This is a textbook for the first year of college calculus.
An edition of First-year calculus
(1968)
First-year calculus
1
by Einar Hille
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Book Details
First Sentence
"This is a textbook for the first year of college calculus."
Table of Contents
1.
Preliminaries
Page 1
1.1.
Sets
Page 1
1.2.
Real numbers
Page 3
1.3.
The coordinate line
Page 7
1.4.
Inequalities on the line
Page 9
1.5.
The coordinate plane
Page 13
1.6.
The straight line
Page 15
1.7.
Pairs of lines
Page 19
1.8.
Some subsets of the plane and their analytic representations
Page 22
1.9.
Symmetry and boundedness
Page 25
1.10.
Functions
Page 29
1.11.
More on functions
Page 34
1.12.
Combining functions
Page 38
1.13.
The axiom of induction
Page 41
2.
The Limit of a Function
Page 44
2.1.
Preliminary intuitive discussion
Page 44
2.2.
Definition of limit
Page 46
2.3.
Limit theorems
Page 54
2.4.
More theorems on limits
Page 61
2.5.
One-sided limits
Page 64
2.6.
Continuity
Page 68
3.
Differentiation
Page 74
3.1.
Motivation
Page 74
3.2.
The derivative
Page 76
3.3.
Some rules for differentiation
Page 78
3.4.
The chain rule
Page 83
3.5.
Derivatives of higher orders; another notation
Page 87
3.6.
The tangent line
Page 88
3.7.
The mean-value theorem
Page 91
3.8.
The derivative and monotonicity
Page 94
3.9.
Extreme values
Page 98
3.10.
Concavity and points of inflection
Page 104
3.11.
Rates of change
Page 107
3.12.
The notations of Newton on Leibniz
Page 113
4.
The Integral of a Continuous Function
Page 115
4.1.
Motivation
Page 115
4.2.
Definition of the definite integral
Page 118
4.3.
The function F(x) = inegral x to a f(t) dt
Page 121
4.4.
The fundamental theorem of integral calculus
Page 126
4.5.
Some properties of the integral
Page 132
5.
The Logarithm and Exponential Functions
Page 137
5.1.
The logarithm function
Page 137
5.2.
The exponential funciton
Page 145
5.3.
The functions p^x and log base p x; estimating e
Page 154
5.4.
Integration by parts
Page 161
5.5.
(Optional) The equation y'(x) + P(x)y(x) = Q(x)
Page 165
6.
The Trigonometric Functions
Page 169
6.1.
The least-upper-bound axiom
Page 169
6.2.
Circular arc length
Page 171
6.3.
The trigonometric functions
Page 178
6.4.
Differentiating the trigonometric functions
Page 185
6.5.
Integrating the trigonometric functions
Page 189
6.6.
The inverse trigonometric functions
Page 193
6.7.
(Optional) The differential equation y" + ay' + by = 0
Page 200
7.
Some Analytic Geometry
Page 202
7.1.
The distance between a point and a line; translations
Page 202
7.2.
The conic sections
Page 206
7.3.
The parabola
Page 206
7.4.
The ellipse
Page 214
7.5.
The hyperbola
Page 219
7.6.
Polar coordinates
Page 225
7.7.
Curves given parametrically
Page 229
7.8.
Rotations: eliminating the xy-term
Page 236
7.9.
Curvature
Page 242
8.
The Technique of Integration
Page 248
8.1.
A short table of integrals; review
Page 248
8.2.
The method of partial fractions
Page 252
8.3.
Integration by substitution
Page 260
8.4.
The indefinite integral notation
Page 266
8.5.
Approximate integration
Page 267
9.
Integration as an Averaging Process: Applications
Page 272
9.1.
The average value of a function
Page 272
9.2.
Area and volume
Page 276
9.3.
Area in polar coordinates
Page 288
9.4.
Arc length
Page 291
9.5.
Area of a surface of revolution
Page 299
9.6.
(Optional) The notion of work
Page 303
9.7.
(Optional) Moment of area
Page 307
9.8.
(Optional) Some remarks on the integral as a set function
Page 312
10.
Sequences and Series
Page 316
10.1.
Sequences of real numbers
Page 316
10.2.
The limit of a sequence
Page 320
10.3.
Some important limits
Page 330
10.4.
Some comments on notation
Page 333
10.5.
Infinite series
Page 334
10.6.
Taylor series
Page 339
10.7.
The logarithm and the arc tangent; computing pi
Page 346
10.8.
Series with nonnegative terms
Page 349
10.9.
Convergence and absolute convergence; alternating series
Page 357
10.10.
Power series
Page 359
11.
Problems on Supplementary Topics
Page 369
11.1.
Limits as x -> +- infinity
Page 369
11.2.
L'Hospital's rule (0/0)
Page 371
11.3.
Infinite limits
Page 373
11.4.
L'Hospital's rule (infinity/infinity)
Page 375
11.5.
Improper integrals
Page 377
11.6.
More on the hyperbolic cosine and hyperbolic sine
Page 382
A.
Appendix
Page 385
A.1.
The intermediate-value theorem
Page 385
A.2.
The maximum-minimum theorem
Page 386
A.3.
The integrability of continuous functions
Page 387
B.
Table of integrals
Page 391
C.
Answers to starred excercises
Page 393
D.
Index
Page 411
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