Syllabus (COMPUTER APPLICATION)

Course Type: ME-2

Semester: 2

Course Code: BBCAMEB12T

Course Title: Basic Algebra, Calculus and Ordinary Differential Equations

(L-P-Tu): 4-0-0

Credit: 4

Practical/Theory: Theory

Course Objective: Course Objectives: 1. Use algebraic methods to solve a variety of problems involving exponential, logarithmic, polynomial, and rational functions, systems of equations and inequalities, sequences. 2. Solve equations by correctly completing several logical steps before arriving at a final answer, and when possible, check solutions. 3. Explain and interpret the meaning of the derivative of a function. 4. Use shortcuts to calculate derivatives efficiently. 5. Use derivatives to solve authentic real-life application problems. 6. Use definite integrals and the Fundamental Theorem of Calculus to find areas and total change. 7. To model mechanical systems using differential equations. 8. To analyse and solve ordinary differential equations. 9. To understand numerical methods for solving ordinary differential equations.

Learning Outcome: Course Outcomes: Upon successful completion of this course, students will be able to- 1. Solve tangent and area problems using concepts of limit, derivatives and integrals. 2. Calculate higher order derivatives and find limit of functions which are of indeterminate form using L’Hospital’s rule. 3. Find the rate in which a curve curves, further to find Asymptotes of curves, envelopes etc. 4. Learn concepts of complex numbers, De’Movire’s theorem and its application. 5. Find number of real (positive & negative) and complex roots of algebraic equations using Descarte’s rule and learn the methods to solve cubic and bi-quadratic equations. 6. Acquire knowledge of inequalities. 7. Solve first order differential equations utilizing the standard techniques for exact, linear, homogeneous, or Bernoulli cases. 8. Solve first order nonlinear differential equations using the standard techniques and get an idea of singular solution. 9. Find the complete solution of a non-homogeneous differential equation as a linear combination of the complementary function and a particular solution.

Syllabus:

Credit: 4 (L 60)

Algebra: Sets, Union and Intersection, Complement, Mapping, Composition, notion of a Group, Ring, Field with simple examples. [L 4]

Complex Number: Modulus and amplitude, De Moiver’s theorem. [L 4]

Polynomials, Division algorithm, Fundamental theorem of classical algebra (Proof not required), Descartes rule of sign and their application, Relation between roots and coefficients; symmetric function of roots, Transformation of polynomial equation, Cardon's solution of cubic equation, Determinants, Addition and Multiplication of Matrices, Inverse of a Matrix ; Solution of linear equations in three variables by Cramer's rule and solution of three line linear equations by matrix inversion methods.
[L 12]

Differential Calculus: Limit of a function and continuity. Fundamental properties of continuous functions (proofs not required); Derivative and Differential-Geometric meaning, Rules of Differentiation. Successive dirrerentiation. Rolle’s theorem, Mean-Value theorems, Taylor’s and Maclaurin’s theorems with Cauchy’s and Lagrange’s forms of remainder; Taylor’s series. Functions of several variables. Partial Derivatives. Total Differential. Euler’s theorem on homogeneous functions of two variables. Application to plane curves. [L 12]

Sequence and Series: Bounded and unbounded sequences, Convergence or divergence of a sequence, Behavior of monotone sequences, Algebra of convergent sequences, Cauchy sequence, Cauchy's general principle of convergence, Infinite series, it's convergence and sum, Alternating Series, Leibniz Test, Absolute convergence. [L 10]

Integral Calculus: Rules of Integration of Indefinite Integrals, Solution of Definite Integrals and their elementary properties. Idea of improper integrals. [L 8]

Differential Equations: order, degree, solution and formation of a differential equation. Standard techniques of solving a linear differential equation with constant coefficients. Cauchy's and Ligendre's Liner Differential Equations with variable coefficients. [L 10]

Reading References:

1. Higher Algebra: Classical-S. K. Mapa
2. A Text book of Algebra- B.K. Lahiri & K. C. Roy
3. Linear Algebra- Das & Roy
4. Co-ordinate Geometry- S. L. Loney
5. Differential Calculus- Das and Mukherjee
6. Integral Calculus - Das and Mukherjee
7. Differential Equations - Sheplay I. ( John Wiley & Sons, Inc )
8. Linear Algebra - Kenneth Hoffman & Ray Kunze ( PHI )
9. Mathematical Analysis - S. C. Malic ( Wiley Eastern Limited )
10. Differential Calculus – Das and Mukherjee
11. Integral Calculus – Das and Mukherjee