Calculus

PART I: Prologue
1 Basic Properties of Numbers
2 Numbers of Various Sorts

PART II: Foundations
3 Functions
Appendix. Ordered Pairs
4 Graphs
Appendix 1. Vectors
Appendix 2. The Conic Sections
Appendix 3. Polar Coordinates
5. Limits
6 Continuous Functions
7 Three Hard Theorems
8 Least Upper Bounds
Appendix. Uniform Continuity

PART III: Derivatives and Integrals
9 Derivatives
10 Differentiation
11 Significance of the Derivative
Appendix. Convexity and Concavity
12 Inverse Functions
Appendix. Parametric Representation of Curves
13 Integrals
Appendix. Riemann Sums
14 The Fundamental Theorem of Calculus
15 The Trigonometric Functions
*16 π is Irrational
*17 Planetary Motion
18 The Logarithm and Exponential Functions
19 Integration in Elementary Terms
Appendix. The Cosmopolitan Integral

PART IV: Infinite Sequences and Infinite Series
20 Approximation by Polynomial Functions
*21 e is Transcendental
22 Infinite Sequences
23 Infinite Series
24 Uniform Convergence and Power Series
25 Complex Numbers
26 Complex Functions
27 Complex Power Series

PART V: Epilogue
28 Fields
29 Construction of the Real Numbers
30 Uniqueness of the Real Numbers

Suggested Reading
Answers (to selected problems)
Glossary of Symbols
Index

AalbmcndoE, 276
Abel, Niels Henrik, 410, 522
Abel summable, 522
Abel's formula for summation by parts, 392
Abel's Lemma, 393
Abel's test, 496
Abel's Theorem, 522
Absolute value, 11
of a complex number, 534
Absolutely convergent, 480, 556
Absolutely summable, 480
Acceleration, 161
Acta Eruditorum, 148
Addition, 3
associative law for, 9
commutative law for, 9
of complex numbers, 531
geometric interpretation of, 535
of vector-valued functions, 245
of vectors, 75
Addition formula
for arcsin, 317
for arctan, 317
for cos, 314
for sin, 313, 314
for tan, 317
Additive identity
existence of, 9
for vectors, 76
Additive inverses
existence of, 9
Algebra, Fundamental Theorem of, 377, 538, 548, 567
Algebraic functions, 363
Algebraic number, 442
Algebraist's real numbers, 598
Almost lower bound, 142
Almost upper bound, 142
Analyst's real numbers, 598
Angle, 303
directed, 303
Antidiagonal, 243
Arabic numerals, multiplication of, 8
Arc length, 278, 283
Arccos, 311
derivative of, 311
Archimedes, 138, 141, 263
Archimedian property
for the rational numbers, 584
for the real numbers, 138
Archimedian spiral, 85, 249
Arcsec, 320, 383
Arcsin, 310
addition formula for, 317
derivative of, 311
Taylor series for, 517
Arctan, 311
addition formula for, 317
derivative of, 311
Taylor polynomials for, 413, 420
remainder term for, 426
Area, 253, 258
Arg cosh, 353
Arg sinh, 353
Arg tanh, 353
Argument, 536
Argument function, 542
discontinuities of, 546
Argument of the hyperbolic functions, 353
Arithmetic mean, 33
Arrow, 75, 76
\x arrow sin(x2)", 45
Associative law
for addition, 9
of vectors, 76
for multiplication, 9
Average velocity, 152
Axis
horizontal, 57
imaginary, 534
real, 533
vertical, 57
Bacon, Francis, vi
Basic properties of numbers, 3
\Bent graphs", 149
Bernoulli, Jakob, 148, 574
Bernoulli numbers, 572
Bernoulli polynomials, 575
Bernoulli's inequality, 32
Big game hunting, mathematical theory of, 552
Binary operation, 581
Binomial coefficient, 27, 437
Binomial series, 495, 518
Binomial theorem, 28
Bisection argument, 142, 552
Bohr, Harold, 394
Bolzano-Weierstrass Theorem, 458, 469, 552
Bound
almost lower, 142
almost upper, 142
greatest lower, 134
least upper, 133, 584
lower, 134
upper, 133
Boundedness criterion, 474
Bounded above, 122, 133, 457, 584
Bounded below, 134, 457
Bourbaki, Nicholas, 148
Cantor, Georg, 448
Cardioid, 89, 250
Cartesian coordinates, 84
Cauchy
Condensation Theorem, 496
criterion, 473
form of the remainder, 437
Mean Value Theorem, 204
product, 493, 513
sequence, 459, 571
equivalence of, 599
Cauchy-Hadamard formula, 569
Cauchy-Schwarz inequality, 281
Cavalieri, Bonaventura, 275
Cesaro summable, 493
Chain Rule, 174 ff.
proof of, 178
Change, rate of, 152
Characteristic (of a field), 586
Circle, 65
\f circleg", 44
unit, 66
Circle of convergence, 560
Classical notation
for derivatives, 154-156, 162, 167, 187, 241-242
for integrals, 265
Cleio, 186
Closed interval, 57
Closed rectangle, 547
Closure under addition, 9
Closure under multiplication, 9
Commutative law
for addition, 9
of vectors, 76
for multiplication, 9
Comparison test, 474
limit, 475
Comparison Theorem, Sturm, 323
Complete induction, 23
Complete ordered field, 584, 603
Completing the square, 17, 379
Complex analysis, 565
Complex function
continuous, 545, 546
differentiable, 550
graph of, 542
limit of, 542
nondifferentiable, 551
Taylor series for, 563
Complex nth root, 536
Complex numbers, 526, 531
absolute value of, 534
addition of, 531
geometric interpretation of, 535
geometric interpretation of, 533-534
imaginary part of, 531
infinite sequence of, 555
infinite series of, 555-557
logarithm of, 570
modulus of, 534
multiplication of, 531
geometric interpretation of, 535-536
real part of, 531
Complex plane, 533
Complex power series, 557
circle of convergence of, 560
radius of convergence of, 559
Complex-valued functions, 541
Composition of functions, 44
Concave function, 220
Conditionally convergent series, 481
Cone, 80
generating line of, 80
surface area of, 404
Conic sections, 80; see also Ellipse, Hyperbola, Parabola
Conjugate, 534, 539
Conjugate function, 541
Constant function, 43
Construction of the real numbers, 588 ff.
Continued fraction, 462
Continuous, uniformly, 144
Continuous at a, 115, 545
Continuous function, 115, 118, 546
nowhere differentiable, 159, 509
Continuous on (a, b), 118
Continuous on [a, b], 118
Contraction, 466
Contraction lemma, 466
Converge
pointwise, 502
uniformly, 502, 506
Convergent sequence, 453, 555
Convergent series, 472, 556
absolutely, 480, 556
conditionally, 481
Convex function, 219
strictly, 228
weakly, 228
Convex subset of the plane, 229, 553
Cooling, Newton's law of, 356
Coordinate
first, 57
second, 57
Coordinate system, 57
cartesian, 84
origin of, 57
Coordinates
polar, 84
\Corner", 60
Cos, 303-304, 306, 321-322, 563
addition formula for, 314
derivative of, 172, 307
inverse of, see Arccos
Taylor polynomials for, 413
remainder term for, 424
Cosh, 353
Cosine, hyperbolic, 353
Cot, 310
derivative of, 310
Countable, 449
Counting numbers, 21
Critical point, 190
Critical value, 190
Csc, 310
derivative of, 310
Cubic equation, general solution, 528-529
Curve
parameterized, tangent line of, 246
parametric representation of, 244
reparameterization of, 247
Cycloid, 250
Darboux's Theorem, 214
De Moivre's Theorem, 536
Decimal expansion, 73, 492, 599
Decreasing function, 195
Decreasing sequence, 457
Dedekind, Richard, 38
Defined implicitly, 241
Definite integral, 365
DEFINITION, 47
Definition, recursive, 23
Degree (of a polynomial), 42
Degree measurement, 304-305
Delicate ratio test, 493
Delicate root test, 493
Dense, 140
Derivative, 149 ff., 151
classical notation for, 154-156, 162, 167, 187, 241-242
higher-order, 161
\infinite", 158
left-hand, 156
Leibnizian notation for, see Derivative, classical notation for
logarithmic, 351
\negative infinity", 158
of f , 151
of f at a, 151
of vector-valued function, 246
right-hand, 156
Schwarzian, 184
second, 161
Schwarz, 438
Derivative of quotient, incantation for, 171
Descartes, René, 84
Determinant, 79
of vector-valued functions, 248
Diagonal, 233
Difference operator, 573
Differentiable, 151, 550
Differential equation, 292, 300, 321, 322-323, 356, 361, 438-440
initial conditions for, 440
Differentiation, 168 ff.
implicit, 241
logarithmic, 351
Differentiation operator, 573
Dini's Theorem, 524
Directed angle, 303
Dirichlet's test, 495
Disc method, 402
Discontinuities of a nondecreasing function, 450
Discontinuity, removable, 121
Disraeli, Benjamin, 2
Distance, 58, 534
shortest between two points, 278
Distributive law, 9
Diverge, 453, 556
Division, 6
Division by zero, 6
Domain, 40, 41, 47, 601
Dot product
of vectors, 78
of vector-valued functions, 246
Double intersection, 165
Double root, 185
Durège, 38
e, 343
irrationality of, 429
relation with π, 448, 564
transcendentality of, 444
value of, 344, 426
Eccentricity of ellipse, 87
Elementary function, 363
Ellipse, 66, 82
axes of, 87
eccentricity of, 87
equation in polar coordinates, 86-87
focus point of, 66, 86
major axis of, 87
minor axis of, 87
Ellipsoid of revolution, 405
Empty collection, 23
Entire function, 567
Epsilon, 18
Equal up to order n, 418
Equality, order of, 418
Equations, differential, see Differential equations
Equivalent Cauchy sequences, 599
Etymology lesson, 82
Euler, Leonhard, 575
Euler's number, 463
Euler-Maclaurin Summation Formula, 576
Even function, 51, 199
Even number, 25
Eventually inside, 555
Exhaustion, method of, 141
Exp, 343 ff., 563
classical approach to, 357
elementary definition of, 468
Taylor polynomials for, 413
remainder term for, 426
Expansion, decimal, 73, 492, 599
Extension of a function, 115-116
Factorial, 23
Factorials, table of, 432
Factorization into primes, 31
Fibonacci, 32
Fibonacci Association, 32
Fibonacci Quarterly, The, 32
Fibonacci sequence, 32, 521, 572
Field, 581
characteristic of, 586
complete ordered, 584, 603
ordered, 583
First coordinate, 57
First Fundamental Theorem of Calculus, 285
Fixed point of a function, 465
Focus point, 66, 86
Force, as vector, 76
Four leaf clover, 88
Fourier series, 318, 320, 323
Fraction, continued, 462
Function, 39, 47
absolute value, 541
argument, 542
discontinuities of, 546
complex valued, 541
composition of, 44
concave, 220
conjugate, 541
constant, 43
continuous, 115 ff.
convex, 219
critical point of, 190
critical value of, 190
decreasing, 195
derivative of, 149 ff.
differentiable, 151, 550
elementary, 363
entire, 567
even, 51, 199
exponential, 343-344
extension of, 115-116
fixed point of, 465
from A to B, 601
from real numbers to the plane, 244
graphs of, 57-65, 198, 542
hyperbolic, 353
identity, 43
imaginary part, 541
implicitly defined, 241
increasing, 195
integrable, 258
integral of, 258
inverse, 231 ff.
linear, 58
local maximum point of, 189
local minimum point of, 189
local strict maximum point of, 218
logarithm, 341, 346
maximum point of, 188
maximum value of, 188
minimum value of, 188
most general definition of, 601
negative part of, 51
nondecreasing, 243
nonincreasing, 243
nonnegative, 51
notation for, 40, 45
odd, 51, 199
one-one, 230
periodic, 71, 164, 298
polynomial, 42
positive part of, 51
power, 60
product of, 43
quotient of, 43
rational, 42
real part, 541
real-valued, 541
\reasonable", 68, 118, 149, 180
regulated, 524
square root, 546-547
step, 278
strict maximum point of, 218
sum of, 43
trigonometric, 303 ff.
value at x, 40
vector-valued, 244
Fundamental Theorem of Algebra, 377, 538, 549, 567
Fundamental Theorem of Calculus
First, 285
Second, 289
Gabriel, 408
Galileo Galilei, 148, 164
Gamma function, 394, 444
Generating line, of a cone, 80
Geometric mean, 33
Geometric series, 473
Global property, 123
Goes to, \x goes to sin(x2)", 45
Graph of polynomial function, 197-198
Graph sketching, 196-201
Graphs, 57-65, 85 ff., 90-91, 197-198, 542
Gravitation, 330
Greatest lower bound, 134
Grin and bear it, 385-386
Gronwall's inequality, 356
Grow
at the same rate as, 361
faster than, 361
Hadamard, see Cauchy-Hadamard formula
Half-life (of radioactive substance), 356
Hermite, Charles, 443
High-school student's real numbers, 599
Higher-order derivatives, 161
Hilbert, David, 443
Horizontal axis, 57
Hyperbola, 67, 82
equation in polar coordinates, 88
Hyperbolic cosine, 353
Hyperbolic functions, 353
Hyperbolic sine, 353
Hyperbolic spiral, 316
Hyperbolic tangent, 353
Identity
additive, 9
multiplicative, 9
Identity function, 43
Identity operator, 573
Imaginary axis, 534
Imaginary part function, 541
Imaginary part of a complex number, 531
Implicit differentiation, 241
Implicitly defined, 241
Improper integral, 301-302, 397-399
Incantation for derivative of quotient, 171
Increasing at a, 217
Increasing function, 195
Increasing sequence, 457
Indefinite integral, 365
Indefinite integrals, short table of, 365-366
Induction, mathematical, 21
complete, 23
Inductive set of real numbers, 34
Inequalities, 9
in an ordered field, 584
Inequality
Bernoulli's, 32
Cauchy-Schwarz, 281
geometric-arithmetic mean, 33
Gronwall's, 356
Jensen's, 228
Liouville's, 448
Schwarz, 17, 33, 281
triangle, 71
Young's, 276
Infimum, 134
\Infinite" derivative, 158
Infinite intervals, 57
Infinite product, 329, 395, 497
Infinite sequence, 452, 555
Infinite series, 471
multiplication of, 486
Infinite sum, 430, 471
Infinite trumpet, 408
Infinitely many primes, 32
\Infinitely small", 155, 264
Infinity, 57
minus, 57
Inflection point, 225
Initial conditions for differential equations, 440
Instantaneous speed, 152
Instantaneous velocity, 152
Integer, 25
Integrable, 258
Integral, 258
classical notation for, 264-265
definite, 365
improper, 301-302, 397-399
indefinite, 365
short table of, 365-366
Leibnizian notation for, see Integral, classical notation for
lower, 295
Mean Value Theorem for, 277
Second Mean Value Theorem for, 391
upper, 295
Integral form of the remainder, 423
Integral sign, 258
Integral test, 478
Integration
by parts, 366 ff.
by substitution, 369 ff.
limits of, 258
of rational functions, 377 ff.
Interest (finance), 355
Intermediate Value Theorem, 124, 131, 135, 299
Interpolation, Lagrange, 49
Intersection of sets, 43
Interval, 56
closed, 57
infinite, 57
open, 56; see also Nested Intervals Theorem
Inverse
additive, 9
multiplicative, 9
Inverse of a function, 231 ff.
Inverse square law, 330
Inverses of trigonometric functions, see Trigonometric functions
Irrational numbers, 25
Isomorphic fields, 602
Isomorphism, 602
Jensen's inequality, 228
Johnson, Samuel, 607
Jump, 60
Kepler, Johannes, 330
Kepler's laws of planetary motion, 330
Lagrange form of the remainder, 423, 436
Lagrange interpolation formula, 49
Large negative, 64
Least upper bound, 133 ff., 584
Least upper bound property, 135
Lebesgue, see Riemann-Lebesgue Lemma
Left-hand derivative, 156
Leibniz, Gottfried Wilhelm, 155, 264
Leibniz's formula, 184
Leibniz's Theorem, 481
Leibnizian notation for derivatives, 154-156, 167, 187, 241
for higher order derivatives, 162
Lemma, 102
Lemniscate, 89
Length, 278, 283
L'Hôpital, Marquis de, 148
L'Hôpital's Rule, 204, 213-214
Limit, 90 ff., 98, 542
at infinity, 106
\does not exist", 101
from above, 106
from below, 106
of a sequence, 453
of vector-valued function, 246, 252
uniqueness of, 100
Limit comparison test, 475
Limit of integration, 258
Limit point, 469, 552
Limit superior, 143, 467
Lindemann, Ferdinand von, 447
Line, real, 56
Line, tangent, see Tangent line
Linear functions, 58
Liouville, Joseph, 448
Liouville's inequality, 448
Liouville's Theorem, 567
Lipschitz of order α, 210
Local maximum point of function, 189
higher-order derivative test for, 417
second derivative test for, 201
Local minimum point of function, 189; see also Local maximum point
Local property, 109, 123, 166
Local strict maximum point, 218
Log, 341, 346
Taylor polynomials for, 413
remainder term for, 427
Logarithm
classical approach to, 357
Napierian, 358
of a complex number, 570
to the base 10, 339
Logarithmic derivative, 351
Lower bound, 134
almost, 142
greatest, 134
Lower integral, 295
Lower limit of integration, 258
Lower sum, 254
Lowest terms, 73
Maclaurin, see Euler-Maclaurin summation formula
Major axis of ellipse, 87
Mass, rate of change of, 152
Mathematical induction, 21
complete, 23
Maximum of two numbers, 16
Maximum point of a function, 188
local, 189; see also Local maximum point
local strict, 218
strict, 218
Maximum value of function, 188
Mean
arithmetic, 33
geometric, 33
Mean Value Theorem, 193, 194
Cauchy, 204
for integrals, 277
Second, 391
Method of exhaustion, 141
Minimum of function, 188
Minimum of two numbers, 16
Minimum point of a function, local, 189; see also Local minimum point
Minor axis of ellipse, 87
Minus infinity, 57
Mirifici logarithmonum canonis description, 358
Modulus of a complex number, 534
Mollerup, Johannes, 394
Multiplication, 5
associative law for, 9
closure under, 9
commutative law for, 9
of arabic numerals, 8
of complex numbers, 531
geometric interpretation, 535-536
of function and vector-valued function, 245
of infinite series, 486
of number and vector, 77
of vectors, 77
Multiplicative identity, existence of, 9
Multiplicative inverses, existence of, 9
Multiplicity (of a root), 130, 185-186
Napier, John, 358
Napierian logarithm, 358
Natural numbers, 21, 34
Negative, large, 64
\Negative infinity", derivative, 158
Negative number, 9
Negative numbers, product of two, 7
Negative part of a function, 51
Nested Interval Theorem, 142
Newton, Isaac, 155, 276, 330
Newton's law of cooling, 356
Newton's laws of motion, 161
Newton's method, 464
Nondecreasing function, 243
Nondecreasing sequence, 457
Nondifferentiable complex functions, 551
Nonincreasing function, 243
Nonincreasing sequence, 457
Nonnegative function, 51
Nonnegative sequence, 474
Norm, 78, 252
Notational nonsense, 573
Nowhere differentiable continuous function, 509
nth root, 71, 536
existence of, 125, 536, 553
primitive, 540
Null set, 23
Number
algebraic, 442
complex, 526, 531
counting, 21
even, 25
imaginary, 526
irrational, 25
natural, 21, 34
odd, 25
prime, 31
rational, 25
real, 25, 534, 589
transcendental, 442
Numbers, basic properties of, 3
Odd function, 51, 199
Odd number, 25
One-one function, 230
Open interval, 56
\Or", 6
Order of equality, 418
Ordered field, 583
complete, 584
Ordered pair, 47 (footnote), 54
Origin (of a coordinate system), 57
Pair, 46
ordered, 47 (footnote), 54
Parabola, 60, 82
area under, 263
equation in polar coordinates, 88
Parallelogram, 76
Parameterized curve, tangent line of, 246
Parametric representation of a curve, 244
Partial fraction decomposition, 378
Partial sums, 471
Partition, 254
Parts
Abel's formula for summation by, 392
integration by, 366 ff.
Pascal's triangle, 27
\Peak", 61
Peak point, 458
Period of a function, 71, 164, 298
Periodic function, 71, 164, 298
Perpendicularity of lines, 70
Petard, H., 552
Pig, yellow, v, 375
Pigheaded, 186
Plane, 58
complex, 533
Planetary motion, Kepler's laws of, 330
Point, 56
Point of contact, 220
Point-slope form of equation of a line, 59, 70
Polar coordinates, 84 ff.
Polynomial function, 42
graph of, 61, 197 ff.
multiplicity of roots, 130, 185-186
Polynomials, Bernoulli, 575
Pope, Alexander, 330
Position, rate of change of, 152
Positive element of R, 593
Positive elements of an ordered field, 583
Positive number, 9
Positive part of a function, 51
Power functions, 60
Power series, 510
complex, 557
centered at a, 510, 564
Powers of 2, table of, 432
Prime number, 31
characteristic of a field, 586
infinitely many of, 32
unique factorization into, 31
Primitive, 363
Primitive nth root, 540
Principia, 276
Product, 5
Cauchy, 493, 513
infinite, 329, 395, 497
of function and vector-valued function, 245
of functions, 43
of number and vector, 77
of two negative numbers, 7
of vectors, 77
Pyramid
surface area of, 403
volume of, 407
Pythagorean theorem, 25, 58
π, 305
Archimedes' approximation of, 141
irrationality of, 326
relation to e, 448, 564
transcendentality of, 447
value of, 433
Viète's product for 2 /π, 329
Wallis' product for π/2, 395
Quaternions, 587
Quotient of functions, 43
Rabbits
growth of population, 32
Radian measure, 63, 304-305
Radioactive decay, 355-356
Radius of convergence of complex power series, 559
Rate of change of mass, 152
Rate of change of position, 152
Ratio test, 476
delicate, 493
Rational functions, 42
integration of, 377 ff.
Rational numbers, 25
Real axis, 533
Real line, 56
Real number (formal definition), 589
Real numbers, 25
algebraist's, 598
analyst's, 598
Archimedian property of, 138
construction of, 588 ff.
high-school student's, 599
inductive set of, 34
Real part function, 541
Real part of a complex number, 531
Real-valued function, 541
Rearrangement of a sequence, 483
\Reasonable" function, 68, 118, 149, 180
Rectangle, closed, 547
Recursive definition, 23
Reduction formulas, 377
Regulated function, 524
Remainder term for Taylor polynomials, 421, 423, 437
Removable discontinuity, 121
Reparameterization, 247
Revolution
ellipsoid of, 405
solid of, 402
Riemann sum, 282
Riemann-Lebesgue Lemma, 320, 391
Right-hand derivative, 156
Rising Sun Lemma, 143
Rolle, Michel, 186
Rolle's Theorem, 193
Root
multiplicity of, 130, 185-186
Root of a polynomial function, 50
double, 185; see also nth roots
Root test, 493
delicate, 493
Same sign, 12
Scalar, 78
Scalar product of vectors, 78
Schwarz, H. A., 217
Schwarz inequality, 17, 33, 281
Schwarz second derivative, 438
Schwarzian derivative, 184
Sec, 310
derivative of, 310
inverse of, see Arcsec
Secant line, 150
Second coordinate, 57
Second derivative, 161
Schwarz, 438
Second derivative test for maxima and minima, 201
Second Fundamental Theorem of Calculus, 289
Second Mean Value Theorem for Integrals, 391
Sequence
absolutely summable, 480
Cauchy, 459
complex, 571
equivalence of, 599
complex numbers, 555
convergent, 453
pointwise, 502
uniformly, 502
decreasing, 457
divergent, 453, 556
Fibonacci, 32, 521, 572
increasing, 457
infinite, 452, 555
limit of, 453
nondecreasing, 457
nonincreasing, 457
nonnegative, 474
rearrangement of, 483
summable, 472
Series
absolutely convergent, 480
conditionally convergent, 481
convergent, 472, 556
Fourier, 318, 320, 323
geometric, 473
power, 510, 557
Taylor, 511
Set, 22
empty, 23
Sets
intersection of, 43
notation for, 43-44
Shadow point, 143
Shell method, 403
Sigma, 24
Sign, 12
Simpson's rule, 401
Sin, 43, 303-304, 306, 321-322, 563
addition formula for, 313, 314
derivative of, 172, 307
inverse of, see Arcsin
Taylor polynomials for, 412
remainder term for, 424
Sine, hyperbolic, 353
Sine function, 43
Sinh, 353
Sketching graphs, 196-201
Skew field, 587
Slope of a straight line, 58
Solid of revolution, 402
Speed, instantaneous, 152
Spiral
Archimedian, 85, 249
hyperbolic, 316
Square root, 12, 527
existence of, 124
Square root function, 546-547
Square root in a field, 586
Squaring the circle, 447
Step function, 278
Stirling's Formula, 578
Straight line
analytic definition, 58
shortest distance between two points, 278
slope of, 58
Strict maximum point, 218
Strictly convex, 228
Sturm Comparison Theorem, 323
Subsequence, 458
Substitution
integration by, 369 ff.
world's sneakiest, 386
Substitution formula, 369
Subtraction, 5
Sum
finite, 3-4
infinite, 430, 471
lower, 254
of an infinite sequence, 472
of an infinite sequence of complex numbers, 555
of functions, 43
of vector-valued functions, 245
of vectors, 75
partial, 471
sigma notation for, 24
upper, 254
Sum of squares, 552
Summable, 472, 556
Abel, 522
absolutely, 480
Cesaro, 493
uniformly, 506
Summation by parts, Abel's formula for, 392
Supremum, 134
Surface area
of cone, 404
of pyramid, 403
of solid of revolution, 404
Swift, Jonathan, 580
Symmetry in graphs, 199
Tan, 310
derivative of, 310
inverse of, see Arctan
Taylor series for, 573
Tangent, hyperbolic, 353
Tangent line, 149, 151
of parameterized curve, 246
point of contact of, 220
\Tangent line", vertical, 158
Tanh, 353
Taylor polynomial, 412 ff.
remainder term of, 421, 423, 427; see also specific functions
Taylor series, 511
Taylor's Theorem, 424
Torus, 405
Transcendental number, 442
Trapezoid rule, 400
Triangle inequality, 71
Trichotomy law, 9
Trigonometric functions, 303 ff., see also cos, cot, csc, sec, sin, tan
integration of, 376-377
inverses of, 310 ff., see also arccos, arcsec, arcsin, arctan
Trumpet
infinite, 408
Truncation of a polynomial, 434
Two-time differentiable, 161
Uniform limit, 502
Uniformly continuous function, 144
Uniformly convergent sequence, 502
Uniformly convergent series, 506
Uniformly distributed sequence, 469
Uniformly summable, 506
Uniqueness
of factorization into primes, 31
of limits, 100
Unit circle, 66
Upper bound, 133, 584
almost, 142
least, 133
Upper integral, 295
Upper limit of integration, 258
Upper sum, 254
\Valley", 61
Value
absolute, see Absolute value
Value of f at x, 40
Vanishing condition, 473
Vector-valued functions, 244
Vector-valued functions
determinant of, 246
derivative of, 246
dot product of, 246
limit of, 246, 252
multiplication of function by, 245
sum of, 245
Vectors, 75
addition of, 75
as forces, 76
dot product of, 78
multiplication by numbers, 77
multiplication of, 77
scalar product of, 78
Velocity
average, 152
instantaneous, 152
Vertical axis, 57
Viète, François, 329
Volume, 402-403
of solid of revolution, 402
Wallis' product, 395
Weakly convex, 228
Weierstrass, see Bolzano-Weierstrass Theorem
Weierstrass M-test, 507
Well-ordering principle, 23
Wright, Edward, 388
Young's inequality, 276
Zahl, 25
Zero, division by, 6