Course StructureBCA 154
Mathematics II
1
Limit and Continuity
Limits of functions, indeterminate forms, continuity, and properties of continuous functions.
Concept of Limits
- Definition of limit (informal and formal)
- Left-hand and right-hand limits
- Limit of a function (epsilon-delta definition)
- Algebra of limits
- Limits involving infinity
Techniques of Evaluating Limits
- Direct substitution method
- Factorization method
- Rationalization method
- L'Hospital's rule
- Special limits (sin x/x, (1+1/x)^x)
Continuity
- Definition of continuity at a point
- Continuity on an interval
- Types of discontinuities (removable, jump, infinite)
- Properties of continuous functions
- Intermediate value theorem
2
Derivatives
Definition, rules of differentiation, chain rule, implicit differentiation, higher order derivatives, and partial derivatives.
Definition and Basic Rules
- Definition of derivative (first principles)
- Differentiability and continuity relationship
- Derivative by first principles
- Power rule and constant rule
- Sum and difference rule
Advanced Differentiation Rules
- Product rule
- Quotient rule
- Chain rule
- Implicit differentiation
- Logarithmic differentiation
Higher Order Derivatives
- Second and higher order derivatives
- Notation for higher order derivatives
- Applications of second derivative
- Partial derivatives
- Total differential
3
Applications of Derivatives
Maxima and minima, tangents and normals, L'Hospital's rule, mean value theorems, Taylor and Maclaurin series.
Maxima and Minima
- Critical points and stationary points
- First derivative test
- Second derivative test
- Absolute maxima and minima
- Applications in optimization problems
Tangents and Normals
- Equation of tangent line
- Equation of normal line
- Slope of tangent and normal
- Angle of intersection of curves
- Length of tangent and normal
Mean Value Theorems
- Rolle's theorem
- Lagrange's mean value theorem
- Cauchy's mean value theorem
- Taylor's theorem
- Maclaurin series expansion
Indeterminate Forms
- Indeterminate forms (0/0, infinity/infinity, 0*infinity)
- L'Hospital's rule application
- Evaluating limits using series expansion
- Form 0^0, 1^infinity, infinity^0
- Techniques for different indeterminate forms
4
Anti-derivative and its Applications
Indefinite and definite integrals, integration by parts, partial fractions, area under curve, surface and volume integrals.
Integration Basics
- Definition of antiderivative
- Indefinite integration
- Basic integration formulas
- Integration by substitution
- Integration by parts
Advanced Integration Techniques
- Integration using partial fractions
- Integration using trigonometric identities
- Trigonometric substitution
- Integration of special functions
- Reduction formulas
Definite Integration
- Definite integral definition
- Fundamental theorem of calculus
- Properties of definite integrals
- Area under a curve
- Area between two curves
Applications of Integration
- Volume of solids of revolution
- Arc length of curves
- Surface area of revolution
- Average value of a function
- Work and energy applications
5
Differential Equations
First and second order ODEs, variable separable, homogeneous, exact equations, and applications.
First Order Differential Equations
- Definition and classification
- Order and degree of differential equations
- Variable separable method
- Homogeneous equations
- Linear first order equations
Advanced Solution Methods
- Exact equations
- Integrating factor method
- Bernoulli's equation
- Orthogonal trajectories
- Applications of first order ODEs
Second Order Differential Equations
- Homogeneous equations with constant coefficients
- Auxiliary equation
- Real and distinct roots
- Real and equal roots
- Complex conjugate roots
Applications of Differential Equations
- Growth and decay models
- Newton's law of cooling
- Simple harmonic motion
- RLC circuits
- Mixing problems
6
Computational Methods
Bisection method, Newton-Raphson method, Gauss elimination, linear programming, and simplex method.
Root Finding Methods
- Bisection method (interval halving)
- Regula-Falsi method (false position)
- Newton-Raphson method
- Secant method
- Comparison of convergence rates
Linear Algebra Methods
- Gaussian elimination
- Gauss-Jordan elimination
- Matrix inversion methods
- LU decomposition
- Iterative methods (Jacobi, Gauss-Seidel)
Linear Programming
- Formulation of linear programming problems
- Graphical method
- Simplex method
- Duality in linear programming
- Transportation problems
Interpolation and Approximation
- Finite differences
- Newton's forward difference
- Newton's backward difference
- Lagrange's interpolation
- Least squares method