Topics
Mathematical Logic
Matrices
Differentiation
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Inverse Functions
- Derivatives of Logarithmic Functions
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Second Order Derivative
- Overview of Differentiation
Applications of Derivatives
Integration
Definite Integration
Applications of Definite Integration
- Standard Forms of Parabola and Their Shapes
- Standard Forms of Ellipse
- Area Under Simple Curves
- Overview of Application of Definite Integration
Differential Equation and Applications
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equation by Eliminating Arbitary Constant
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Overview of Differential Equations
Commission, Brokerage and Discount
- Commission and Brokerage Agent
- Concept of Discount
- Overview of Commission, Brokerage and Discount
Insurance and Annuity
- Insurance
- Types of Insurance
- Annuity
- Overview of Insurance and Annuity
Linear Regression
- Regression
- Types of Linear Regression
- Fitting Simple Linear Regression
- The Method of Least Squares
- Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
- Properties of Regression Coefficients
- Overview: Linear Regression
Time Series
- Introduction to Time Series
- Uses of Time Series Analysis
- Components of a Time Series
- Mathematical Models
- Measurement of Secular Trend
- Overview of Time Series
Index Numbers
- Weighted Aggregate Method
- Cost of Living Index Number
- Method of Constructing Cost of Living Index Numbers - Aggregative Expenditure Method
- Overview of Index Numbers
- Method of Constructing Cost of Living Index Numbers - Family Budget Method
- Uses of Cost of Living Index Number
Linear Programming
- Introduction of Linear Programming
- Linear Programming Problem (L.P.P.)
- Mathematical Formulation of Linear Programming Problem
- Overview of Linear Programming
Assignment Problem and Sequencing
- Assignment Problem
- Hungarian Method of Solving Assignment Problem
- Special Cases of Assignment Problem
- Sequencing Problem
- Types of Sequencing Problem
- Finding an Optimal Sequence
- Overview of Assignment Problem and Sequencing
Probability Distributions
- Poisson Distribution
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
- Overview of Binomial Distribution
- Simple curves: lines, parabolas, polynomial functions
Notes
We consider the easy way of finding the area bounded by the curve y = f(x), x-axis and the ordinates x = a and x = b. Fig.
The area under the curve as composed of large number of very thin vertical strips. Consider an arbitrary strip of height y and width dx, then dA (area of the elementary strip)= ydx, where, y = f(x).
This area is called the elementary area which is located at an arbitrary position within the region which is specified by some value of x between a and b. The total area A of the region between x-axis, ordinates x = a, x = b and the curve y = f (x) as the result of adding up the elementary areas of thin strips across the region PQRSP. Symbolically, we express
A = `int_a^b dA = int_a^b ydx = int_a^b f(x) dx`
The area A of the region bounded by the curve x = g (y), y-axis and the lines y = c, y = d is given by
A = `int_c^d x dy = int_c^d g(y) dy`
we consider horizontal strips as shown in the following Fig.
Remark: If the position of the curve under consideration is below the x-axis, then since f (x) < 0 from x = a to x = b, as shown in following Fig .
The area bounded by the curve, x-axis and the ordinates x = a, x = b come out to be negative. But, it is only the numerical value of the area which is taken into consideration. Thus, if the area is negative, we take its absolute value, i.e., `|int_a^b f(x) dx|`.
Generally, it may happen that some portion of the curve is above x-axis and some is below the x-axis as shown in the following Fig .
Here , `A_1` < 0 and `A_2`> 0.Therefore, the area A bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b is given by A = |`A_1`|+`A_2`.
