Basic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9
Dr. Atul Kumar Srivastava
Learning Objectives Understand the use of this Branch of mathematics in various branches of science and Humanities
Understand the basics of differentiation and integration
Know how to compute derivative of a function by the first principal, derivative of a function by the use of various formulae and higher order differentiation
Make familiar with various techniques of integration
Understand the concept of definite integrals of functions and its properties.
What is Differential Calculus – An Introduction Derivative or Deferential Coefficient (First Principal Definition) Basic Formulas Laws for Differentiation (Algebra of Derivative of Functions) Derivative of A Function of Function (Chain Rule) .Derivative of Implicit Function .Derivative of Function In Parametric Form
What is differential calculus – an 4 introduction One of the most fundamental operations in calculus is that of differentiation. In the study of mathematics, there are many problems containing two quantities such that the value of one quantity depends upon the other. A variation in the value of any ones produces a variation in the value of the other. For example the area of a square depends upon it's side. The area of circle and volume of sphere depend upon their radius etc.
Differential calculus is the Branch of Mathematics which studies changes
To express the rate of change of any function we introduce concept of derivative. The concept involves a very small change in the dependent variable with reference to a very small change in the independent variable
y=f(x) • •
x - Independent variable y - Dependent Variable
Thus differentiation is a process of finding the derivative of a continuous function. It is defined as the limiting value of the ratio of the change in the function corresponding to small change in the independent variable as the later tends to zero.
Derivative or differential coefficient (First principal definition) Derivative of
is defined as (1)
is a function
is small increment in x corresponding increment in y or f(x) (1) Is denoted as known as differential coefficient of
The derivative of f(x) is also w.r.t. x
The above process of differentiation is called the first principal
Examples of differentiation from first principal: Example :1 We have
Example-2 f(x)=a where a is fixed real number
Following are some of the standard derivative:-
Laws for differentiation
(Algebra of derivative of functions)
Let f(x) and g(x) be two functions such that their derivatives are defined in a common domain. Then 1.Sum Rule
Derivative of implicit function Until now we have been differentiating various function given in the form y=f(x) But it is not necessary that functions are always expressed in this form. For example consider one of the following relationship between x and y
Continued…. In the first case we can solve y and rewrite the relationship as In second case it does not seem easy to solve for Y. When it is easy to express the relation as y=f(x) we say that y is given as an explicit function of x, otherwise it is an implicit function of x
Now we will attempt to find
for implicit function
Examples Example 1
Differentiating on both sides
Derivative of functions in parametric forms If relation between two variables is expressed via third variable. The third variable is called parameter. More precisely a relation expressed between two variables x and y in the form x=f(t),y=g(t) is said to be parametric form with t is a parameter.
In order to find derivative of function in such form, we have by chain rule.
Example 1 Find Given That
We differentiate such functions by taking logarithm on both sides. This process in called logarithmic differentiate.
Taking logarithms on both sides
Differentiate both sides w.r.t x
Higher Order differentiation
If is differentiable, we may differentiate it w.r.t. x. The LHS becomes which is called the second order derivative of or and is denoted by . It is also denoted by . If we remark that higher order derivatives may be defined similarly.
Example-1 Given that =
Gradient or slope of the curve
Let y=f(x)be a curve. The derivative of f(x) at a point x represents the slope of the tangent to the curve y=f(x)at the point x. Sometime the derivative is called gradient of the curve.
Find the gradient of the curve
The gradient of the curve at point X=0 is -12
Example-2 to the curve
Find the slope of the tangent
Miscellaneous Examples Example:1
Find derivative of
Example-4: Find dy /dx differentiate implicitly w.r.t. x ., we get
By cross multiplication
Integral Calculus What Is Integration (Definition) • Basic Formulas
Method Of Substitution (Change Of Variable) Integration By Parts Method Of Partial Fraction Definite Integration Important Properties Miscellaneous Examples
What is Integration (definition) Integration is inverse process of differentiation. Integral calculus deals with integration and its application. It was invented in attempt to solve the problems of finding areas under curves and volumes of solids of revolution.
Also we can define integration as the inverse process of differentiation .
Constant of integration
and c is an arbitrary constant we also have
Evidently the integral of
is obtained by giving different
values to C . Here 'C' is called constant of integration
The process of finding the integral is called integration. The function which is integrated is called the integrand.
Two Simple Theorem 1.
Since integration and differentiation are inverse process we have Example-1
Examples Example-6 Evaluate
Example-8 By simple division
Integration By Substitution The given integral can be transformed into another form by changing the independent variable x to t by substituting Consider
Usually we make a substitution for a function whose derivation also occur in the integrand.
adx = dt
Example 2 .
Important standard formulas 1.
Integration by Parts It is useful method to find integration of product of function. Integration of product of two function
(Solve first integral only)
Methods of Partial Fractions
Find the partial fraction of
we put x=2
we put x=3 and get
Comparing coefficients of
and constant term on both sides
Solving we get
Consider indefinite integral Now consider Here a = b =
lower limit of integration upper limit of integration
is called definite integral of f(x) from a to b
Miscellaneous examples find
Example-3 integration by parts
Example 4. solve
Example:5 by simple division
Example-6 First simplify integrand
Example 10. I
Example11. Find the equation of the curve where slope at (x,y) is 9x which passes through origin Given
since it passes through the origin (o,o) Then or
SUMMARY OF THE CHAPTER Differential calculus
(a < b < c)
HINT- Logarithmic differentiation then Apply product rule
HINT- (Apply Quotient rule)
HINT - (Apply chain rule)
HINT - (Apply product rule and chain rule)
HINT- (Quotient rule)
HINT- (Take log both sides then apply quotient rule)
HINT- (Differentiation of implicit function)
HINT- (Logarithmic differentiation)
Question.15 The value of p and q are.
HINT- Integrate By Parts
HINT- Integration by substitution let
HINT- [Integration by substitution, let t
HINT- [Integration by substitution let
HINT- (Integration by substitution let 7x + 5 = t)
HINT- Divide Numerator by Denominator then Integrate
Practice makes a man perfect and that’s what mathematics demands. .
So, students ‘all the best’ for your upcoming examinations . keep practicing.