• Asymptotes – any lines that the curve approaches as it goes to infinity?
• Points – any points of special interest which are worth plotting?
• Humps and hollows – any stationary points or points of inflection?

Working through each of these (not necessarily in this order) should provide a good idea of
the shape and major features of the curve. We describe each in turn. S. We look for any
symmetry of the curve. For example, if it is an even function, f (x) = f (-x) then it is
symmetric about the y-axis, and we only need to sketch half the curve. This may reduce
considerably the amount of work we have to do in sketching the curve.

G. Look for points where the curve crosses the axes, or ‘gateways’. It crosses the y-axis
at points where x = 0, i.e. at the value(s) y = f (0). It crosses the x-axis at points where y = 0,
i.e. at solutions of the equation f (x) = 0.

R. Consider any restrictions on either the domain or the range of the function – that is,
forbidden regions where the curve cannot exist. The simplest such case occurs when we
have discontinuities. Thus, for example

1
y
x 1

does not exist at the point x = −1 i.e. there is a break in the curve at this point. See
Figure 15

Figure 15
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Any rational function will have a number of such points equal to the number of real
roots of the denominator. Irrational functions such as √1 − x2 exhibit whole sets of
values that x can‘t take, because for the square root to exist 1 − x2 must be positive,
hence the curve y = √1 − x2 does not exist for | x| >1. Thus, the curve is confined
to the region | x| ≤ 1.

A. Look for asymptotes. These are lines to which the curve becomes infinitely close as
1
x or y tend to infinity. Thus, the function y  has the x-axis as a horizontal
x 1
asymptote, since y → 0 as x → ±∞. It also has the line x = −1 as an asymptote to which the
curve tends as y → ±∞. Of course the curve can never coincide with the line because of the
restriction.

Also, look for the behaviour of the curve for very small and very large values of x or
y. This is often very instructive. For example, the curve y = x3 + x behaves like the curve y
= x for small values of x, allowing us to approximate the curve near the origin by a straight
line at 45° to the x-axis. For very large (positive or negative) values of x it behaves like the
cubic curve y = x3.

P. Consider any special points, apart from ‗gateways‘ in the axes. They may literally
be simply specific points you plot to determine which side of an asymptote the curve
approaches from, for example points of particular importance include maximum and
minimum values (humps and hollows) and also points of inflection. While we do already
know how to find these, it may not in fact be necessary. Information already available may
hint strongly at certain turning values – for example a continuous curve which ends up
going in the same direction for two different values of x must have passed through at least
one turning value in between. Also, it is not always necessary to determine points of
inflection, especially those with a non-zero gradient, unless we want a really accurate
picture of the graph.

Exercise
Sketch the graph for each of the following function: assuming the domain for
each one of them is the whole set R of real numbers.
a). y = 2 x − 3
b. ) y = 4x2 − 12x
c). y = x3 − 3 x + 1

7.4. TOPIC SUMMARY

The derivative of any given function is given by
dy y  2( x  x)  2 x 
  f ( x)  lim  lim   . We can follow a systematic procedure
 x 0  x x
dx  x 0
 
for sketching graphs which can be nicely summarized in the acronym S(ketch) GRAPH:
• Symmetry – does the curve have symmetry about the x- or y-axes
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• Gateways – where does the curve cross the axes?

• Restrictions – are there any limits on the variable or function values?
• Asymptotes – any lines that the curve approaches as it goes to infinity?
• Points – any points of special interest which are worth plotting?
• Humps and hollows – any stationary points or points of inflection?

7.5. NOTE
Points of particular importance include maximum and minimum values (humps and
hollows) and also points of inflection. A continuous curve which ends up going in the
same direction for two different values of x must have passed through at least one turning
value in between. Also, it is not always necessary to determine points of inflection,
especially those with a non-zero gradient, unless we want a really accurate picture of the
graph.

7.6 TOPIC ACTIVITIES

The student to determine the higher derivatives in questions 1 and 2 below.

7.7 FURTHER READING

References:

 Thomas, G. B. and Finney, R. L. (1998). Calculus and Analytic Geometry; Narosa

Publishing House, 6th Edition.
 Swokowski, E. W.: (1983). Calculus with Analytic Geometry, Alternate Edition,
PWS Publishers.
 Larry, J. G, et-al1993). Calculus and its Applications; Prentice-Hall International
Press, London.

 Wolfram MathWorld
http://mathworld.wolfram.com/calculus.html
• Read this entry for differentiation.
• Follow links to explain specific concepts as you need to.

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 Wikipedia
http://www.wikipedia.org/
• Follow links to explain specific concepts as you need to.

7.8 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.

7.9.1 Assessments and Solutions

(A). Differentiate the following functions using the first principles.

(B). Differentiate the functions below using the rules of differentiation

(C ). Sketch the graphs of the following functions

1. y = x4
1
2. y  x 
x

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Solutions to Assessments 7.9.1

A. (1). 1 (2). 7, ( 3). -8x. (4). 15x2. (5). -4x+3. (6). 0. (7). -9. (8). -2/3.

(9). -18x, (10). -21x2. (11). -2x+5. (12). 0. (13). 12x2. (14). 6x.

3
1
1 3 2 1 1
B. (1). 28x3 (2). (3). t . (4). . (5). 3  
1
2 x4 3 3
2x 2
2x 2
x 2

10 7
(6). 3
 9 . 7. 6(t  2) . (8). 6(t  2) .
x
2x 2
C.
1. y = x4

1
2. y  x 
x

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7.9.2 Further Assessments

NB: The icons will be put later in consultation with ICT.

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TOPIC 8 TITLE: STATISTICS: RAW DATA ORGANIZATION, MEASURES OF

CENTRAL TENDENCY AND MEASURES OF DISPERSION.

LECTURE ONE
8.1 INTRODUCTION
STATISTICS
Statistics is a discipline of study dealing with the collection, analysis, interpretation, and
presentation of data. Pollsters who sample our opinions concerning topics ranging from arts
to zoology utilize statistical methodology. Statistical methodology is also utilized by
business and industry to help control the quality of goods and services that they produce.
Social scientists and psychologists use statistical methodology to study our behaviors.

DESCRIPTIVE STATISTICS
The use of graphs, charts, and tables and the calculation of various statistical measures to
organize and summarize information is called descriptive statistics. Descriptive statistics
help to reduce our information to a manageable size and put it into focus

INFERENTIAL STATISTICS: POPULATION AND SAMPLE

The population is the complete collection of individuals, items, or data under consideration
in a statistical study. The portion of the population selected for analysis is called the
sample. Inferential statistics consists of techniques for reaching conclusions about a
population based upon information contained in a sample.

8.2 LECTURE OBJECTIVES

 Explain the basic terms in statistics.

 Summarize and present data in various forms
 Work out the measures of central tendency and measures of dispersion.

NOMINAL, ORDINAL, INTERVAL, AND RATIO LEVELS OF MEASUREMENT

Data can be classified into four levels of measurement or scales of measurements.

The nominal scale applies to data that are used for category identification. The nominal
level of measurement is characterized by data that consist of names, labels, or categories
only. Nominal scale data cannot be arranged in an ordering scheme. The arithmetic
operations of addition, subtraction, multiplication, and division are not performed for
nominal data. Table 4 gives examples of nominal scale.

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Table 4

The ordinal scale applies to data that can be arranged in some order, but differences
between data values either cannot be determined or are meaningless. The ordinal level of
measurement is characterized by data that applies to categories that can be ranked. Ordinal
scale data can be arranged in an ordering scheme. Table 5 gives examples of the ordinal
scale.

Table 5

The interval scale applies to data that can be arranged in some order and for which
differences in data values are meaningful. The interval level of measurement results from
counting or measuring. Interval scale data can be arranged in an ordering scheme and
differences can be calculated and interpreted. The value zero is arbitrarily chosen for
interval data and does not imply an absence of the characteristic being measured. Ratios are
not meaningful for interval data.

EXAMPLE Temperatures represent interval level data. The high temperature on February
1 equaled 25ºF and the high temperature on March 1 equaled 50ºF. It was warmer on
March 1 than it was on February 1. That is, temperatures can be arranged in order. It was
25º warmer on March 1 than on February 1. That is, differences may be calculated and
interpreted. We cannot conclude that it was twice as warm on March 1 than it was on
February 1. That is, ratios are not readily interpretable. A temperature of 0ºF does not
indicate an absence of warmth.

EXAMPLE Test scores represent interval level data. Lana scored 80 on a test and
Christine scored 40 on a test. Lana scored higher than Christine did on the test; that is, the
test scores can be arranged in order. Lana scored 40 points higher than Christine did on the
test; that is, differences can be calculated and interpreted. We cannot conclude that Lana
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knows twice as much as Christine about the subject matter. A test score of 0 does not
indicate an absence of knowledge concerning the subject matter.

The ratio scale applies to data that can be ranked and for which all arithmetic operations
including division can be performed. Division by zero is, of course, excluded. The ratio
level of measurement results from counting or measuring. Ratio scale data can be arranged
in an ordering scheme and differences and ratios can be calculated and interpreted. Ratio
level data has an absolute zero and a value of zero indicates a complete absence of the
characteristic of interest.

EXAMPLE The grams of fat consumed per day for adults in the United States is ratio
scale data. Joe consumes 50 grams of fat per day and John consumes 25 grams per day. Joe
consumes twice as much fat as John per day, since 50/25 = 2. For an individual who
consumes 0 grams of fat on a given day, there is a complete absence of fat consumed on
that day. Notice that a ratio is interpretable and an absolute zero exists.

Summation Notation
Many of the statistical measures discussed in the following chapters involve sums of
various types. Suppose the number of 911 emergency calls received on four days were 411,
375, 400, and 478. If we let x represent the number of calls received per day, then the
values of the variable for the four days are represented as follows: x1= 411, x2 = 375, x3 =
400, and x4 = 478. The sum of calls for the four days is represented as x1 + x2 + x3 + x4,
which equals 411 + 375 + 400 + 478 or 1664. The symbol ΣX, read as ―the summation of
x,‖ is used to represent x1 + x2 + x3 + x4. The uppercase Greek letter Σ (pronounced sigma)
corresponds to the English letter S and stands for the phrase ―the sum of.‖ Using the
summation notation, the total number of 911 calls for the four days would be written as
ΣX = 1664.

EXAMPLE The following five values were observed for the variable X: x1 = 4, x2 = 5, x3
= 0, x4 = 6, and x5 = 10. The following computations illustrate the usage of the summation
notation.
ΣX = x1 + x2 + x3 + x4 + x5 = 4 + 5 + 0 + 6 + 10 = 25

8.3.1 SUBTOPIC 1: DATA PRESENTATION

Raw data
Raw data is information obtained by observing values of a variable. Data obtained by
observing values of a qualitative variable are referred to as qualitative data. Data obtained
by observing values of a quantitative variable are referred to as quantitative data.
Quantitative data obtained from a discrete variable are also referred to as discrete data and
quantitative data obtained from a continuous variable are called continuous data.
Data are obtained largely by two methods:
(a) By counting — for example, the number of stamps sold by a post office in equal periods
of time, and
(b)By measurement — for example, the heights of a group of people.
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As noted above, when data are obtained by counting and only whole numbers are possible,
the data are called discrete whereas measured data can have any value within certain limits
and are called continuous .
A set is a group of data and an individual value within the set is called a member of the
set. Thus, if the masses of five people are measured correct to the nearest 0.1 kilogram and
are found to be 53.1 kg, 59.4 kg, 62.1 kg, 77.8 kg and 64.4 kg, then the set of masses in
kilograms for these five people is: {53.1, 59.4, 62.1, 77.8, 64.4} and one of the members of
the set is 59.4
A set containing all the members is called a population. Some member selected at random
from a population are called a sample. Thus all car registration numbers form a population,
but the registration numbers of, say, 20 cars taken at random throughout the country
are a sample drawn from that population. The number of times that the value of a member
occurs in a set is called the frequency of that member. Thus in the set: {2, 3, 4, 5, 4, 2, 4, 7,
9}, member 4 has a frequency of three, member 2 has a frequency of 2 and the other
members have a frequency of one.
The relative frequency with which any member of a set occurs is given by the ratio:
frequency of member total frequency of all members For the set: {2, 3, 5, 4, 7, 5, 6, 2, 8},
the relative frequency of member 5 is 29 . Often, relative frequency is expressed as a
percentage and the percentage relative frequency is: (relative frequency × 100)%.

Frequency distribution table

A frequency distribution table is at a table showing the various classes and their
frequencies.
EXAMPLE 2.7 Group the following weights into the classes 100 to under 125, 125 to
under 150, and so forth:
111 120 127 129 130 145 145 150 153 155 160
161 165 167 170 171 174 175 177 179 180 180
185 185 190 195 195 201 210 220 224 225 230
245 248
The weights 111 and 120 are tallied into the class 100 to under 125. The weights 127, 129,
130, 145, and 145are tallied into the class 125 to under 150 and so forth until the
frequencies for all classes are found. The frequency distribution for these weights is given
in Table 6.

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Table 6
When forming a frequency distribution, the following general guidelines should be
followed:
1. The number of classes should be between 5 and 15
2. Each data value must belong to one, and only one, class.
3. When possible, all classes should be of equal width.

Class limits, Class boundaries, Class marks, and Class width

The frequency distribution given in table 7 is composed of five classes. The classes are:
80–94, 95–109, 110–124, 125–139, and 140–154. Each class has a lower class limit and an
upper class limit. The lower class limits for this distribution are 80, 95, 110, 125, and 140.
The upper class limits are 94, 109, 124, 139, and 154.

If the lower class limit for the second class, 95, is added to the upper class limit for the first
class,94, and the sum divided by 2, the upper boundary for the first class and the lower
boundary for the second class are determined.

Table 7 gives a frequency distribution of the Stanford−Benet intelligence test scores for 75
adults

Table 7

Table 8 gives all the boundaries for Table 7.If the lower class limit is added to the upper
class limit for any class and the sum divided by 2,the class mark for that class is obtained.
The class mark for a class is the midpoint of the class and is sometimes called the class
midpoint rather than the class mark. The class marks for Table 7 are shown in Table 8. The
difference between the boundaries for any class gives the class width for a distribution. The
class width for the distribution in Table 7 is 15.

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Table 8

For continuous data, the class limits and the class boundaries are the same.

DATA PRESENTATION

BAR GRAPH
A bar graph is a graph composed of bars whose heights are the frequencies of the different
categories. A bar graph displays graphically the same information concerning qualitative or
quantitative data that a frequency distribution shows in tabular form.

EXAMPLE 1 The distribution of the primary sites for cancer is given in Table 9 for the
residents of Dalton County.

Table 9

To construct a bar graph, the categories are placed along the horizontal axis and
frequencies are marked along the vertical axis. A bar is drawn for each category such that
the height of the bar is equal to the frequency for that category. A small gap is left between
the bars (figure 16). Bar graphs can also be constructed by placing the categories along the
vertical axis and the frequencies along the horizontal axis.

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Figure 16

PIE CHART
A pie chart is also used to graphically display qualitative and quantitative data. To
construct a pie chart, a circle is divided into portions that represent the relative frequencies
or percentages belonging to different categories.

EXAMPLE 2: To construct a pie chart for the frequency distribution in table 9 above.
construct a table that gives angle sizes for each category. The table below shows the
determination of the angle sizes for each of the categories. The 360º in a circle are divided
into portions that are proportional to the category sizes. The pie chart for the frequency
distribution is shown in figure 17

Table 9

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Figure 17
HISTOGRAMS
A histogram is a graph that displays the classes on the horizontal axis and the frequencies
of the classes on the vertical axis. The frequency of each class is represented by a vertical
bar whose height is equal to the frequency of the class. A histogram is similar to a bar
graph. However, a histogram utilizes classes or intervals and frequencies while a bar graph
utilizes categories and frequencies.
EXAMPLE 3. A frequency distribution for aspirin prices is shown in table 10

.
Table 10

Histogram for the aspirin prices is shown in figure 18

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Figure 18

CUMULATIVE FREQUENCY DISTRIBUTIONS

A cumulative frequency distribution gives the total number of values that fall below
various class boundaries of a frequency distribution. The cumulative frequency
distribution for table 11 is shown in table 12.

Table 11

Table 12
OGIVES
An ogive is a graph in which a point is plotted above each class boundary at a height equal
to the cumulative frequency corresponding to that boundary. Ogives can also be
constructed for a cumulative relative frequency distribution as well as a cumulative
percentage distribution.
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EXAMPLE 4 The ogive corresponding to the cumulative frequency distribution in

Table 11 is shown in Figure 19.

Figure 19

8.3.2 MEASURES OF CENTRAL TENDENCY.

TYPES OF DATA
Often times, we are interested in a typical numerical value to help us describe a data set.
This typical value is often called an average value or a measure of central tendency. We are
looking for a single number that is in some sense representative of the complete data set.
Measures of central tendency are mean, mode and median.

MEAN, MEDIAN, AND MODE FOR UNGROUPED DATA

A data set consisting of the observations for some variable is referred to as raw data or
ungrouped data. Data presented in the form of a frequency distribution are called grouped
data. There are many different measures of central tendency. The three most widely used
measures of central tendency are the mean, median, and mode.

The mean for a sample consisting of n observations is

EXAMPLE 1. The number of 911 emergency calls classified as domestic disturbance

calls in a large metropolitan location were sampled for thirty randomly selected 24-hour
periods with the following results. Find the mean number of calls per 24-hour period.

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25 46 34 45 37 36 40 30 29 37 44 56
50 47 23 40 30 27 38 47 58 22 29 56
40 46 38 19 49 50

The median of a set of data is a value that divides the bottom 50% of the data from the top
50%of the data. To find the median of a data set, first arrange the data in increasing order.
If the number of observations is odd, the median is the number in the middle of the ordered
list. If the number of observations is even, the median is the mean of the two values closest
to the middle of the ordered list.

EXAMPLE 2. To find the median number of domestic disturbance calls per 24-hour
period for the data in example 1 first arrange the data in increasing order.

19 22 23 25 27 29 29 30 30 34 36 37
37 38 38 40 40 40 44 45 46 46 47 47
49 50 50 56 56 58

The two values closest to the middle are 38 and 40. The median is the mean of these two
values or 39

The mode is the value in a data set that occurs the most often. If no such value exists, we
say that the data set has no mode. If two such values exist, we say the data set is bimodal. If
three such values exist, we say the data set is tri-modal. There is no symbol that is used to
represent the mode.

EXAMPLE 3 Find the mode for the data given in example 1. Often it is helpful to arrange
the data in increasing order when finding the mode. The data, in increasing order is shown
below.

19 22 23 25 27 29 29 30 30 34 36 37
37 38 38 40 40 40 44 45 46 46 47 47
49 50 50 56 56 58

When the data are examined, it is seen that 40 occurs three times, and that no other value
occurs that often. The mode is equal to 40.

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GROUPED DATA

Statistical data are often given in grouped form, i.e., in the form of a frequency distribution,
and in most cases the raw data corresponding to the grouped data are not available or may
be difficult to obtain.

Example

Consider the following data relating to the weights of 100 students as shown in table 13.

Classes 0-9 10-19 20- 29 30- 39 40-49 50-59

Frequency 5 15 25 30 15 10

Table 13

To obtain the mean, median and mode, table 13 is modified as shown in table 14.

Classes Xi Frequency fiX i CF

(mid f
point)
0-9 4.5 5 22.5 5
10 – 19 14.4 15 217.5 20
20 – 29 24.5 25 612.5 45
30 – 39 34.5 30 1035 75
40 - 49 44.5 15 667.5 90
50 - 59 54.5 10 545 100
6 6


i 1
f =  fiX
i 1
i =

100 3100

Table 14

The mean of grouped data is given by the formulae

 fiX i
3100
X  
100
f i

The median for grouped data is given by

 TCF 
 2  CF
i

L i
 f 
 

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 100 
 2  45 
Median 29.5   10  31.7
 30 
 

The mode for grouped data is given by =

 f  fi   30  25 
L  i  29.5   10  32
 2 f  f   fi   30  25  15 

8.3.3 MEASURES OF POSITION: PERCENTILES, DECILES, AND QUARTILES

Measures of position are used to describe the location of a particular observation in relation
to the rest of the data set. Percentiles are values that divide the ranked data set into 100
equal parts. The pth percentile of a data set is a value such that at least p percent of the
observations take on this value or less and at least (100 – p) percent of the observations
take on this value or more. Deciles are values that divide the ranked data set into 10 equal
parts. Quartiles are values that divide the ranked data set into four equal parts. The 2nd
quartile is the same as the median which is equal to the 50th percentile. The techniques for
finding the various measures of position will be illustrated by using the data in table 13.

Percentiles are usually given by the formulae

 p i 
 100 TCF  CF 
L i
 f 
 

 70  45 
Using table 13, the 70th percentile is given by  29.5   10  37.83
 30 

Deciles are given by the formulae

 d 
 10 TCF  CF
i

L i
 f 
 

 30  20 
Using table 13 the 3rd decile is given by  19.5   10  23
 25 

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The 2nd quartile is given by the formulae

 TCF 
 2  CF
i

L i
 f 
 

 100 
 2  45 
Q2 = 29.5   10  31.7
 30 
 

8.3.4 MEASURES OF DISPERSION

In addition to measures of central tendency, it is desirable to have numerical values to
describe the spread or dispersion of a data set. Measures that describe the spread of a data
set are called measures of dispersion.

RANGE, VARIANCE, AND STANDARD DEVIATION FOR UNGROUPED DATA

The range for a data set is equal to the maximum value in the data set minus the minimum
value in the data set. It is clear that the range is reflective of the spread in the data set since
the difference between the largest and the smallest value is directly related to the spread in
the data. The variance and the standard deviation of a data set measures the spread of the
data about the mean of the data set. The variance of a sample of size n is represented by s2.

Using the same data in table 13, the measures of dispersion are discussed below.

The variance and standard deviation are discussed using table 14.

Classes Xi Frequency fiX i ( X i  X )2 fi( X i  X )2 CF

(mid f
point)
0-9 4.5 5 22.5 702.25 3511.25 5
10 – 19 14.4 15 217.5 272.25 4083.75 20
20 – 29 24.5 25 612.5 42.5 1056.25 45
30 – 39 34.5 30 1035 12.25 367.5 75
40 - 49 44.5 15 667.5 182.25 2733.75 90
50 - 59 54.5 10 545 552.25 5522.5 100
6 6 n

f =
i 1
 fiX
i 1
i =  fi( X
i 1
i  X )2

100 3100 =17275

Table 14

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 fiX i
3100
Mean X  
100
f i

Variance is given by the formulae

 fi( X i  X )2
Variance =
f
17275
Using the data in table 14, variance =  172.75
100

Standard deviation is the square root of the variance

 fi( X i  X )2
S=  172.75  13.14
f

8.4 TOPIC SUMMARY

Statistics is a discipline of study dealing with the collection, analysis, interpretation, and
presentation of data. Data can be presented in various ways including bar charts, pie
charts, histograms and ogives.

Measures of central tendency help us describe a data set. This typical value is often
called an average value or a measure of central tendency. This involves looking for a single
number that is in some sense representative of the complete data set. Measures of central
tendency are mean, mode and median.

Measures of position are used to describe the location of a particular observation in

relation to the rest of the data set. Percentiles are values that divide the ranked data set into
100 equal parts. Deciles are values that divide the ranked data set into 10 equal parts.
Quartiles are values that divide the ranked data set into four equal parts. The 2nd quartile is
the same as the median which is equal to the 50th percentile.

Measures of dispersion gives numerical values describing the spread or dispersion of a

data set. The range for a data set is equal to the maximum value in the data set minus the

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minimum value in the data set. The variance and the standard deviation of a data set
measures the spread of the data about the mean of the data set.

8.5 NOTE
An ogive curve can also be obtained by joining the co-ordinates of cumulative frequency
(vertically) against upper class boundary (horizontally). Grouped data can also be
presented diagrammatically by using a frequency polygon, which is the graph produced by
plotting frequency against class midpoint values and joining the coordinates with straight
lines.
There are other formulas for calculating the mean and variance as shown below

 fx 2
 ( fx) 2
Variance =
f

 fd
The assumed mean formulae is given by: Mean x  A  -
f
8.6 TOPIC ACTIVITIES
Data are obtained on the topics given below. State whether they are discrete or
continuous data.
(a) The number of days on which rain falls in a month for each month of the year.
(b) The mileage travelled by each of a number of salesmen.
(c) The time that each of a batch of similar batteries lasts.
(d) The amount of money spent by each of several families on food.

8.7 FURTHER READING

References:

 Murray S., Schiller J. and Srinivasan A. (2001). Probability and Statistics, Mc

Graw – Hill.

 Larry J. Stephens, (2009). Beginning statistics, 2nd edition, Mc Graw Hill.

 Wolfram MathWorld
http://mathworld.wolfram.com/statistics.html
• Read this entry for statistics.
• Follow links to explain specific concepts as you need to.

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8.8 SELF ASSESSMENT

The self-assessments exercises below are meant to aid the learners understand the topic and
course content.

8.8.1 Assessments and Solutions

BAR GRAPHS AND PIE CHARTS

1. The subjects in an eating disorders research study were divided into one of three
different groups. The table below gives the frequency distribution for these three
groups.

(a) Construct a bar graph.

Table 1a

(b). Construct a table showing the relative frequency and the size of the angles
for the frequency distribution given in Table 1a.

2. A survey of 500 randomly chosen individuals is conducted. The individuals are

asked to name their favorite sport. The pie chart in Figure 2a summarizes the
results of this survey.

Figure 2a

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(i). How many individuals in the 500 gave baseball as their favorite sport?
(ii) How many gave a sport other than basketball as their favorite sport?
(iii) How many gave hockey or golf as their favorite sport?

HISTOGRAMS

3. The manager of a convenience store records the number of gallons of gasoline

purchased for a sample of customers chosen over a one-week period. Table 3a lists
the raw data. Construct a frequency distribution having five classes, each of width
4. Use 0.000 as the lower limit of the first class.

Table 3a

4. The Food Guide Pyramid divides food into the following six groups: Fats, Oils, and
Sweets Group; Milk, Yogurt, and Cheese Group; Vegetable Group; Bread, Cereal,
Rice, and Pasta Group; Meat, Poultry, Fish, Dry Beans, Eggs, and Nuts Group;
Fruit Group. One question in a nutrition study asked the individuals in the study to
give the number of groups included in their daily meals. The results are given below

6 4 5 4 4 3 4 5 5 5 6
5 4 3 6 6 6 5 2 3 4 5
6 4 5 5 5 6 5 6 5

Give a frequency distribution for these data.

OGIVES
5. Table 4a below gives the cumulative frequency distribution for the daily breast-milk
production in grams for 25 nursing mothers in a research study. Construct an ogive for this
distribution.

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Table 4a

6(a).The data given below refer to the gain of each of a batch of 40 transistors, expressed
correct to the nearest whole number. Form a frequency distribution for these data having
seven classes
81 83 87 74 76 89 82 84
86 76 77 71 86 85 87 88
84 81 80 81 73 89 82 79
81 79 78 80 85 77 84 78
83 79 80 83 82 79 80 77
6(b).Construct a histogram for the data given in the table above.

7.The amount of money earned weekly by 40 people working part-time in a factory, correct
to the nearest £10, is shown below. Form a frequency distribution having 6 classes for these
data.
80 90 70 110 90 160 110 80 140 30 90 50
100 110 60 100 80 90 110 80 100 90 120 70
130 170 80 120 100 110 40 110 50 100 110 90 100
70 110 80
Solutions to Assessments 8.8.1
1.(a)

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1.(b).

2,

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(i) How many individuals in the 500 gave baseball as their favorite sport? = 150
(ii) How many gave a sport other than basketball as their favorite sport? = 400
(ii) How many gave hockey or golf as their favorite sport? = 100
. (i) .3 × 500 = 150 (ii) .8 × 500 = 400 (iii) .2 × 500 = 100

3.

4.

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5.

6(a) The range of the data is the value obtained by taking the value of the smallest member
from that of the largest member. Inspection of the set of data shows that, range = 89 − 71 =
18. The size of each class is given approximately by range divided by the number of
classes. Since 7 classes are required, the size of each class is 18/7, that is, approximately 3.
To achieve seven equal classes spanning a range of values from 71 to 89,the class intervals
are selected as: 70–72, 73–75, and so on.

6(b)
Class class mid-point Frequency
70–72 71 1
73–75 74 2
76–78 77 7
79–81 80 12
82–84 83 9
85–87 86 6
88–90 89 3

The histogram is shown below. The width of the rectangles corresponds to the upper class
boundary values minus the lower class boundary values and the heights of the rectangles
correspond to the class frequencies. The easiest way to draw a histogram is
to mark the class mid-point values on the horizontal scale and draw the rectangles
symmetrically about the appropriate class mid-point values and touching one another.

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Inspection of the set given shows that the majority of the members of the set lie between
£80 and £110 and that there are a much smaller number of extreme values ranging from
£30 to £170. If equal class intervals are selected, the frequency distribution obtained does
not give as much information as one with unequal class intervals. Since the majority of
members are between £80 and £100, the class intervals in this range are selected to be
smaller than those outside of this range.

NB: The icons will be put later in consultation with ICT.

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