## 2.11.22

UNIT – I: Sequences, Series and Mean value theorems:
Sequences and Series: Convergences and divergence – Ratio test – Comparison tests – Integral test – Cauchy’s root test – Alternate series– Leibnitz’s rule.
Mean Value Theorems (without proofs): Rolle’s Theorem – Lagrange’s mean value theorem – Cauchy’s mean value theorem – Taylor’s and Maclaurin’s theorems with remainders, Problems and applications on the above theorem.

UNIT – II: Differential equations of first order and first degree
Linear differential equations– Bernoulli’s equations –Exact equations and equations reducible to exact form.
Applications: Newton’s Law of cooling– Law of natural growth and decay– Orthogonal trajectories– Electrical circuits

UNIT – III: Linear differential equations of higher order
Homogeneous and Non-homogeneousdifferential equations of higher order with constant coefficients – with non-homogeneous term of the type eax, sin ax, cos ax, polynomials in xn, eaxV(x) and xnV(x) – Method of Variation of parameters, Cauchy and Legendre’s linear equations.
Applications: LCR circuit, Simple Harmonic motion.h and decay– Orthogonal trajectories– Electrical circuits

UNIT – IV: Partial differentiation
Introduction – Homogeneous function – Euler’s theorem– Total derivative– Chain rule– Jacobian – Functional dependence –Taylor’s and MacLaurin’s series expansion of functions of two variables.
Applications: Maxima and Minima of functions of two variables without constraints and Lagrange’s method.

UNIT – V: Multiple integrals
Double and Triple integrals – Change of order of integration in double integrals – Change of variables to polar, cylindrical and spherical coordinates.
Applications: Finding Areas and Volumes.

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