Differentiation and Its Application
Chapter One
Purpose Of The Study
The purpose of this project is to introduce the operational principles of differentiation in calculus. Also to analyse many problems that have long been considered by mathematicians and scientists.
CHAPTER TWO
FUNDAMENTALS OF CALCULUS
FUNCTIONS OF SINGLE VARIABLE AND THEIR GRAPHS
Functions are essential ingredients in the study of calculus. Their absence means an incomplete study of this branch of mathematics which is more or less the care of many aspects of mathematics.
Calculus by its basis is the notion of correspondence for example, the surface area (A) of a sphere relates to its radius by the formula
A= 4πr2
The sphere volume (V) of a given mass of gas is related to the pressure (P) of the gas.
These example give us an idea what a function is. Conclusion from them could imply that if the value of one quantity, say y depends on the value of another quantity, say x, then for every value of x there corresponds one and only the value one of y.
On this basis we say that y is a function of x. thus, A is a function of r and V is a function of p as above.
Though, the same element of y may correspond to different element of x. for example, two different books may have the same number of pages.
Definition 2.0: A function f from a set x to a set y is a correspondence that assigns to each element x of X a unique element y of y.
The set of values of X is called the domain of the function. The element y is called the image X under f and is denoted by f (x). the range of the function consists of all images of elements of X.
Definition 2.1 .A function f is said to be defined at x or f(x) exists if x is in the domain. Put differently, a function is defined for a certain value b of x, if a definite value of f (b) corresponds to the value of x.
On the other hand, f is undefined at x if x is not in the domain of f or at a given value of x the function is meaningless, since division by zero is not a valid operation.
Example2.2 : for what value of x is the function y = 5x-5 defined? What is the domain and co- domain?
Solution: y=5x-5 is defined for every value of x since for every value of x, we obtain one value for y.
The domain are the real numbers and the co-domain is the set of all the real numbers.
Y or f depends on the value of x hence the domain is called the DEPENDENT VARIABLE while
X is called the INDEPENDENT VARIABLE.
DEFINITION2.2 a function f form x to y is one to one function. (injective)
Example2.3 : determine if the function f is one to one if : (i) f (x) =2x+9, (ii) f(x) =3x2 at x = 3 and 4 for (i) and at x=-2 and 2 for (ii)
Solution: (i) if a≠ b then with a =3 and b = 4 we have 2.3+9 ≠ 2.4+9 or 15≠17 hence f is a one to one function
CHAPTER THREE
DIFFERENTIATION
Differentiation as a limit of rate of change of elementary function
In this section we shall discuss about the gradient of a line and a curve, gradient function also called the derived function and finally, the main topic differentiation as a limit of rate of change of elementary function.
Gradient: Straight line and curve
As we rightly said in the study of rate of change of a function, the gradient of a line is measured by taking the ratio of the increase in Y and the increase in x in moving from one point to another on the line. Such that if (x1, y1,) and (x2, y2,) are two point on a line, the gradient is then taken as.
CHAPTER FOUR
APPLICATION OF DIFFERENTIATION
In this chapter we shall be looking at the area such as: the tangent and normal to a curve, the stationary values of simple functions. Also in this chapter we shall consider the minima, maxima and points of inflection and finally look at curve sketching.
CHAPTER FIVE
SUMMARY AND CONCLUSION
Calculus has been an important aspect of Mathematics that the significance of its study cannot be over-emphasized.
This aspect of mathematics is divided into two part:
DIFFERENTIATION AND INTEGRATION.
However our emphasis is on differentiation.
Most companies, private organization and factories mobilize their staff to produce one goods or the other with respect to time. So this aspect of calculus called differentiation serves as a very good method use in maximing production and the distribution of raw and finished goods as at when needed.
In conclusion, this aspect of calculus (differentiation) is a needed factor that must be in mind in the course of any production.
REFERENCES
- Aashikpelokhai U.S.U et al: An introductory course in higher Mathematics Calculus, Vector and Mechanics 2009 PON publisher Ltd, Edo State, Nigeria.
- Boyer Carl The history of calculus New York, Dover publication 1949.
- Frank Ayres Jr. et al: Schaum’s outlines fifth edition (2009) The McGraw-Hill publishing Companies USA
- George B Thomas Jr. et al: Calculus and Analytic Geometry 9th edition 1998 Addison-Wesley publishing company USA
- Harper et al : The calculus with Analytical Geometry fifth edition 1986 Harper and Row publishers. New York
- Mary W. Gray: Calculus with finite mathematics 1972 Addison-Wesley Publishing company London
- Murray R Epiegel : Schaum’s Outline 1971 Advance Mathematic for Engineers and Scientist. Mc Graw-Hill, New York
- Raymond A. Barnett et al: Applied Calculus 3rd edition 1987, Dellen Publishing company San Francisco California
