College of Technology, Pantnagar

The sign of the first derivative, concavity and point of inflexion. Asymptotes and symmetry, Rolle’s Theorem, Mean Value Theorem, Extended Mean Value Theorem, Taylor’s formula, Estimating approximation errors. Taylor Theorem with remainder and estimating the remainder. Newton’s method for approximating solution of equation. Inverse functions and Picard’s Method. The first and second fundamental theorem of Integral Calculus. Leibnitz’s rule. Approximating finite sums with integrals. Rules for approximating definite integrals with the help of Trapezoidal and Simpson’s rule and their error estimation. Convergence and divergence of infinite series non-negative terms with the help of comparison test, integral test, Limit comparison test, Ratio test, Root’s test. Limit and continuity of function of two or more variables. Partial Derivatives, chain rules for functions of two or more variables. Linear approximation of z = f(x,y) and their increment estimation. Maximum, minimum and saddle points for function for two or more variables, Lagrange’s multipliers method. Convergence and divergence of improper integrals. Calculating volume by slicing, volume modeled with shells and washers. Length of a plane curve. Area of a surface of revolution. Polar coordinates; Polar equations of conics and other curves, Area of plane curves, Arc length and surface area. Multiple Integrals : Double integrals, Area bounded by curves, First and second moments, Polar moment of intertia, Radius of gyration, changing double integrals from cartesion to polar co-ordinates. Evaluation of triple integrals. Parametric equations in analytical geometry and idea of spherical and cylindrical co-ordinates. Hyperbolic Functions : definition and identities, derivatives and integrals, Inverse hyperbolic functions.