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Calculus

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Overview

The fundamental idea of calculus is to study change by studying “instantaneous” change, by which we mean changes over tiny intervals of time. It turns out that such changes tend to be lots simpler than changes over finite intervals of time. This means they are lots easier to model. Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. It has two basic applications: differential calculus and integral calculus. The simplest introduction to differential calculus involves an explicit series of numbers. Differential calculus essentially lets one calculate the rate of change. So say variable ‘y’; is a function of variable ‘x’. Integrals, together with derivatives, are the fundamental objects of calculus. The process of computing an integral is called integration. Among the disciplines that utilize calculus include physics, engineering, economics, statistics, and medicine. It is used to create

Syllabus

Weeks Weekly Lecture Topics (Module Titles)1 Day 1 Module 1 : Transcendental functions- Inverse trigonometric function and logarithmic function Day 2 Module 2 : The exponential functions Day 3 Module 3 : Hyperbolic functions and inverse hyperbolicfunctions Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
2 Day 1 Module 4 : Successive differentiation- n th derivatives Day 2 Module 5 : Taylor series and Maclaurin series expansions
Day 3 Module 6 : Concavity, points of inflexion and multiplepoints Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
3 Day 1 Module 7 : Curvature and evolutes Day 2 Module 8 : Asymptotes Day 3 Module 9 : Tracing of curves Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
4 Day 1 Module 10 : Partial differentiation Day 2 Module 11 : Young’s theorem and Schwarz’s theorem Day 3 Module 12 : Partial differentiation of implicit, homogeneous and composite functions Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
5 Day 1 Module 13 : Integration by successive reduction Day 2 Module 14 : Definite integrals and their evaluation Day 3 Module 15 : Application of integration- rectification and quadrature Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
6 Day 1 Module 16 : Application of integration- Volume of solids ofrevolution Day 2 Module 17 : Infinite series Day 3 Module 18 : Series of arbitrary terms Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
7 Day 1 Module 19 : Tests for positive term series Day 2 Module 20 : Fourier series representation of a periodicfunction Day 3 Module 21 : Fourier series of a function having period 2C Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
8 Day 1 Module 22 : Fourier series representation of even and oddfunctions Day 2 Module 23: Half range Fourier series expansion offunctions Day 3 Module 24 : Differential equations- Introductions Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
9 Day 1 Module 25 : Variable separable and homogeneousdifferential equations Day 2 Module 26 : Linear and exact differential equationsObjectives Day 3 Module 27: Equations reducible to exact differentialequations Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
10 Day 1 Module 28 : Application of differential equations- orthogonal trajectories
Day 2 Module 29 : Differential equation of the first order but not of the first degree Day 3 Module 30 : Higher order differential equations Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
11 Day 1 Module 31 : Second order differential equations with variable coefficients
Day 2 Module 32 : Method of undetermined coefficients, method of variation of parameters and normal form Day 3 Module 33: Cauchy’s differential equation and particular integrals Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
12 Day 1 Module 34 : Partial differential equations- introduction
Day 2 Module 35 : Non-linear partial differential equations of first order Day 3 Module 36 : Charpit’s method Day 4 Interaction based on the three Modules covered. Day 5 Deadline for submitting assignments.
13 Day 1 Module 37 : Linear partial differential equations of higher order with constant coefficients Day 2 Module 38 : separation of variables. Day 3 Module 39 : Application of partial differential equations- Vibrations of a stretched string Day 4 Module 40 : Application of partial differential equations- One dimensional heat flow Day 5 Interaction based on the three Modules covered.
14 Day 1 Module 41 : Two dimensional heat equation and wave equation
Day 2 Module 42 : Laplace transform and inverse Laplace transform Day 3 Module 43 : Laplace transform- some special results Day 4 Module 44 : Application of Laplace transforms andLaplace transforms of some useful functions Day 5 Deadline for submitting assignments.

Dr. Mansoor P