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Up: Math 111 Calculus I
Previous: B. Proofs of Some
 A
 absolute value
 3.3
 6.1
 absolute value (derivative of)
 11.1
 acceleration
 15.2
 acceleration due to gravity
 15.2
 addition law
 9.1
 9.1
 addition of points
 4.1
 addition rule for area
 5.2
 addition theorem for area
 B.
 additivity of area
 5.1
 almost disjoint sets
 5.2
 Analyst
 10.1
 and (logical) operator
 3.1
 antiderivative
 12.4
 17.1
 antiderivative theorem
 12.4
 antidifferentiation, Maple commands
 17.1
 Apollonius (c 260170 B.C.)
 10.1
 approachable point
 10.2
 approximation
 6.2
 approximation (strong approximation theorem)
 6.2
 approximation to decimals
 6.2
 arc length
 9.1
 arccos
 14.6
 arccot
 14.6
 Archimedes(287212 B.C.)
 0.1
 2.2
 2.3
 2.4
 9.1
 9.2
 9.2
 Archimedes and
 0.3
 Archimedes and
 9.3
 Archimedes and
 0.3
 Archimedes and area of parabolic segment
 2.3
 arcsin
 14.6
 arctan
 14.6
 area = integral
 8.6
 area as a number
 1.
 area (assumptions about)
 5.1
 area between graphs
 8.6
 area function
 5.1
 5.2
 area of box
 1.
 area of circle
 8.5
 area of circular sector
 17.5
 area of ellipse
 8.5
 area of parabola
 2.1
 2.3
 area of parabolic segment
 2.3
 area of snowflake
 2.6
 area of triangle
 5.3
 area under power function
 7.2
 area, basic assumptions
 5.1
 Aristotle(384322 B.C.)
 1.
 10.1
 10.3
 Aryabhata (circa 510)
 9.1
 associative law for points
 4.1
 assumptions about area
 5.1
 assumptions: additivity of area
 5.1
 assumptions: symmetry invariance of area
 5.1
 assumptions: normalization of area
 5.1
 assumptions: translation invariance of area
 5.1
 average velocity
 10.3
 B
 Babylonians
 2.2
 2.2
 2.4
 4.3
 Babylonians
 0.3
 Banach, Stefan(18921945)
 5.1
 Berkeley, George (16851753)
 10.1
 Bernoulli, Jacob (16541705)
 2.2
 6.3
 6.7
 Bernoulli, Daniel(17001782)
 14.6
 between, is between and .
 14.1
 bijective
 14.3
 Bolzano, Bernard (17811848)
 10.3
 14.1
 bound for a function
 5.1
 bounded function
 5.1
 bounded set
 5.1
 box
 1.
 box (circumscribed)
 2.1
 Brouncker, William(16201684)
 5.5
 C
 Cantor, Georg (18451918)
 1.
 Cartesian product.
 3.2
 Cauchy, Augustin (17891857)
 3.4
 10.3
 Cavalieri, Bonaventura (15981647)
 2.5
 chain rule
 11.3
 change of scale for integrals
 8.5
 chord of an arc
 9.1
 circle (area of)
 8.5
 circle (definition of)
 4.3
 circle (unit)
 4.3
 circular sector (area of)
 17.5
 circumscribed box
 2.1
 circumscribed hexagon for snowflake
 2.6
 clockwise
 9.1
 closed interval
 1.
 codomain
 3.3
 commutative law for points
 4.1
 composition of continuous functions
 12.1
 composition of functions
 11.3
 composition problem
 11.3
 compound interest
 6.7
 computer calculation of area
 5.6
 congruence problem
 5.1
 conic section
 10.1
 conservation of energy
 15.2
 constant sequence
 6.4
 constructivists
 3.1
 continuity defined
 12.1
 continuity on a set
 12.1
 contrapositive
 3.1
 convergent sequence
 6.3
 6.3
 convex downward (spills water)
 15.3
 convex upward (holds water)
 15.3
 cosh
 14.6
 cosine defined
 9.1
 cosine, integral of
 9.3
 9.3
 cot
 11.2
 counterclockwise
 9.1
 critical point
 12.3
 critical point theorem I
 12.3
 critical point theorem II
 12.3
 critical set
 12.3
 csc
 11.2
 D
 decreasing function
 5.3
 derivative
 0.1
 0.1
 derivative defined
 10.3
 derivative (fractional)
 15.1
 derivative (higher order)
 15.1
 derivative of absolute value
 11.1
 derivative of
 14.6
 derivative of arccot
 14.6
 derivative of
 14.6
 derivative of
 14.6
 derivative of
 14.6
 derivative of
 14.6
 derivative of exponential
 14.6
 derivative of logarithm
 11.1
 derivative of order 0
 15.1
 derivative of powers
 11.1
 derivative of reciprocal
 11.2
 derivative of and
 11.1
 derivative of
 14.6
 derivative of trigonometric functions
 11.2
 derivative of
 14.6
 derivative (second)
 15.1
 derived function
 0.1
 Descartes, Rene (15961660)
 10.1
 difference of sets
 1.
 differential
 0.1
 0.1
 11.1
 differentiation
 0.1
 differentiation, logarithmic
 11.3
 Dirichlet,P.G.Lejenue(18051859)
 8.3
 discontinuous function
 12.1
 disjoint sets
 5.1
 disjoint (almost disjoint sets)
 5.2
 distance between numbers
 6.1
 distance between points
 4.3
 distributive law for points
 4.1
 divergent sequence
 6.3
 domain
 3.3
 double angle formulas
 9.1
 dummy index
 1.
 dummy variable
 3.4
 8.2
 10.2
 E
 e
 5.4
 e (numerical calculatioin of)
 6.7
 ellipse (area of)
 8.5
 Emperor Yu(c. 21st century B.C.)
 4.3
 end points of interval
 1.
 entertainment
 6.7
 entertainment
(Calculation of sines)
 9.1
 entertainment
(Discontinuous derivative problem)
 15.3
 entertainment
(Falling bodies problem)
 10.1
 entertainment (Archimedes sine integral)
 9.3
 entertainment (Calculate )
 5.4
 entertainment (Composition problem)
 11.3
 entertainment (Area of a triangle)
 5.3
 entertainment (Square root problem)
 0.3
 entertainment (Calculate )
 5.4
 entertainment (Pine Tree problem)
 2.4
 entertainment (Snowflake problem)
 2.6
 entertainment (Bernoulli's problem)
 2.2
 entertainment (Calculation of )
 0.3
 entertainment (Congruence problem)
 5.1
 equal functions
 3.3
 equal propositions
 3.1
 equivalent proposition
 3.1
 error function
 17.1
 etymology of corollary
 5.2
 etymology of sine
 9.1
 Euclid ( fl. c. 300 B.C.)
 1.
 2.4
 10.1
 10.3
 Euler, Leonard (17071783)
 3.3
 6.3
 Euler (summation notation)
 3.4
 even function
 12.4
 exponential function
 14.3
 14.4
 exponential function (derivative of)
 14.6
 exponential function (properties of)
 14.4
 extreme point
 12.3
 extreme point (local)
 12.3
 extreme value property
 12.3
 F
 fluxion
 0.1
 11.1
 Fourier, Joseph (17681830)
 8.1
 15.1
 fractional derivatives
 15.1
 function
 0.1
 function (bounded)
 5.1
 function (defined)
 3.3
 function (Euler's definition)
 3.3
 function (increasing)
 5.3
 functions equal
 3.3
 functions (operations on)
 8.2
 fundamental theorem of the calculus
 0.1
 fundamental theorem of calculus I
 16.
 fundamental theorem of calculus II
 16.
 Fundamental theorem of calculus (Leibniz statement of)
 16.
 G
 Galileo (15641642)
 10.1
 10.1
 geometric series
 6.6
 geometric series (finite)
 2.4
 Gougu
 4.3
 graph
 3.3
 graphs (area between)
 8.6
 gravity (acceleration due to)
 15.2
 H
 half angle formulas
 9.1
 Hausdorff, Felix (18681942)
 5.1
 height of box
 1.
 Heine, Heinrich Eduard (18211881)
 10.3
 Heron (sometime between 250B.C and 150 A.D.)
 13.3
 higher order derivatives
 15.1
 hyperbolic functions
 14.6
 I
 IbnalHaitham (circa 1000 A.D.)
 2.2
 image of under
 3.3
 image of
 3.3
 implies
 3.1
 increasing function
 5.3
 indefinite integral
 9.4
 induction
 3.5
 inequality rule for limits of functions
 10.2
 inequality rule for sequences
 6.4
 inequality theorem for integrals
 8.2
 inflection point
 15.3
 injective
 14.3
 inner snowflake
 2.6
 instantaneous velocity
 10.3
 integrable function
 8.1
 integral
 8.1
 integral (as area under a curve)
 8.6
 integral (Ei = exponential integral)
 17.1
 integral indefinite
 9.4
 integral of
 9.3
 9.3
 integral of

9.3
 9.3
 integral (Si = sine integral)
 17.1
 integral (change of scale in)
 8.5
 integration
 0.1
 integration by parts
 17.3
 integration by substitution
 17.4
 integration of rational functions
 17.7
 interior point
 10.2
 intermediate value property
 14.1
 14.1
 14.1
 intersection of sets
 1.
 2.6
 interval
 1.
 inverse function
 14.3
 inverse function theorem
 14.5
 K
 Katyayana (c. 600 BC or 500BC??)
 4.3
 kinetic energy
 15.2
 Koch, Helga von(18701924)
 2.6
 12.2
 L
 l'Hôpital,Guillaume François (16611701)
 6.3
 Lagrange, Joseph Louis(17361813)
 0.1
 11.1
 Leibniz, Gottfried (16461716)
 0.1
 0.1
 0.1
 3.3
 8.1
 8.2
 10.1
 11.1
 13.3
 15.1
 16.
 Leibniz (notation for sums)
 3.4
 Leibniz (proof of product rule)
 11.2
 length of arc
 9.1
 limit of a sequence
 6.3
 limit (onesided)
 13.1
 limits (infinite)
 13.1
 lines
 4.1
 Liouville, Joseph (18091882)
 15.1
 ln(2) calculation
 5.6
 ln(a)
 0.1
 local maximum
 12.3
 local minimum
 12.3
 localization rule
 11.1
 logarithm
 5.4
 5.4
 7.2
 logarithm (derivative of)
 11.1
 logarithmic differentiation
 11.3
 M
 Maple
 5.6
 17.1
 17.6
 Maple
leftsum
 5.6
 Maple
rightsum
 5.6
 Maple
average
 5.6
 Maple calculation of
 6.7
 Maple mypi
 9.2
 Maple routine
 9.2
 Maple sinsq
 9.2
 Maple, approximate integration
 5.6
 Maple, symbolic antidifferentiation
 17.1
 Maple: integration
 9.4
 maximum (local)
 12.3
 maximum of function
 12.3
 mean value theorem
 12.4
 mean value theorem for integrals
 16.
 Mercator, Nicolaus (16201687)
 5.5
 mesh of partition
 5.3
 minimum (local)
 12.3
 minimum of function
 12.3
 monotonic function
 5.3
 monotonic (piecewise)
 8.2
 monotonicity of area
 5.2
 B.
 N
 th root rule for sequences
 6.4
 th power theorem
 6.5
 Napier, John(15501632)
 5.4
 Newton, Isaac(16421727)
 0.1
 0.1
 10.3
 11.1
 Newton's law (F=ma)
 0.1
 nice function
 16.
 nonintegrable function
 8.3
 normalization property of area
 5.1
 not (logical) operator
 3.1
 nowhere differentiable function
 12.2
 Nullsequence rule
 6.4
 number as area
 1.
 O
 objects in a set
 1.
 odd function
 12.4
 open interval
 1.
 operations on functions
 8.2
 optical illusion
 9.1
 optimization problems
 13.2
 or (logical) operator
 3.1
 ordered pair
 3.2
 outer snowflake
 2.6
 P
 pair (ordered pair)
 3.2
 parabola (area of)
 2.1
 2.3
 parabolic segment (area of)
 2.3
 parallelogram
 4.1
 partition
 5.3
 partition regular
 5.3
 partitionsample sequence
 8.2
 Pascal, Blaise (16231662)
 2.2
 peicewise monotonic function
 8.2
 piecewise monotonic (example of nonpiecewise monotonic function)
 12.2
 pi
 0.1
 0.3
 0.3
 5.6
 8.5
 pi, computer calculation
 5.6
 point in plane
 1.
 point of inflection
 15.3
 points (in
)
 4.1
 points in a set
 1.
 points (addition of)
 4.1
 potential energy
 15.2
 power function
 14.4
 power (th power theorem)
 6.5
 powersums (list of)
 2.2
 prerequisites
 0.2
 product rule for derivatives
 11.2
 product rule for limits
 10.2
 product rule for limits of functions
 10.2
 product rule for sequences
 6.4
 product (Cartesian)
 3.2
 proofs without words
 2.2
 proposition
 3.1
 proposition (set defined by)
 3.2
 proposition form
 3.1
 propositions (equality of)
 3.1
 propositions (equivalence of)
 3.1
 Ptolemy, Claudius (fl 127151)
 9.1
 Pythagoras (f. 530510 B.C.)
 4.3
 Pythagoreans
 2.2
 Q
 quadratic formula
 4.3
 quotient rule for derivatives
 11.2
 quotient rule for limits
 10.2
 quotient rule for limits of functions
 10.2
 quotient rule for sequences
 6.4
 R
 Ramanujan, Srinivasa (18871920)
 0.3
 rate of change
 13.3
 rational functions
 17.7
 rational number (definition)
 1.
 real number
 0.1
 reciprocal (derivative of)
 11.2
 reflection
 2.3
 4.2
 reflection law for sin and cos
 9.1
 reflection theorem
 14.3
 regular partition
 5.3
 Rhind Papyrus
 0.3
 Riemann sum
 7.1
 Riemann, Bernhard (18261866)
 8.1
 right triangle
 5.3
 rituals for integration
 17.4
 17.4
 17.5
 17.6
 17.6
 Rolle, Michel (16521719)
 12.4
 Rolle's theorem
 12.4
 rotation
 4.2
 ruler function
 8.4
 S
 SaintVincent, Grégoire de, (15841667)
 6.6
 sample
 7.1
 Sarasa, Alfons Anton de (16181667)
 5.4
 schizophrenia
 3.1
 sec
 11.2
 sector (circular)
 17.5
 segment
 4.2
 segments
 4.1
 sequence
 3.3
 sequence (constant sequence rule)
 6.4
 sequence (convergent)
 6.3
 6.3
 sequence (divergent)
 6.3
 sequence (null sequence rule)
 6.4
 sequence (constant)
 6.4
 sequence (limit of)
 6.3
 sequence (product rule for)
 6.4
 sequence (sum rule for)
 6.4
 sequence (translate of)
 6.4
 sequence (translation rule for)
 6.4
 sequence (th root rule for)
 6.4
 sequences (inequality rule for)
 6.4
 sequences (quotient rule for)
 6.4
 set
 1.
 set (bounded)
 5.1
 sets defined by propositions
 3.2
 sine (definition)
 9.1
 sine (etymology of)
 9.1
 sine (integral of)
 9.3
 9.3
 sine integral (Si)
 17.1
 sinh
 14.6
 snowflake
 2.6
 12.2
 snowflake (area of)
 2.6
 spike function
 8.2
 8.2
 10.2
 square root problem
 0.3
 squeezing rule for limits of functions
 10.2
 squeezing rule for sequences
 6.4
 stretch
 8.5
 Stringham, Irving
 5.4
 subadditivity of area
 5.2
 subinterval of a partition
 5.3
 subset
 1.
 substitution in integrals
 17.6
 substitution (trigonometric)
 17.5
 sum rule for derivatives
 11.2
 sum rule for limits
 10.2
 sum rule for limits of functions
 10.2
 sum rule for sequences
 6.4
 sum theorem for derivatives
 17.2
 sum theorem for indefinite integrals
 9.4
 sum theorem for integrals
 8.2
 summation formula
 2.2
 2.2
 2.2
 2.2
 2.2
 surjective
 14.3
 symmetric set
 12.4
 symmetry invariance
 5.1
 symmetry of square
 4.2
 T
 tangent
 10.1
 10.1
 10.3
 tan
 11.2
 translate of a sequence
 6.4
 translate of set
 4.2
 translation invariance
 5.1
 translation rule for sequences
 6.4
 triangle inequality
 6.1
 triangle (area of)
 5.3
 triangle (right)
 5.3
 trick
 6.5
 trigonometric functions (definition)
 9.1
 trigonometric functions derivative of
 11.1
 11.2
 trigonometric identities
 9.1
 9.1
 9.1
 trigonometric substitution
 17.5
 U
 union of sets
 1.
 2.6
 uniqueness of inverses
 14.3
 Uniqueness theorem for convergence
 6.3
 unit circle
 4.3
 V
 velocity
 0.1
 10.1
 10.1
 velocity (average)
 10.3
 velocity (instantaneous)
 10.3
 W
 Weierstrass, Karl(18151897)
 6.1
 12.2
 width of box
 1.
 work
 0.1
 Z
 zeroarea set
 5.2
 8.6
 zeroth order derivative
 15.1
 Zu Chongzhi (429500 A.D.)
 0.3
Ray Mayer
20070907