Calculus I

Syllabus

Unit I: Limit Continuity and Derivatives (5 hrs)
1.1 Introduction
1.2 Limit, continuity and differentiability
1.3 Higher order derivatives by Leibnitz method. 

Unit II: Applications of Derivatives (8 hrs)
2.1 Mean value theorems: Rolle’s theorem, Lagrange’s 
Theorem (Geometrical interpretation and verification) and 
applications
2.2 Higher order mean value theorem: Taylor’s Series, 
Maclaurin’s Series expansion of function.
2.3 Asymptotes to Cartesian curves up to four degrees.
2.4 Curve tracing in Cartesian form and parametric form
2.5 Curvature

Unit III: Integral Calculus (6 hrs)
3.1 Introduction
3.2 Review on Indefinite Integral and fundamental theorem of integral calculus.
3.3 Definite integral and its properties
3.4 Improper Integrals; comparison test.
3.5 Reduction formula, Beta Gamma function

Unit IV: Application of Integral (6 hrs)
4.1 Application of integrals for finding area beneath a curve 
and between two curves and arc length
4.2 Surface and volume of solid of revolution in the plane for 
Cartesian and parametric curves.

Unit V: Partial Differentiation (3 hrs)
5.1 Introduction 
5.2 Partial Derivatives
5.3 Homogeneous function and Euler’s theorem for the 
function of two and three variables
5.4 Total Derivatives and Differentiation of Implicit 
functions.

Unit VI: Application of Partial Differentiation (4 hrs)
6.1 Extrema of functions of two and three variables. 
6.2 Lagrange’s method of undetermined Multipliers (up to 2 
multipliers)

Unit VII: First Order Ordinary Differential Equations 
(6 hrs)
7.1 Review of separable, homogeneous and exact differential 
equation with engineering applications 
7.2 Linear, Bernoulli equation and Riccati’s equation with 
engineering application.
7.3 Mathematical modeling of engineering problems using 
first order equation.

Unit VIII: Second Order Ordinary Differential Equations 
(7 hrs)
8.1 Second order Homogeneous ODE with constant and 
variable coefficients, Euler-Cauchy equation. 
8.2 Existence and uniqueness of solutions, Wronskian and 
general solutions for solving ODE. 
8.3 Non-homogeneous second order ODE and Solution by 
undetermined coefficients and variation of parameters and 
engineering application