KENYATTA UNIVERSITY
DIGITAL SCHOOL OF VIRTUAL AND OPEN LEARNING
IN COLLABORATION WITH
SCHOOL OF PURE & APPLIED SCIENCES
DEPARTMENT: MATHEMATICS AND ACTUARIAL SCIENCE
SMA 104: Calculus I
Written by Dr. Bunyasi Magero and Dr. Lydia Njuguna
Vetted by Dr. Kangogo Ruto Moses
Introduction
, Welcome to this module. In your earlier studies of mathematics, you encountered equations of
straight line from which you could read its gradient easily. If the line is not straight, can you
think of what the gradient would be? Are there real life situations where such kind of
mathematics can be applied? To this end, the module will equip you with basic mathematical
skills and notations necessary in understanding such and more advanced concepts.
In this is interactive instructional module, we use both action and collaborative learning styles
that provide you with diverse online learning experiences and effective learning processes. The
main purpose of this module is to equip you with basic mathematical skills which build the
foundation of mathematics. Calculus I (Differential calculus) gives you a framework for
modeling systems in which there is change, and a way to deduce the predictions of such models.
The idea is to study change by studying "instantaneous" change, which implies changes over tiny
intervals of time. Such changes tend to be lots simpler than changes over finite intervals of time.
For more incite in learning of calculus, read the introduction on this link.
In this module, we shall define the concept of limit. The understanding of this concept will aid
you in comprehending the next concept of continuity of functions. Once you can determine
continuity of a function, then you will be set to determine its gradient function through a process
called differentiation. Just like there are varied types of functions, so are the methods of
differentiating. You will be exposed to these method giving examples on how the process is done
and practice questions to ensure you get the skill in depth. The last section of the module will
lead you into appreciating applications of differential calculus. Several such applications are
given.
Calculus I Flow Chat
WEEK TOPIC
WEEK 0: Motivation and prior knowledge
WEEK 1: Introduction to functions
WEEK 2: Composition and inverse functions
WEEK 3: Continuity of functions
WEEK 4: Differentiation from first principles
,WEEK 5: Product rule, quotient rule and chain rule.
WEEK 6: Differentiation of algebraic and exponential functions
WEEK 7: Differentiation of logarithmic, trigonometric functions, hyperbolic and inverse
hyperbolic functions
WEEK 8: Hyperbolic, Inverse hyperbolic, Implicit and Parametric differentiation
WEEK 9: Higher order derivatives, tangents and normals
WEEK 10: Stationery points and Asymptotes
WEEK 11: Curve sketching and Small change
WEEK 12: Rate of change and Kinematics
WEEK 13 & 14 EXAMINATION
Overview of the course
Week 0: Motivation and prior knowledge
This lesson is intended to help you acclimatize to e-learning and to create a community of
learners who will motivate each other during the course. You will be required to introduce
yourself to your lecturer and colleagues online before other academic interactions start. You will
also converse about netiquette and ground rules. Lastly, any issue you may be having regarding
Calculus I: differential calculus will be clarified in preparation to the unit contents.
Week 1: Introduction to functions
In our daily today life, we face changes. This may be a simple change in the weather, water level
in a dam, a slope of a given section of road or change of seasons. These changes affect us and we
respond to them differently. Thus, the saying the only constant in life is change is valid.
This unit is designed to expose you to the understanding of such changes. The study of
differential calculus (SMA104: Calculus I) deals with rates at which quantities change. The
various changes in quantities are described using functions. Thus, this lesson will first make the
function concept concrete to you before delving in differentiation process.
Week 2: Composition and inverse functions
In this lesson, your attention will be drawn to the concepts of function of function(s) and inverse
of functions. We will find definition of the concepts, looking at examples and solving questions
involving these concepts. Later in the lesson, you will be exposed to the concept of limits. This
will prepare you for differentiation process.
, Week 3: Continuity of functions
This lesson will enable you apply the knowledge of limits in demystifying
continuity concept. The definition of continuity is given axiomatically and the
concept concretized by way of examples. Discontinuity is also looked at.
Continuity of a function is important in differentiation process.
Week 4: Differentiation from first principles
In this lesson, you will learn differentiation process. The formal technique for finding the
gradient of a tangent is known as Differentiation from First Principles. By taking two points on
the curve that lie very closely together, the straight line between them will have approximately
the same gradient as the tangent there. Evidently, the closer these points are together, the better
the approximation.
Week 5: Product rule, quotient rule and chain rule
There are several short cut methods of determining derivatives which are depended on the nature
of the function. In this lesson, you will learn that the derivative of a product is not the product of
the derivatives. A similar thing happens with quotients. For composite function (function of
functions), the method applied is called chain rule.
Week 6: Differentiation of algebraic and exponential functions
Since an algebraic function is a function that involves only algebraic operations, like, addition,
subtraction, multiplication, and division, as well as fractional or rational exponents, combination
of the differentiation rules learned in the previous lesson comes in handy here. Using the
linearity of the derivative and the aforementioned rules, we can differentiate any algebraic
function. The derivative of an algebraic function is another algebraic function. In the last section
of the lesson, you will encounter functions whose powers are variable to be differentiated.
Week 7: Differentiation of logarithmic and trigonometric functions
In this lesson, you will learn differentiation of inverse functions of exponential functions called
logarithmic functions where bases and powers are at play. In the last section of this lesson, you
will look at the derivatives of the trigonometric functions; , and
from first principles.
Week 8: Hyperbolic, Inverse hyperbolic, Implicit and Parametric
differentiation
In this lesson, you will learn that hyperbolic functions are analogues of the ordinary
trigonometric functions defined for the hyperbola rather than on the circle. Therefore, from the
definition of trigonometric functions, one can define hyperbolic and inverse hyperbolic
trigonometric function. Differentiation of these function follow a similar path as those of
trigonometric functions. In the last section of the lesson, you will look at functions in which y is
given implicitly by an equal such as x y y 1. In this case, we cannot express y in terms of
4
x. Later on, we look at parametric equations.
DIGITAL SCHOOL OF VIRTUAL AND OPEN LEARNING
IN COLLABORATION WITH
SCHOOL OF PURE & APPLIED SCIENCES
DEPARTMENT: MATHEMATICS AND ACTUARIAL SCIENCE
SMA 104: Calculus I
Written by Dr. Bunyasi Magero and Dr. Lydia Njuguna
Vetted by Dr. Kangogo Ruto Moses
Introduction
, Welcome to this module. In your earlier studies of mathematics, you encountered equations of
straight line from which you could read its gradient easily. If the line is not straight, can you
think of what the gradient would be? Are there real life situations where such kind of
mathematics can be applied? To this end, the module will equip you with basic mathematical
skills and notations necessary in understanding such and more advanced concepts.
In this is interactive instructional module, we use both action and collaborative learning styles
that provide you with diverse online learning experiences and effective learning processes. The
main purpose of this module is to equip you with basic mathematical skills which build the
foundation of mathematics. Calculus I (Differential calculus) gives you a framework for
modeling systems in which there is change, and a way to deduce the predictions of such models.
The idea is to study change by studying "instantaneous" change, which implies changes over tiny
intervals of time. Such changes tend to be lots simpler than changes over finite intervals of time.
For more incite in learning of calculus, read the introduction on this link.
In this module, we shall define the concept of limit. The understanding of this concept will aid
you in comprehending the next concept of continuity of functions. Once you can determine
continuity of a function, then you will be set to determine its gradient function through a process
called differentiation. Just like there are varied types of functions, so are the methods of
differentiating. You will be exposed to these method giving examples on how the process is done
and practice questions to ensure you get the skill in depth. The last section of the module will
lead you into appreciating applications of differential calculus. Several such applications are
given.
Calculus I Flow Chat
WEEK TOPIC
WEEK 0: Motivation and prior knowledge
WEEK 1: Introduction to functions
WEEK 2: Composition and inverse functions
WEEK 3: Continuity of functions
WEEK 4: Differentiation from first principles
,WEEK 5: Product rule, quotient rule and chain rule.
WEEK 6: Differentiation of algebraic and exponential functions
WEEK 7: Differentiation of logarithmic, trigonometric functions, hyperbolic and inverse
hyperbolic functions
WEEK 8: Hyperbolic, Inverse hyperbolic, Implicit and Parametric differentiation
WEEK 9: Higher order derivatives, tangents and normals
WEEK 10: Stationery points and Asymptotes
WEEK 11: Curve sketching and Small change
WEEK 12: Rate of change and Kinematics
WEEK 13 & 14 EXAMINATION
Overview of the course
Week 0: Motivation and prior knowledge
This lesson is intended to help you acclimatize to e-learning and to create a community of
learners who will motivate each other during the course. You will be required to introduce
yourself to your lecturer and colleagues online before other academic interactions start. You will
also converse about netiquette and ground rules. Lastly, any issue you may be having regarding
Calculus I: differential calculus will be clarified in preparation to the unit contents.
Week 1: Introduction to functions
In our daily today life, we face changes. This may be a simple change in the weather, water level
in a dam, a slope of a given section of road or change of seasons. These changes affect us and we
respond to them differently. Thus, the saying the only constant in life is change is valid.
This unit is designed to expose you to the understanding of such changes. The study of
differential calculus (SMA104: Calculus I) deals with rates at which quantities change. The
various changes in quantities are described using functions. Thus, this lesson will first make the
function concept concrete to you before delving in differentiation process.
Week 2: Composition and inverse functions
In this lesson, your attention will be drawn to the concepts of function of function(s) and inverse
of functions. We will find definition of the concepts, looking at examples and solving questions
involving these concepts. Later in the lesson, you will be exposed to the concept of limits. This
will prepare you for differentiation process.
, Week 3: Continuity of functions
This lesson will enable you apply the knowledge of limits in demystifying
continuity concept. The definition of continuity is given axiomatically and the
concept concretized by way of examples. Discontinuity is also looked at.
Continuity of a function is important in differentiation process.
Week 4: Differentiation from first principles
In this lesson, you will learn differentiation process. The formal technique for finding the
gradient of a tangent is known as Differentiation from First Principles. By taking two points on
the curve that lie very closely together, the straight line between them will have approximately
the same gradient as the tangent there. Evidently, the closer these points are together, the better
the approximation.
Week 5: Product rule, quotient rule and chain rule
There are several short cut methods of determining derivatives which are depended on the nature
of the function. In this lesson, you will learn that the derivative of a product is not the product of
the derivatives. A similar thing happens with quotients. For composite function (function of
functions), the method applied is called chain rule.
Week 6: Differentiation of algebraic and exponential functions
Since an algebraic function is a function that involves only algebraic operations, like, addition,
subtraction, multiplication, and division, as well as fractional or rational exponents, combination
of the differentiation rules learned in the previous lesson comes in handy here. Using the
linearity of the derivative and the aforementioned rules, we can differentiate any algebraic
function. The derivative of an algebraic function is another algebraic function. In the last section
of the lesson, you will encounter functions whose powers are variable to be differentiated.
Week 7: Differentiation of logarithmic and trigonometric functions
In this lesson, you will learn differentiation of inverse functions of exponential functions called
logarithmic functions where bases and powers are at play. In the last section of this lesson, you
will look at the derivatives of the trigonometric functions; , and
from first principles.
Week 8: Hyperbolic, Inverse hyperbolic, Implicit and Parametric
differentiation
In this lesson, you will learn that hyperbolic functions are analogues of the ordinary
trigonometric functions defined for the hyperbola rather than on the circle. Therefore, from the
definition of trigonometric functions, one can define hyperbolic and inverse hyperbolic
trigonometric function. Differentiation of these function follow a similar path as those of
trigonometric functions. In the last section of the lesson, you will look at functions in which y is
given implicitly by an equal such as x y y 1. In this case, we cannot express y in terms of
4
x. Later on, we look at parametric equations.
