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
Formula: Standard Integration Formulae
| Integral | Result |
|---|---|
| \[\int x^ndx\] | \[\frac{x^{n+1}}{n+1}+C\] |
| \[\int(ax+b)^ndx\] | \[\frac{(ax+b)^{n+1}}{a(n+1)}+C\] |
| \[\int\frac{1}{x}dx\] | \[log/x/+c\] |
| \[\int\frac{1}{ax+b}dx\] | \[\frac{\log\left|ax+b\right|}{a}+c\] |
| \[\int a^xdx\] | \[\frac{a^x}{\log a}+C\] |
| \[\int a^{px+q}dx\] | \[\frac{a^{px+q}}{p\log a}+C\] |
| \[\int e^xdx\] | \[e^{x}+C\] |
| \[\int e^{px+q}dx\] | \[\frac{e^{px+q}}{p}+C\] |
Key points: Rules of Integration
Sum Rule:
\[\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx\]
Difference Rule:
\[\int[f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx\]
Constant Multiple Rule:
\[\int kf(x)dx=k\int f(x)dx\]
Substitution Result:
If \[\int f(x)dx=F(x)+C\] Then \[\int f(ax+b)dx=\frac{1}{a}F(ax+b)+C\]
Theorem: Change of Variable
If x = ϕ(t), then \[\int f(x)dx=\int f(\phi(t))\phi^{\prime}(t)dt\]
Formula: Change of Variable
- \[\int
\begin{bmatrix}
f(x)
\end{bmatrix}^nf^{\prime}(x)dx=\frac{
\begin{bmatrix}
f(x)
\end{bmatrix}^{n+1}}{(n+1)}+c\] -
\[\int\left[\frac{f^{\prime}(x)}{f(x)}\right]dx=\log f(x)+c\]
-
\[\int\left[\frac{f^{\prime}(x)}{\sqrt{f(x)}}\right]dx=2\sqrt{f(x)}+c\]
-
\[\int\left[\frac{f^{\prime}(x)}{\sqrt[n]{f(x)}}\right]dx=\frac{n\sqrt[n]{\left[f(x)\right]^{n-1}}}{n-1}+c\]
Formula: Standard Integrals of Quadratic Forms
-
\[\int\frac{1}{x^2-a^2}dx=\frac{1}{2a}log\left|\frac{x-a}{x+a}\right|+c\]
-
\[\int\frac{1}{a^2-x^2}dx=\frac{1}{2a}log\left|\frac{a+x}{a-x}\right|+c\]
-
\[\int\frac{1}{\sqrt{x^2+a^2}}dx=log\left|x+\sqrt{x^2+a^2}\right|+c\]
-
\[\int\frac{1}{\sqrt{x^2-a^2}}dx=log\left|x+\sqrt{x^2-a^2}\right|+c\]
Theorem: Integration by Parts
If u and v are two functions of x, then
\[\int u.vdx=u\int vdx-\int\left[\int vdx.\frac{du}{dx}\right]dx\]
Key Points: LAE Rule
L → A → E
-
L = Logarithmic (log x)
-
A = Algebraic (x, x², polynomial)
-
E = Exponential (eˣ)
Formula: Partial Fractions
| Type | Rational Form | Partial Fraction Form |
|---|---|---|
| 1 | \[\frac{px\pm q}{(x-a)(x-b)}\] | \[\frac{A}{x-a}+\frac{B}{x-b}\] |
| 2 | \[\frac{px^2\pm qx\pm r}{(x-a)(x-b)(x-c)}\] | \[\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\] |
| 3 | \[\frac{px\pm q}{\left(x-a\right)^2}\] | \[\frac{A}{x-a}+\frac{B}{\left(x-a\right)^2}\] |
| 4 | \[\frac{px^2\pm qx\pm r}{(x-a)^2(x-b)}\] | \[\frac{A}{x-a}+\frac{B}{\left(x-a\right)^2}+\frac{C}{x-b}\] |
| 5 | \[\frac{px^2\pm qx\pm r}{(x-a)^3(x-b)}\] | \[\frac{A}{x-a}+\frac{B}{\left(x-a\right)^2}+\frac{C}{\left(x-a\right)^3}+\frac{D}{x-b}\] |
| 6 | \[\frac{px^2\pm qx\pm r}{(x-a)(ax^2\pmb x\pm c)}\] | \[\frac{A}{x-a}+\frac{Bx+C}{ax^2 \pm b\pmb x\pm c}\] |
