Enginineering Mathematics I (RMK)

Cover
About Pearson
Title
Copyright
Dedication
Contents
Roadmap to the Syllabus
Preface
About the Authors
Chapter 1 Differential Calculus
	1.0 Introduction
	1.1 Function
		1.1.1 Methods of representing a function
		1.1.2 Arrow diagram
		1.1.3 Domain and range
		1.1.4 Intervals
		1.1.5 Solution of quadratic inequalities
			Worked Examples
		1.1.6 The vertical line test
			Worked Examples
			Absolute value function
			Worked Examples
		1.1.7 Even and odd function
			Worked Examples
			Exercise 1.1
			Answers to Exercise 1.1
	1.2 New functions from given functions
		Worked Examples
		1.2.1 Inverse function
		1.2.2 Procedure to find inverse function
			Worked examples
			Exercise 1.2
			Answers to Exercise 1.2
	1.3 Limit of a function
		Worked Examples
		1.3.1 One-sided limits
			Worked Examples
		1.3.2 Extended real number system
		1.3.3 Infinite limits
			Worked Examples
		1.3.4 Limits with trigonometric functions
			Worked Examples
		1.3.5 Limits with exponential and logarithmic functions
			Worked Examples
			Exercise 1.3
			Answers to Exercise 1.3
	1.4 Continuity
		Worked Examples
		1.4.1 Types of discontinuity
			Exercise 1.4
			Answers to Exercise 1.4
	1.5 Derivative
		1.5.1 One-side derivative
			Worked Examples
		1.5.2 Derivative as a function
		1.5.3 Leibnitz notation
		1.5.4 Differentiability in an interval
		1.5.5 Higher order derivatives
		1.5.6 Power function
		1.5.7 Exponential function
			Worked Examples
			Exercise 1.5
			Answers to Exercise 1.5
	1.6 Derivative Problems Based on General Rules of Differentiation
		Worked Examples
	1.7 Chain Rule or Derivative of Composite Function
		Exercise 1.7
		Answers to Exercise 1.7
	1.8 Derivative of Inverse Functions
		1.8.1 Derivative of inverse trigonometric functions
			Worked Examples
			Exercise 1.8
			Answers to Exercise 1.8
	1.9 Implicit Differentiation
		Worked Examples
	1.10 Logarithmic Differentiation
		Worked Examples
	1.11 Differentiation of Parametric Equations
		Worked Examples
		Exercise 1.11
		Answers to Exercise 1.11
	1.12 Hyperbolic Functions
		1.12.1 Hyperbolic identities
		1.12.2 Derivatives of hyperbolic functions
		1.12.3 Inverse hyperbolic functions
		1.12.4 Derivatives of inverse hyperbolic functions
			Worked Examples
			Exercise 1.12
			Answers to Exercise 1.12
	1.13 Geometrical interpretation of derivative
		1.13.1 Geometrical interpretation of derivative
		1.13.2 Equation of the tangent and the normal to the curve y = f (x)
			Worked Examples
			Exercise 1.13
			Answers to Exercise 1.13
	1.14 Maxima and Minima of a Function of One Variable
		1.14.1 Geometrical meaning
		1.14.2 Tests for maxima and minima
			Worked Examples
			Exercise 1.14
			Answers to Exercise 1.14
			Part A – Questions and Answers
Chapter 2 Functions of Several Variables
	2.0 Introduction
	2.1 Limit and Continuity
		Neighbourhood of a point in the plane
		Limit of a function
		Repeated limits or iterated limits
		Continuity of a function
		Worked Examples
		Exercise 2.1
		Answers to Exercise 2.1
	2.2 Partial Derivatives
		2.2.1 Geometrical meaning of dz/dx, dz/dy
		2.2.2 Partial derivatives of higher order
		2.2.3 Homogeneous functions and Euler’s theorem
			Worked Examples
		2.2.4 Total derivatives
			Worked Examples
			Exercise 2.2
			Answers to Exercise 2.2
	2.3 Jacobians
		2.3.1 Properties of Jacobians
			Worked Examples
		2.3.2 Jacobian of implicit functions
			Exercise 2.3
			Answers to Exercise 2.3
	2.4 Taylor’s Expansion for Function of Two Variables
		Worked Examples
		Exercise 2.4
		Answers to Exercise 2.4
	2.5 Maxima and Minima for Functions of Two Variables
		2.5.1 Necessary conditions for maximum or minimum
		2.5.2 Sufficient conditions for extreme values of f (x, y)
		2.5.3 Working rule to find maxima and minima of f (x, y)
			Worked Examples
		2.5.4 Constrained maxima and minima
			Lagrange’s method of (undetermined) multiplier
		2.5.5 Method to decide maxima or minima
			Worked Examples
			Exercise 2.5
			Answers to Exercise 2.5
			Part A – Questions and Answers
Chapter 3 Integral Calculus
	3.0 Introduction
	3.1 Definite Integral (Rectangle method of finding area)
		Worked Examples
	3.2 Indefinite Integral
		Worked Examples
		3.2.1 Integration by substitution
			Exercise 3.1
			Answers to Exercise 3.1
		3.2.2 Special type: reciprocal form
			Worked Examples
		3.2.3 Integration of trigonometric functions of products and powers
			Worked Examples
			Type 1
			Worked Examples
			Type 2
			Worked Examples
			Type 3
			Worked Examples
			Exercise 3.2
			Answers to Exercise 3.2
		3.2.3 Integration of irrational functions by trigonometric substitutions
			Worked Examples
			Exercise 3.3
			Answers to Exercise 3.3
		3.2.4 Integration of rational algebraic functions
			Worked Examples
		3.2.5 Integration by partial fractions
			Exercise 3.4
			Answers to Exercise 3.4
			Worked Examples
			Type 1
			Worked Examples
			Type 2
			Worked Examples
			Exercise 3.5
			Answers to Exercise 3.5
	3.3 Integration by Parts
		Worked Examples
		3.3.1 Bernoulli’s formula
		3.3.2 Special integrals
			Worked Examples
		3.3.3 Reduction formula
			Worked Examples
			Exercise 3.6
			Answers to Exercise 3.6
		3.3.4 Properties of definite integrals
			Worked Examples
		3.3.5 Leibnitz rule for derivative of integral
			Worked Examples
			Exercise 3.7
			Answers to Exercise 3.7
		3.3.6 Definite integral (fx) dx as a limit of a sum
			Worked Examples
			Exercise 3.8
			Answers to Exercise 3.8
	3.4 Improper Integrals
		3.4.1 Kinds of improper integrals and their convergence
			Worked Examples
			Exercise 3.9
			Answers to Exercise 3.9
		3.4.2 Tests of convergence of improper integrals
			Worked Examples
			Exercise 3.10
			Answers to Exercise 3.10
			Part A – Questions and Answers
Chapter 4 Multiple Integrals
	4.1 Double Integration
		4.1.1 Double integrals in cartesian coordinates
		4.1.2 Evaluation of double integrals
			Worked Examples
			Exercise 4.1
			Answers to Exercise 4.1
		4.1.3 Change of order of integration
			Worked Examples
			Exercise 4.2
			Answers to Exercise 4.2
		4.1.4 Double integral in polar coordinates
			Worked Examples
		4.1.5 Change of variables in double integral
			Worked Examples
			Exercise 4.3
			Answers to Exercise 4.3
		4.1.6 Area as double integral
			Worked Examples
			Exercise 4.4
			Answers to Exercise 4.4
			Worked Examples
			Exercise 4.4(A)
			Answers to Exercise 4.4(A)
	4.2 Triple Integral in Cartesian Coordinates
		Worked Examples
		4.2.1 Change of variables in triple integral
		4.2.2 Volume as triple integral
			Worked Examples
			Exercise 4.5
			Answers to Exercise 4.5
			Part A – Questions and Answers
Chapter 5 Differential Equations
	5.0 Introduction
	5.1 Linear Differential Equation with Constant Coefficients
		5.1.1 Complementary function
		5.1.2 Particular integral
			Type 1
			Worked Examples
			Type 2
			Worked Examples
			Type 3
			Worked Examples
			Type 4
			Type 5
			Worked Examples
			Exercise 5.1
			Answers to Exercise 5.1
	5.2 Linear Differential Equations with Variable Coefficients
		5.2.1 Cauchy’s homogeneous linear differential equations
			Worked Examples
		5.2.2 Legendre’s linear differential equation
			Worked Examples
			Exercise 5.2
			Answers to Exercise 5.2
	5.3 Simultaneous Linear Differential Equations with Constant Coefficients
		Worked Examples
		Exercise 5.3
		Answers to Exercise 5.3
	5.4 Method of Variation of Parameters
		5.4.1 Working rule
			Worked Examples
			Exercise 5.4
			Answers to Exercise 5.4
	5.5 Method of Undetermined Coefficients
		Worked Examples
		Exercise 5.5
		Answers to Exercise 5.5
		Part A – Questions and Answers
Appendix: Important Formulae
Index