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Syllabus - B.Tech 1st semester Mathematics

Infinite Series: Sequence, Convergence and Divergence of Infinite series – and typical examples of convergent and divergent series, Comparison test (statement only) and related problems, Ratio test (statement only) and related problems, Cauchy’s root test (statement only) and related problems, Alternating series, Leibnitz’s theorem (without proof), absolute convergence and related problems.
Calculus of Functions of One Variable: Review of limit and continuity and differentiability, Successive differentiation, Leibnitz’s theorem (without proof but with problems of the type of recurrence relations in derivatives of different orders and also to find (yn)0 , Rolle’s theorem (statement only); Mean Value Theorems—Lagrange & Cauchy (statement only), Taylor’s theorem (without proof and problems in respect of direct use and applications of the theorem only), Expansions of functions by Taylor and Maclaurin series. Maclaurin’s expansion in infinite series of the functions: log (1+x), ex , sinx, cosx, (a+x)n , n being a negative integer or a fraction L’Hospital’s Rule (statement only) and related problems. Reduction Formula and applications, Rectification
Calculus of Functions of Several Variables: Introduction of Function of several variables and examples, Knowledge of limit and continuity, Partial derivative & related problems. Homogeneous Functions and Euler’s Theorem (statement only) & Problems upto 3 variables, Chain rules and related problems, Differentiation of implicit functions & related problems, Total differentials and related problems, Maxima, minima and saddle points – definition, condition of extrema & problems for two variables. Lagrange’s multiplier method – problems related to two variables only, Line Integral, Double Integrals, Triple Integral – Discussion w.r.t. different types of limits and problems; Moment of Inertia, Centre of Gravity, Jacobian – Definition and related problems for two variables, Applications to areas and volumes, surface area of revolution.
Three Dimensional Geometry (Cartesian): Direction Cosine, Direction Ratio; Equation of a Plane (general form, normal form and intercept form); Equation of a Straight Line passing through one point and two points; Pair of intersecting planes representing a straight line, Elementary ideas of surfaces like sphere, Right Circular Cone and Right Circular Cylinder (through Geometrical configuration) and equations in standard forms.
Vector Calculus: Scalar and Vector fields – Definition and Terminologies; Products: dot, cross, box, vector triple product, Gradient, directional derivative, divergence, curl. (with problems), Tangent planes and normals and related problems, Statements of Green’s theorem, Divergence theorem, Stokes’ theorem with applications