week 1
elementary function of a single re
variable and their graphs
, History- why calculus?
Calculus is a branch of mathematics concerned with the study of such concepts as the rate of change of o
quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maxim
minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, a
and geometry, it is the basis of that part of mathematics called analysis.
Calculus is widely employed in the physical, biological, and social sciences. It is used, for example, in the
sciences to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of dec
radioactive material. In the biological sciences a problem such as the rate of growth of a colony of bacter
function of time is easily solved using calculus. In the social sciences calculus is widely used in the study o
and probability.
Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, t
slowest, or the least. These maximum or minimum amounts may be described as values for which a certa
change (increase or decrease) is zero. By using calculus it is possible to determine how high a projectile w
finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero. M
principles governing the behavior of physical processes are formulated almost invariably in terms of rates
is also possible, through the insights provided by the methods of calculus, to resolve such problems in log
famous paradoxes posed by the Greek philosopher Zeno. The fundamental concept of calculus, which di
from other branches of mathematics and is the source from which all its theory and applications are deve
theory of limits of functions of variables.
(Source: Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.)
03-May-20 MTH 121- General Mathematics II (Calculus)
, Intended learning outcomes (ILOs)
At the end of this lesson, every students should be able to:
(i) Define the concepts of functions, range, and domain
(i) Diagrammatically represent, Range, Domain relationship using maps.
(ii) Differentiate between one-to-one mapping and many-to-one mapping
(i) State different kinds of functions.
, Functions
The notion of correspondence occurs frequently in everyday life. Some examples are g
the following illustrations:
• To each book in the library there corresponds the number of pages in the book.
• To each human being there correspond a date of birth.
• If the temperature of the air is recorded throughout the day, then to each instant o
there corresponds a temperature.
Definition
If a quantity y is called a ‘function of x’ written y=f(x) it means that for every value of x
which the function is defined) there is a corresponding value of y. x is called the argum
independent variable and y the dependent variable of the function.
Given the following:
1. 𝑓 𝑥 = 2𝑥 − 3 ∶ Linear function
2.𝑓 𝑥 = 𝑥 2 + 2𝑥 − 3 : Quadratic function
3. 𝑓 𝑥 = 𝑥 3 : Cubic function
4. f x = sin 𝑥: Trigonometric Function.
the plots for the functions as displayed
03-May-20 below
MTH 121- General Mathematics II (Calculus)
elementary function of a single re
variable and their graphs
, History- why calculus?
Calculus is a branch of mathematics concerned with the study of such concepts as the rate of change of o
quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maxim
minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, a
and geometry, it is the basis of that part of mathematics called analysis.
Calculus is widely employed in the physical, biological, and social sciences. It is used, for example, in the
sciences to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of dec
radioactive material. In the biological sciences a problem such as the rate of growth of a colony of bacter
function of time is easily solved using calculus. In the social sciences calculus is widely used in the study o
and probability.
Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, t
slowest, or the least. These maximum or minimum amounts may be described as values for which a certa
change (increase or decrease) is zero. By using calculus it is possible to determine how high a projectile w
finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero. M
principles governing the behavior of physical processes are formulated almost invariably in terms of rates
is also possible, through the insights provided by the methods of calculus, to resolve such problems in log
famous paradoxes posed by the Greek philosopher Zeno. The fundamental concept of calculus, which di
from other branches of mathematics and is the source from which all its theory and applications are deve
theory of limits of functions of variables.
(Source: Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.)
03-May-20 MTH 121- General Mathematics II (Calculus)
, Intended learning outcomes (ILOs)
At the end of this lesson, every students should be able to:
(i) Define the concepts of functions, range, and domain
(i) Diagrammatically represent, Range, Domain relationship using maps.
(ii) Differentiate between one-to-one mapping and many-to-one mapping
(i) State different kinds of functions.
, Functions
The notion of correspondence occurs frequently in everyday life. Some examples are g
the following illustrations:
• To each book in the library there corresponds the number of pages in the book.
• To each human being there correspond a date of birth.
• If the temperature of the air is recorded throughout the day, then to each instant o
there corresponds a temperature.
Definition
If a quantity y is called a ‘function of x’ written y=f(x) it means that for every value of x
which the function is defined) there is a corresponding value of y. x is called the argum
independent variable and y the dependent variable of the function.
Given the following:
1. 𝑓 𝑥 = 2𝑥 − 3 ∶ Linear function
2.𝑓 𝑥 = 𝑥 2 + 2𝑥 − 3 : Quadratic function
3. 𝑓 𝑥 = 𝑥 3 : Cubic function
4. f x = sin 𝑥: Trigonometric Function.
the plots for the functions as displayed
03-May-20 below
MTH 121- General Mathematics II (Calculus)
