📐 Mathematics II BCA 154 | Semester II
📖 Course Description
The course covers the concepts of Limit and Continuity, Derivatives of Algebraic, Trigonometric, Exponential, and Logarithmic functions, and their applications. It also covers Integration and its Applications, Volume and Surface Integral of some selected functions, and Numerical Integration using Trapezoidal and Simpson’s Rule. Ordinary and Partial Differential Equations, their examples and uses are also discussed. Other contents include Optimization Problems like Simplex Method for two variables, and Gauss-Seidel, Gauss Elimination, Matrix Inversion, Bisection, and Newton-Raphson Method for solving linear and non-linear equations.
🎯 Course Objectives
- Understand the concept of limit and continuity of functions and their connection to the derivative
- Differentiate different types of functions, geometrical meaning of derivative, and applications to real-world problems
- Integrate functions, understand their meaning and applications, including surface and volume integrals
- Solve ordinary and partial differential equations and their connections to real-world problems
- Calculate optimization problems, describe and interpret graphical and numerical solutions
📚 Detailed Syllabus
Unit 1: Limit and Continuity
7 Hrs
- Definition of limit including epsilon-delta condition, right and left hand limit, and its interpretation
- Algebraic properties of limit
- Definition and conditions of continuity and discontinuity
- Continuity of algebraic, trigonometric, and exponential functions, examples and counterexamples
Unit 2: Derivatives
7 Hrs
- Definition and geometrical meaning of derivatives
- First principle method to differentiate algebraic, trigonometric, exponential, and logarithmic functions
- Rules of derivatives: sum, product, power, chain, and quotient rule
- Derivatives of inverse circular, hyperbolic functions and implicit functions
- Higher order derivatives
- Relation between derivative and continuity
- Definition and examples of partial derivatives
Unit 3: Applications of Derivatives
8 Hrs
- Increasing and decreasing functions
- Equation of tangents and normals using first derivatives
- L’Hospital’s rule
- Angle between two lines
- Maxima and minima, absolute maxima and minima, concavity, stationary points and points of inflection
- Statement and geometrical interpretation of Rolle’s theorem, Cauchy Mean-value theorem and Generalized Mean-value theorem
- Taylor’s theorem, Maclaurin theorem (without proof) and its use in expansion of simple functions
- Applications of derivatives in Economics
- Rate measures
Unit 4: Anti-derivative and its Applications
8 Hrs
- Definition and geometrical meaning of integration
- Basic integration formulas for algebraic, trigonometric, exponential, and logarithmic functions
- Trigonometric substitution for basic functions
- Integration by parts (Product rule for integration)
- Partial fractions
- Improper integral
- Definite integral in terms of Riemann sum, and fundamental theorem of integral calculus
- Applications of definite integral: Area under curve, area between curves, Quadrature & rectification
- Surface and volume integrals
- Trapezoidal and Simpson’s Rule for numerical integration
Unit 5: Differential Equations
8 Hrs
- Definition, order, and degree of differential equations
- Differential equations of first order and first degree
- Variables separable, homogeneous, exact, and linear differential equations
- Reducible to linear form
- Partial differential equations with some basic examples
Unit 6: Computational Methods
10 Hrs
- Linear programming problems
- Linear inequalities in two variables and their graphical solutions
- Simplex Method (up to 3 variables), Duality problems
- Matrix inversion method
- Gauss Elimination, Gauss-Seidel method
- Bisection method and Newton-Raphson Method for non-linear equations
🧪 Laboratory Work
Tools: Python, MATLAB, or Mathematica
Implement limits and continuity verification numerically
Derivative computation using symbolic libraries (SymPy)
Plotting functions and tangent lines, maxima/minima visualization
Numerical integration: Trapezoidal and Simpson’s rule
Solving differential equations (ODE solvers)
Linear programming: Simplex method implementation
Gauss elimination and Gauss-Seidel for linear systems
Bisection and Newton-Raphson for root finding
📌 Students will solve numerical problems from each unit using computational tools and compare results with pen-and-paper methods.
📊 Examination Scheme:
- Internal Assessment (Theory + Practical): 20 + 20 marks
- External Assessment (Theory – 3 hrs): 60 marks
- Total: 100 marks
📚 Required Readings & References
- Bajracharya, B. C. (2025) – Basic Mathematics, Sukunda Pustak Bhawan
- Budnick, F. S. (2017) – Applied Mathematics for Business Economics and Social Sciences, McGraw-Hill
- Chand, K. B., Sapkota, B. P. (2022) – Principles of Mathematical Analysis, Pinnacle Publication
- Lay, D. C. (2003) – Linear Algebra and Its Applications, Pearson Education
- Stewart, J., Clegg, D., Watson, S. (2020) – Calculus, Cengage
- Thomas G. B., Finney, R. L. (1995) – Calculus and Analytical Geometry, Narosa Publishing House