Differential Calculus ( अवकल गणित )

Syllabus

PAPER I: DIFFERENTIAL CALCULUS

Unit I

Limit, continuity and differentiability of function of single variable, Cauchy’s definition, Heine’s definition, Uniform continuity, Borel’s theorem, boundedness theorem, Bolzano’s theorem, Intermediate value theorem, extreme value theorem, Darboux’s intermediate value theorem for derivatives, Chain rule, indeterminate forms.

Unit II

Rolle’s theorem, Lagrange and Cauchy Mean value theorems, mean value theorems of higher order, Taylor’s theorem with various forms of remainders, Successive differentiation, Leibnitz theorem, Maclaurin’s and Taylor’s series, Limit and Continuity of functions of two variables, Differentiation of function of two variables, Necessary and sufficient condition for differentiability of functions two variables.

Unit III

Partial differentiation, Euler’s theorem on homogeneous function, Schwarz’s and Young theorem, Taylor’s theorem for functions of two variables with examples, Maxima and minima for functions of two variables, Lagrange multiplier method, Jacobians, Inverse function theorem and implicit function theorem.

Unit IV

Tangents and normals, Asymptotes, Curvature, Envelops and evolutes, Tests forconcavity and convexity, Points of inflexion, Multiple points, Parametric representation of curves and tracing of parametric curves, Tracing of curves in Cartesian and Polar forms.

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