## LNMU B.Sc. Part-2 Syllabus of Mathematics Honours-

Paper-1
12 questions to be set and 6 to be answered.

### Differential Calculus:

Definition of the limit of a function, Basic properties of limits Continuous functions and classification of discontinuous. Differentiability Successive differentiation. Leibnitz’s theorem, Maclaurin and Taylor series expansions, Partial differentiation Euler’s theorem, Tangents and normals. Curvature. [4 questions]

#### Integral Calculus:

Integration of irrational algebraic functions and transcendental functions, Reduction formulae, Definite integrals. Quadrature. Rectification. Volumes and surfaces of solids of revolution, Continuity, Sequential continuity Properties of continuous function. Uniform continuity. Chain rule of differentiability. Mean value theorems and their geometrical interpretations. Darboux’s intermediate value theorem for derivatives. Taylor’s theorem with various forms of remainders.

Maxima and minima of functions of two variables, Lagrange’s multiplier method. Beta and Gamma functions, Double and triple integrals, Dirichlof’s integrals, Change of the order of integration in double integrals. [5 questions]

#### Sequence & Series:

Definition of a sequence, Theorem on limits of sequences. Bounded and monotonic sequence, Cauchy’s convergence criterion. Series of non-negative terms, Comparison tests, Ratio tests. Rabbe’s tests. Logarithmic tests, Higher Logarithmic test. Gauss tests, Kummer test, Cauchy’s condensation test. De morgan and Bertrand’s test. Alternating series, Leibnitz’s theorem Absolute and conditional convergence. [3 questions]

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