Analytic Geometry
and Calculus
Lecture Notes
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Contents
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
2.7 Continuity of a Function
2.8 Mean Value Theorems
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
3.3 Trigonometric Integrals
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
3.11 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
5.6 Higher Order Derivatives
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
6.2 Linear Equations of the Second Order
6.3 Solution by Series
6.4 Practice Problems
References
References
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
2.7 Continuity of a Function
2.8 Mean Value Theorems
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
3.3 Trigonometric Integrals
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
3.11 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
5.6 Higher Order Derivatives
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
6.2 Linear Equations of the Second Order
6.3 Solution by Series
6.4 Practice Problems
References
References