1 Analytic Geometry
1.1 Conic Sections
1.2 Parabola
1.3 Ellipse
1.4 Hyperbola
1.5 Translation of Axes
1.6 Eccentricity of a Conic Section
1.7 Asymptotes of a Hyperbola
1.8 Rotation of Axes
1.9 Conics in Polar Coordinates
1.10 Practice Problems
2 Differential Calculus of Functions of one Variable
2.1 Tangents and Normals to General Plane Curves
2.2 Limits
2.3 Properties of Limits
2.4 Limits at Infinity
2.5 Infinite Limits
2.6 Indeterminate Forms of Limits
2.6.1 Indeterminate types of the form \(0^0\), \(\infty ^0\) and \(1^{\infty }\)
2.7 Continuity of a Function
2.7.1 Right and Left \(-\) hand continuity
2.7.2 Continuity in an Interval
2.7.3 Theorems on Continuity
2.7.4 Piece-wise Continuity
2.8 Mean Value Theorems
2.8.1 Rolle’s Theorem
2.8.2 Mean Value Theorem
2.8.3 Generalized Mean Value Theorem
2.8.4 Extended Mean Value Theorem
2.9 Series
2.10 Infinite Sequences
2.11 Infinite Series
2.12 Tests for Convergence of Series
2.13 Power Series
2.14 Series Expansion of Functions
2.15 Curvature
2.16 Theorem (Curvature in Rectangular Coordinates)
2.17 Curvature when a Curve is given in Parametric Form
2.18 The Circle of Curvature
2.19 Intrinsic Coordinates
2.20 Practice Problems
3 Integral Calculus of Functions of one Variable
3.1 Indefinite Integrals
3.2 Integration Formulas
3.2.1 Integration by Substitution
3.2.2 Integration by Parts
3.2.3 Reduction Formula
3.2.4 The Method of Undetermined Coefficient
3.3 Trigonometric Integrals
3.3.1 Method of Trigonometric Substitution
3.4 The Hyperbolic Substitution
3.5 Integration by Partial Fractions
3.6 Radicals of Polynomial Expression
3.7 Rational Functions of Trigonometric Functions
3.8 The Definite Integrals
3.9 The Fundamental Theorem of Calculus
3.10 Applications of Definite Integrals
3.10.1 Area Under the Curve
3.10.2 Area Between Two Curves
3.10.3 Volume of the Solid of Revolution
3.11 The Length of a Plane Curve
3.12 Area of Surface of Revolution
3.13 Practice Problems
4 Vector Analysis
4.1 Three Dimensional Coordinate System
4.2 Vectors
4.3 Direction of a Vector
4.4 The Scalar or Dot Product of Two Vectors
4.5 Projections
4.6 Lines in Three Dimensional Space
4.7 The Vector Product of Two Vectors
4.8 Triple Products
4.9 Planes
4.10 Vector Functions and their Derivatives
4.11 Derivatives and Integrals of Vector Functions
4.12 Arc Length
4.13 Curvature
4.14 The Normal and Binormal Vectors
4.15 Motion in Space: Velocity and Acceleration
4.16 Tangential and Normal Components of Acceleration
4.17 Practice Problems
5 Differential Calculus of Functions of Several Variables
5.1 Level Curves
5.2 Limits and Continuity
5.3 Continuity
5.4 Partial Derivatives
5.5 The Chain Rule
5.5.1 The Chain Rule of Functions of Three Variations
5.6 Higher Order Derivatives
5.6.1 The mixed Derivative Theorem
5.6.2 The Implicit Function Theorem
5.7 Applications of Partial Derivatives
5.8 Maximum and Minimum Values
5.9 Tangent Planes
5.10 Linear Approximations
5.11 Predicting Change (Differentials)
5.12 Euler’s Theorem of Homogeneous Functions
5.13 Practice Problems
6 Ordinary Differential Equations
6.1 First Order Equations of the First Degree
6.1.1 Separable Equations
6.1.2 Homogeneous Equations
6.1.3 Linear Equations
6.1.4 Exact Equations
6.1.5 Integrating Factors
6.1.6 Bernoulli Equations
6.2 Linear Equations of the Second Order
6.2.1 Homogeneous Equations with Constant Coefficients
6.2.2 The Method of Undetermined Coefficients
6.2.3 The Method of Variation of Parameters
6.3 Solution by Series
6.4 Practice Problems
References