Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra

Preface
Contents
1. INTRODUCTION
	Part 1. Historical Introduction
		I 1.1 The two basic concepts of calculus
		I 1.2 Historical background
		I 1.3 'The method of exhaustion for the area of a parabolic segment
		*I 1.4 Exercises
		I 1.5 A critical analysis of Archimedes' method
		I 1.6 The approach to calculus to be used in this book
	Part 2. Some Basic Concepts of the Theory of Sets
		I 2.1 Introduction to set theory
		I 2.2 Notations for designating sets
		I 2.3 Subsets
		I 2.4 Unions, intersections, complements
		I 2.5 Exercises
	Part 3. A Set of Axioms for the Real-Number System
		I 3.1 Introduction
		I 3.2 The field axioms
		*I 3.3 Exercises
		I 3.4 The order axioms
		*I 3.5 Exercises
		I 3.6 Integers and rational numbers
		I 3.7 Geometric interpretation of real numbers as points on a line
		I 3.8 Upper bound of a set, maximum element, least upper bound (supremum)
		I 3.9 The least-Upper-bound axiom (completeness axiom)
		I 3.10 The Archimedean property of the real-number system
		I 3.11 Fundamental properties of the supremum and infimum
		*I 3.12 Exercises
		*I 3.13 Existence of square roots of nonnegative real numbers
		*I 3.14 Roots of higher order. Rational powers
		*I 3.15 Representation of real numbers by decimals
	Part 4. Mathematical Induction, Summation Notation, and Related Topics
		I 4.1 An example of a proof by mathematical induction
		I 4.2 The principle of mathematical induction
		*I 4.3 The well-ordering principle
		I 4.4 Exercises
		*I 4.5 Proof of the well-ordering principle
		I 4.6 The summation notation
		I 4.7 Exercises
		I 4.8 Absolute values and the triangle inequality
		I 4.9 Exercises
		*I 4.10 Miscellaneous exercises involving induction
1. THE CONCEPTS OF INTEGRAL CALCULUS
	1.1 The basic ideas of Cartesian geometry
	1.2 Functions. Informal description and examples
	*1.3 Functions. Formal definition as a set of ordered pairs
	1.4 More examples of real functions
	1.5 Exercises
	1.6 The concept of area as a set function
	1.7 Exercises
	1.8 Intervals and ordinate sets
	1.9 Partitions and step functions
	1.10 Sum and product of step functions
	1.11 Exercises
	1.12 The definition of the integral for step functions
	1.13 Properties of the integral of a step function
	1.14 Other notations for integrals
	1.15 Exercises
	1.16 The integral of more general functions
	1.17 Upper and lower integrals
	1.18 The area of an ordinate set expressed as an integral
	1.19 Informal remarks on the theory and technique of integration
	1.20 Monotonic and piecewise monotonic functions. Definitions and examples
	1.21 Integrability of bounded monotonic functions
	1.22 Calculation of the integral of a bounded monotonic function
	1.23 Calculation of the integral $\int_0^b x^p dx$ when p is a positive integer
	1.24 The basic properties of the integral
	1.25 Integration of polynomials
	1.26 Exercises
	1.27 Proofs of the basic properties of the integral
2. SOME APPLICATIONS OF INTEGRATION
	2.1 Introduction
	2.2 The area of a region between two graphs expressed as an integral
	2.3 Worked examples
	2.4 Exercises
	2.5 The trigonometric functions
	2.6 Integration formulas for the sine and cosine
	2.7 A geometric description of the sine and cosine functions
	2.8 Exercises
	2.9 Polar coordinates
	2.10 The integral for area in polar coordinates
	2.11 Exercises
	2.12 Application of integration to the calculation of volume
	2.13 Exercises
	2.14 Application of integration to the concept of work
	2.15 Exercises
	2.16 Average value of a function
	2.17 Exercises
	2.18 The integral as a function of the upper limit. Indefinite integrals
	2.19 Exercises
3. CONTINUOUS FUNCTIONS
	3.1 Informal description of continuity
	3.2 The definition of the limit of a function
	3.3 The definition of continuity of a function
	3.4 The basic limit theorems. More examples of continuous functions
	3.5 Proofs of the basic limit theorems
	3.6 Exercises
	3.7 Composite functions and continuity
	3.8 Exercises
	3.9 Balzano's theorem for continuous functions
	3.10 The intermediate-value theorem for continuous functions
	3.11 Exercises
	3.12 The process of inversion
	3.13 Properties of functions preserved by inversion
	3.14 Inverses of piecewise monotonic functions
	3.15 Exercises
	3.16 The extreme-value theorem for continuous functions
	3.17 The small-span theorem for continuous functions (uniform continuity)
	3.18 The integrability theorem for continuous functions
	3.19 Mean-value theorems for integrals of continuous functions
	3.20 Exercises
4. DIFFERENTIAL CALCULUS
	4.1 Historical introduction
	4.2 A problem involving velocity
	4.3 The derivative of a function
	4.4 Examples of derivatives
	4.5 The algebra of derivatives
	4.6 Exercises
	4.7 Geometric interpretation of the derivative as a slope
	4.8 Other notations for derivatives
	4.9 Exercises
	4.10 The chain rule for differentiating composite fu nctions
	4.11 Applications of the chain rule. Related rates and implicit differentiation
	4.12 Exercises
	4.13 Applications of differentiation to extreme values of functions
	4.14 The mean-value theorem for derivatives
	4.15 Exercises
	4.16 Applications of the mean-value theorem to geometric properties of functions
	4.17 Second-derivative test for extrema
	4.18 Curve sketching
	4.19 Exercises
	4.20 Worked examples of extremum problems
	4.21 Exercises
	*4.22 Partial derivatives
	*4.23 Exercises
5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
	5.1 The derivative of an indefinite integral. The first fundamental theorem of calculus
	5.2 The zero-derivative theorem
	5.3 Primitive functions and the second fundamental theorem of calculus
	5.4 Properties of a function deduced from properties of its derivative
	5.5 Exercises
	5.6 The Leibniz notation for primitives
	5.7 Integration by substitution
	5.8 Exercises
	5.9 Integration by parts
	5.10 Exercises
	*5.11 Miscellaneous review exercises
6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
	6.1 Introduction
	6.2 Motivation for the definition of the natural logarithm as an integral
	6.3 The definition of the logarithm. Basic properties
	6.4 The graph of the natural logarithm
	6.5 Consequences of the functional equation L(ab) = L(a) + L(b)
	6.6 Logarithms referred to any positive base b ≠ 1
	6.7 Differentiation and integration formulas involving logarithms
	6.8 Logarithmic differentiation
	6.9 Exercises
	6.10 Polynomial approximations to the logarithm
	6.11 Exercises
	6.12 The exponential function
	6.13 Exponentials expressed as powers of e
	6.14 The definition of e^x for arbitrary real x
	6.15 The definition of a^x for a > 0 and x real
	6.16 Differentiation and integration formulas involving exponentials
	6.17 Exercises
	6.18 The hyperbolic functions
	6.19 Exercises
	6.20 Derivatives of inverse functions
	6.21 Inverses of the trigonometric functions
	6.22 Exercises
	6.23 Integration by partial fractions
	6.24 Integrals which can be transformed into integrals of rational functions
	6.25 Exercises
	6.26 Miscellaneous review exercises
7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS
	7.1 Introduction
	7.2 The Taylor polynomials generated by a function
	7.3 Calculus of Taylor polynomials
	7.4 Exercises
	7.5 Taylor's formula with remainder
	7.6 Estimates for the error in Taylor' s formula
	*7.7 Other forms of the remainder in Taylor' s formula
	7.8 Exercises
	7.9 Further remarks on the error in Taylor' s formula. The o-notation
	7.10 Applications to indeterminate forms
	7.11 Exercises
	7.12 L'Hôpital's rule for the indeterminate form 0/0
	7.13 Exercises
	7.14 The symbols +∞ and -∞. Extension of L'Hôpital's rule
	7.15 Infinite limits
	7.16 The behavior of log x and e^x for large x
	7.17 Exercises
8. INTRODUCTION TO DIFFERENTIAL EQUATIONS
	8.1 Introduction
	8.2 Terminology and notation
	8.3 A first-order differential equation for the exponential function
	8.4 First-order linear differential equations
	8.5 Exercises
	8.6 Some physical problems leading to first-order linear differential equations
	8.7 Exercises
	8.8 Linear equations of second order with constant coefficients
	8.9 Existence of solutions of the equation y" + by= 0
	8.10 Reduction of the general equation to the special case y" + by = 0
	8.11 Uniqueness theorem for the equation y" + by = 0
	8.12 Complete solution of the equation y" + by = 0
	8.13 Complete solution of the equation y" + ay' + by = 0
	8.14 Exercises
	8.15 Nonhomogeneous linear equations of second order with constant coefficients
	8.16 Special methods for determining a particular solution of the nonhomogeneous equation y" + ay' + by = R
	8.17 Exercises
	8.18 Examples of physical problems leading to linear second-order equations with constant coefficients
	8.19 Exercises
	8.20 Remarks concerning nonlinear differential equations
	8.21 Integral curves and direction fields
	8.22 Exercises
	8.23 First-order separable equations
	8.24 Exercises
	8.25 Homogeneous first-order equations
	8.26 Exercises
	8.27 Some geometrical and physical problems leading to first-order equations
	8.28 Miscellaneous review exercises
9. COMPLEX NUMBERS
	9.1 Historical introduction
	9.2 Definitions and field properties
	9.3 The complex numbers as an extension of the real numbers
	9.4 The imaginary unit i
	9.5 Geometric interpretation. Modulus and argument
	9.6 Exercises
	9.7 Complex exponentials
	9.8 Complex-valued functions
	9.9 Examples of differentiation and integration formulas
	9.10 Exercises
10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS
	10.1 Zeno's paradox
	10.2 Sequences
	10.3 Monotonic sequences of real numbers
	10.4 Exercises
	10.5 Infinite series
	10.6 The linearity property of convergent series
	10.7 Telescoping series
	10.8 The geometric series
	10.9 Exercises
	*10.10 Exercises on decimal expansions
	10.11 Tests for convergence
	10.12 Comparison tests for series of nonnegative terms
	10.13 The integral test
	10.14 Exercises
	10.15 The root test and the ratio test for series of nonnegative terms
	10.16 Exercises
	10.17 Alternating series
	10.18 Conditional and absolute convergence
	10.19 The convergence tests of Dirichlet and Abel
	10.20 Exercises
	*10.21 Rearrangements of series
	10.22 Miscellaneous review exercises
	10.23 Improper integrals
	10.24 Exercises
11. SEQUENCES AND SERIES OF FUNCTIONS
	11.1 Pointwise convergence of sequences of functions
	11.2 Uniform convergence of sequences of functions
	11.3 Uniform convergence and continuity
	11.4 Uniform convergence and integration
	11.5 A sufficient condition for uniform convergence
	11.6 Power series. Circle of convergence
	11.7 Exercises
	11.8 Properties of functions represented by real power series
	11.9 The Taylor' s series generated by a function
	11.10 A sufficient condition for convergence of a Taylor's series
	11.11 Power-series expansions for the exponential and trigonometric functions
	*11.12 Bernstein's theorem
	11.13 Exercises
	11.14 Power series and differential equations
	11.15 The binomial series
	11.16 Exercises
12. VECTOR ALGEBRA
	12.1 Historical introduction
	12.2 The vector space of n-tuples of real numbers
	12.3 Geometric interpretation for n \leq 3
	12.4 Exercises
	12.5 The dot product
	12.6 Length or norm of a vector
	12.7 Orthogonality of vectors
	12.8 Exercises
	12.9 Projections. Angle between vectors in n-space
	12.10 The unit coordinate vectors
	12.11 Exercises
	12.12 The linear span of a finite set of vectors
	12.13 Linear independence
	12.14 Bases
	12.15 Exercises
	12.16 The vector space V_n(C) of n-tuples of complex numbers
	12.17 Exercises
13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
	13.1 Introduction
	13.2 Lines in n-space
	13.3 Some simple properties of straight lines
	13.4 Lines and vector-valued functions
	13.5 Exercises
	13.6 Planes in Euclidean n-space
	13.7 Planes and vector-valued functions
	13.8 Exercises
	13.9 The cross product
	13.10 The cross product expressed as a determinant
	13.11 Exercises
	13.12 The scalar triple product
	13.13 Cramer's rule for solving a system of three linear equations
	13.14 Exercises
	13.15 Normal vectors to planes
	13.16 Linear Cartesian equations for planes
	13.17 Exercises
	13.18 The conic sections
	13.19 Eccentricity of conic sections
	13.20 Polar equations for conic sections
	13.21 Exercises
	13.22 Conic sections symmetric about the origin
	13.23 Cartesian equations for the conic sections
	13.24 Exercises
	13.25 Miscellaneous exercises on conic sections
14. CALCULUS OF VECTOR-VALUED FUNCTIONS
	14.1 Vector-valued functions of a real variable
	14.2 Algebraic operations. Components
	14.3 Limits, derivatives, and integrals
	14.4 Exercises
	14.5 Applications to curves. Tangency
	14.6 Applications to curvilinear motion. Velocity, speed, and acceleration
	14.7 Exercises
	14.8 The unit tangent, the principal normal, and the osculating plane of a curve
	14.9 Exercises
	14.10 The definition of arc length
	14.11 Additivity of arc length
	14.12 The arc-length function
	14.13 Exercises
	14.14 Curvature of a curve
	14.15 Exercises
	14.16 Velocity and acceleration in polar coordinates
	14.17 Plane motion with radial acceleration
	14.18 Cylindrical coordinates
	14.19 Exercises
	14.20 Applications to planetary motion
	14.21 Miscellaneous review exercises
15. LINEAR SPACES
	15.1 Introduction
	15.2 The definition of a linear space
	15.3 Examples of linear spaces
	15.4 Elementary consequence of the axioms
	15.5 Exercises
	15.6 Subspaces of a linear space
	15.7 Dependent and independent sets in a linear space
	15.8 Bases and dimension
	15.9 Exercises
	15.10 Inner products, Euclich planes, norms
	15.11 Orthogonality in a Euclidean space
	15.12 Exercises
	15.13 Construction of orthogonal sets. The Gram-Schmidt process
	15.14 Orthogonal complements. Projections
	15.15 Best approximation of elements in a Euclidean space by elements in a finite dimensional subspace
	15.16 Exercises
16. LINEAR TRANSFORMATIONS AND MATRICES
	16.1 Linear transformations
	16.2 Null space and range
	16.3 Nullity and rank
	16.4 Exercises
	16.5 Algebraic operations on linear transformations
	16.6 Inverses
	16.7 One-to-one linear transformations
	16.8 Exercises
	16.9 Linear transformations with prescribed values
	16.10 Matrix representations of linear transformations
	16.11 Construction of a matrix representation in diagonal form
	16.12 Exercises
	16.13 Linear spaces of matrices
	16.14 Isomorphism between linear transformations and matrices
	16.15 Multiplication of matrices
	16.16 Exercises
	16.17 Systems of linear equations
	16.18 Computation techniques
	16.19 Inverses of square matrices
	16.20 Exercises
	16.21 Miscellaneous exercises on matrices
Answers to exercises
I1.4-I4.7
I4.9-1.15
1.26-2.8
2.11-2.17
2.19-3.6
3.8-4.6
4.9
4.12
4.15-4.19
4.21-4.23
5.5-5.8
5.10-6.9
6.17
6.25
6.26-7.8
7.11-8.5
8.7-8.14
8.17-8.19
8.22-8.28
9.6-9.10
10.4-10.14
10.16-10.22
10.24-11.13
11.16
12.4-12.11
12.15-13.5
13.8-13.17
13.21-13.24
13.25-14.4
14.7-14.13
14.15-14.19
14.21-15.9
16.12-16.4
16.8
16.12
16.16
16.20
16.21
Index