Sequences and Series of real numbers: Sequences of real numbers, their convergence, and limits. Cauchy sequences and their convergence. Monotonic sequences and their limits. Limits of standard sequences. Infinite series and its convergence, and divergence. Convergence of series with non-negative terms. Tests for convergence and divergence of a series. Comparison test, limit comparison test,
D’Alembert’s ratio test, Cauchy’s 𝑛𝑡ℎ root test, Cauchy’s condensation test and integral test. Absolute convergence of series. Leibnitz’s test for the convergence of alternating series. Conditional convergence. Convergence of power series and radius of convergence.

Differential Calculus of one and two real variables: Limits of functions of one real variable. Continuity and differentiability of functions of one real variable. Properties of continuous and differentiable functions of one real variable. Rolle’s theorem and Lagrange’s mean value theorems. Higher order derivatives, Lebnitz’s rule and its applications. Taylor’s theorem with Lagrange’s and Cauchy’s form of remainders. Taylor’s and Maclaurin’s series of standard functions. Indeterminate forms and L’ Hospital’s rule. Maxima and minima of functions of one real variable, critical points, local maxima and minima, global maxima and minima, and point of inflection. Limits of functions of two real variables. Continuity and differentiability of functions of two real variables. Properties of continuous and differentiable functions of two real variables. Partial differentiation and total differentiation. Lebnitz’s rule for successive differentiation. Maxima and minima of functions of two real variables. Critical points, Hessian matrix, and saddle points. Constrained optimization techniques (with Lagrange multiplier).