2426:
49:
1064:, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one.
2445:
2437:
1107:
2892:
2828:, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms
1926:
2480:
can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function
2464:(the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a
900:
1769:
2817:. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.
2605:
1079:(or points where the derivative equals zero). However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the
498:
1774:
1660:
784:
2917:
425:
696:
4298:
2408:
2301:
1738:
4286:
1211:
729:
388:
352:
306:
2843:
If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of
1603:
1564:
662:
2188:
1986:
1278:
at −1 and +1. From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. This function has no global maximum or minimum.
2691:
2332:
2159:
1693:
2126:
2016:
2072:
2043:
2375:
2254:
1460:
2211:
1523:
789:
2352:
2233:
2100:
1949:
1761:
1500:
1480:
749:
4408:
4293:
3350:
4276:
4271:
2452:
For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a
1921:{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}
4281:
4266:
3380:
1098:, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is largest (or smallest).
3568:
3274:
4261:
1072:
2745:, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-
2485:
defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the
3878:
3632:
3187:
3151:
3119:
2503:
1029:. Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.
621:, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:
2626:
If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a
4444:
3430:
4376:
4235:
2993:
2746:
2425:
3790:
3706:
3510:
4371:
4303:
3928:
3783:
3751:
3327:
3311:
1040:
domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded
4439:
4004:
3696:
2477:
2420:
31:
3981:
3336:
4094:
4032:
3827:
3701:
3373:
3099:
430:
1609:
4403:
3580:
3558:
3307:
2075:
1088:
1076:
4388:
3768:
3590:
132:
4154:
2930:
of {−4.49, 4.49}, approximately, because it has 2 global minimum values of approximately −0.217 at
3773:
3543:
3135:
3013:
2486:
151:
3331:
4192:
4139:
2998:
1222:
1128:
1057:
754:
3600:
4308:
4079:
3627:
3366:
2627:
1068:
42:
38:
2902:
4074:
3746:
2473:
4202:
4084:
3905:
3853:
3659:
3637:
3505:
3003:
2962:
2954:
2631:
2494:
2079:
1041:
393:
163:
89:
81:
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4328:
4187:
4099:
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1080:
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69:
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or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a
2381:
4144:
3763:
3610:
3271:
2988:
2966:
170:
101:
4164:
4089:
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3933:
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3450:
2490:
1187:
701:
360:
319:
273:
1570:
1531:
1462:
feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where
628:
4434:
4393:
4217:
4149:
3971:
3948:
3822:
3815:
3718:
3533:
3425:
3257:
3183:
3147:
3115:
3008:
2958:
2167:
1956:
1263:
618:
128:
84:
are, respectively, the largest and smallest value taken by the function. Known generically as
2667:
2308:
2132:
1666:
4351:
4134:
4047:
4027:
3958:
3868:
3810:
3802:
3736:
3649:
3410:
3405:
3167:
2766:
2650:
2105:
1992:
1275:
1176:
113:
895:{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}
4413:
4398:
4182:
4037:
4017:
3986:
3963:
3943:
3837:
3493:
3440:
3278:
2983:
2734:
2661:
2051:
2022:
1084:
48:
2862:) of the set occasionally has a minimum and a maximum, in which case they are called the
2360:
2239:
1445:
1294:= 0 that cannot be found by taking derivatives, because the derivative does not exist at
2497:). In two and more dimensions, this argument fails. This is illustrated by the function
2196:
1505:
4323:
4222:
4069:
4022:
3923:
3726:
3179:
3104:
2844:
2337:
2218:
2085:
1934:
1746:
1485:
1465:
734:
2444:
4428:
4197:
4052:
3938:
3642:
3617:
2753:
2730:
2429:
1037:
4207:
4177:
4042:
3605:
3018:
2465:
511:
136:
2938:
of {−1.43, 1.43}, approximately, because their global minima occur at
3455:
3397:
3143:
3111:
2946:
2821:
2775:
2643:
2436:
1045:
140:
4172:
4104:
3858:
3731:
3595:
3585:
3528:
2448:
Counterexample: The red dot shows a local minimum that is not a global minimum
147:
124:
117:
3229:
3204:
2456:
maximum are similar to those of a function with only one variable. The first
4366:
4114:
4109:
3420:
3171:
1095:
17:
1345:
Infinitely many local maxima and minima, but no global maximum or minimum.
1106:
2760:(i.e., one that is smaller than all others) should not be confused with a
4361:
3863:
3741:
3389:
2970:
2742:
2718:
3343:
4212:
3465:
2950:
2884:
2738:
2600:{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}
2432:, a counterexample to some criteria of local maxima of the 19th century
4381:
3445:
3251:
3249:
3037:
2934: = ±4.49. However, the normalised sinc function (blue) has
2610:
whose only critical point is at (0,0), which is a local minimum with
1075:
states that local extrema in the interior of a domain must occur at
116:
was one of the first mathematicians to propose a general technique,
2891:
2468:. For use of these conditions to solve for a maximum, the function
3460:
3337:
Application of Maxima and Minima with sub pages of solved problems
2642:
Maxima and minima can also be defined for sets. In general, if an
2443:
2435:
2424:
1105:
905:
The definition of local minimum point can also proceed similarly.
47:
2614:(0,0) = 0. However, it cannot be a global one, because
308:, and the value of the function at a minimum point is called the
2896:
1060:. If a function is continuous on a closed interval, then by the
503:
The definition of global minimum point also proceeds similarly.
3362:
27:
Largest and smallest value taken by a function at a given point
2942: = ±1.43, even though the minimum value is the same.
2919:
of {0} because both attain their global maximum value of 1 at
2847:
has no maximum, though it has a minimum. If an infinite chain
1442:
For a practical example, assume a situation where someone has
262:. The value of the function at a maximum point is called the
2357:
Therefore, the greatest area attainable with a rectangle of
1167:
No global minima or maxima. Although the first derivative (3
354:
for clarity). Symbolically, this can be written as follows:
3358:
37:"Maximum" and "Minimum" redirect here. For other uses, see
2778:
of the set which is contained within the set, whereas a
2741:. The maximum and minimum function for sets are used in
1236:
Unique global maximum over the positive real numbers at
2905:
2670:
2506:
2384:
2363:
2340:
2311:
2265:
2242:
2221:
2199:
2170:
2135:
2108:
2088:
2054:
2025:
1995:
1959:
1937:
1772:
1749:
1702:
1669:
1612:
1573:
1534:
1508:
1488:
1468:
1448:
1270:. Setting the first derivative to 0 and solving for
1190:
908:
In both the global and local cases, the concept of a
792:
757:
737:
704:
670:
631:
433:
396:
363:
322:
276:
493:{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}
30:"Extreme value" redirects here. For other uses, see
4341:
4252:
4245:
4163:
4125:
3997:
3904:
3836:
3717:
3519:
3474:
3396:
1655:{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}
3103:
2911:
2685:
2599:
2402:
2369:
2346:
2326:
2295:
2248:
2227:
2205:
2182:
2153:
2120:
2094:
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2037:
2010:
1980:
1943:
1920:
1755:
1732:
1687:
1654:
1597:
1558:
1517:
1494:
1474:
1454:
1205:
894:
778:
743:
723:
690:
656:
492:
419:
382:
346:
300:
120:, for finding the maxima and minima of functions.
52:Local and global maxima and minima for cos(3π
3354:. Vol. 17 (11th ed.). pp. 918–920.
2895:As an example, both unnormalised and normalised
2671:
1375: = 0.1 (a boundary), a global minimum near
1056:Finding global maxima and minima is the goal of
323:
277:
1403:+ 1 defined over the closed interval (segment)
3374:
2102:is restricted. Since width is positive, then
1318:, ..., and infinitely many global minima at ±
8:
88:, they may be defined either within a given
2949:, the arguments of the maxima (abbreviated
779:{\displaystyle (\exists \varepsilon >0)}
4249:
3381:
3367:
3359:
3328:Thomas Simpson's work on Maxima and Minima
2440:The global maximum is the point at the top
1387: = 1.0. (See figure at top of page.)
854:
850:
4409:Regiomontanus' angle maximization problem
3258:"Minimization and maximization refresher"
2926:The unnormalised sinc function (red) has
2904:
2669:
2590:
2589:
2567:
2545:
2532:
2505:
2383:
2362:
2339:
2310:
2264:
2241:
2220:
2198:
2169:
2134:
2107:
2087:
2053:
2024:
1994:
1958:
1936:
1881:
1848:
1805:
1777:
1773:
1771:
1748:
1701:
1668:
1631:
1613:
1611:
1572:
1533:
1507:
1487:
1467:
1447:
1196:
1191:
1189:
865:
835:
816:
811:
791:
756:
736:
709:
703:
684:
683:
669:
645:
630:
463:
452:
432:
410:
409:
395:
368:
362:
321:
275:
2890:
2630:), then the extremum is found using the
1136:
613:. A similar definition can be used when
3757:Differentiating under the integral sign
3176:Thomas' Calculus: Early Transcendentals
3091:
3030:
2912:{\displaystyle \operatorname {argmax} }
1179:. (2nd derivative is 0 at that point.)
1383: = 0.6, and a local minimum near
1310:Infinitely many global maxima at 0, ±2
1091:, given sufficient differentiability.
390:is a global maximum point of function
3633:Inverse functions and differentiation
2082:by determining the interval to which
731:is a local maximum point of function
7:
3290:For clarity, we refer to the input (
420:{\displaystyle f:X\to \mathbb {R} ,}
135:in the set, respectively. Unbounded
2415:Functions of more than one variable
691:{\displaystyle f:X\to \mathbb {R} }
150:, the corresponding concept is the
3431:Free variables and bound variables
2764:(nothing is smaller). Likewise, a
2747:decomposable aggregation functions
2709:with (respect to order induced by
1379: = 0.3, a local maximum near
796:
761:
437:
25:
4236:The Method of Mechanical Theorems
2994:Limit superior and limit inferior
1094:For any function that is defined
3791:Partial fractions in integration
3707:Stochastic differential equation
3342:Jolliffe, Arthur Ernest (1911).
2883:This section is an excerpt from
2622:Maxima or minima of a functional
2403:{\displaystyle 50\times 50=2500}
1432: = 2 and global minimum at
573:. Similarly, the function has a
219:. Similarly, the function has a
3929:Jacobian matrix and determinant
3784:Tangent half-angle substitution
3752:Fundamental theorem of calculus
3106:Calculus: Early Transcendentals
2576:
1743:The derivative with respect to
664:be a metric space and function
127:, the maximum and minimum of a
4005:Arithmetico-geometric sequence
3697:Ordinary differential equation
3272:The Unnormalized Sinc Function
2697:is a subset of an ordered set
2680:
2674:
2564:
2551:
2522:
2510:
2478:second partial derivative test
2421:Second partial derivative test
2290:
2278:
1835:
1823:
1727:
1715:
886:
880:
871:
858:
851:
841:
822:
808:
793:
773:
758:
680:
651:
632:
484:
478:
469:
456:
449:
434:
406:
341:
338:
332:
326:
295:
292:
286:
280:
143:, have no minimum or maximum.
32:Extreme value (disambiguation)
1:
3828:Integro-differential equation
3702:Partial differential equation
916:can be defined. For example,
3138:; Edwards, Bruce H. (2009).
2969:, the output is part of its
2664:of the set, also denoted as
1089:higher-order derivative test
1036:real-valued function with a
3982:Generalized Stokes' theorem
3769:Integration by substitution
2729:. Similar results hold for
2705:is the greatest element of
2296:{\displaystyle xy=x(100-x)}
1733:{\displaystyle xy=x(100-x)}
924:strict global maximum point
133:greatest and least elements
4466:
3511:(ε, δ)-definition of limit
3014:Sample maximum and minimum
2961:, respectively. While the
2882:
2487:intermediate value theorem
2418:
1206:{\displaystyle {\sqrt{x}}}
973:strict local maximum point
724:{\displaystyle x_{0}\in X}
383:{\displaystyle x_{0}\in X}
347:{\displaystyle \min(f(x))}
316:of the function, (denoted
301:{\displaystyle \max(f(x))}
152:sample maximum and minimum
100:extrema) or on the entire
36:
29:
4445:Mathematical optimization
4404:Proof that 22/7 exceeds π
4319:
4193:Gottfried Wilhelm Leibniz
4140:e (mathematical constant)
3345:"Maxima and Minima"
2999:Maximum-minimums identity
2752:In the case of a general
1598:{\displaystyle 2y=200-2x}
1559:{\displaystyle 2x+2y=200}
1216:Unique global maximum at
1153:Unique global minimum at
1058:mathematical optimization
657:{\displaystyle (X,d_{X})}
270:of the function, denoted
4155:Stirling's approximation
3628:Implicit differentiation
3576:Rules of differentiation
2183:{\displaystyle x<100}
1981:{\displaystyle 0=100-2x}
1226:. (See figure at right)
1069:differentiable functions
112:extrema) of a function.
43:Minimum (disambiguation)
39:Maximum (disambiguation)
4389:Euler–Maclaurin formula
4294:trigonometric functions
3747:Constant of integration
3351:Encyclopædia Britannica
3281:", University of Sydney
2959:maximized and minimized
2878:Argument of the maximum
2686:{\displaystyle \max(S)}
2327:{\displaystyle 2500,0,}
2192:Plug in critical point
2154:{\displaystyle x=100-y}
1688:{\displaystyle y=100-x}
1048:(see the graph above).
534:, if there exists some
4358:Differential geometry
4203:Infinitesimal calculus
3906:Multivariable calculus
3854:Directional derivative
3660:Second derivative test
3638:Logarithmic derivative
3611:General Leibniz's rule
3506:Order of approximation
3004:Mechanical equilibrium
2943:
2913:
2687:
2632:calculus of variations
2618:(2,3) = −5.
2601:
2449:
2441:
2433:
2404:
2371:
2348:
2328:
2297:
2250:
2229:
2207:
2184:
2155:
2122:
2121:{\displaystyle x>0}
2096:
2068:
2039:
2012:
2011:{\displaystyle 2x=100}
1982:
1945:
1931:Setting this equal to
1922:
1757:
1734:
1689:
1656:
1599:
1560:
1519:
1496:
1476:
1456:
1428:/3, global maximum at
1207:
1134:
1110:The global maximum of
1085:second derivative test
896:
780:
745:
725:
692:
658:
494:
421:
384:
348:
302:
65:
4440:Mathematical analysis
4277:logarithmic functions
4272:exponential functions
4188:Generality of algebra
4066:Tests of convergence
3692:Differential equation
3676:Further applications
3665:Extreme value theorem
3655:First derivative test
3549:Differential operator
3521:Differential calculus
3234:mathworld.wolfram.com
3209:mathworld.wolfram.com
2965:are defined over the
2914:
2899:functions above have
2894:
2851:is bounded, then the
2772:partially ordered set
2688:
2602:
2447:
2439:
2428:
2405:
2372:
2349:
2329:
2298:
2251:
2230:
2215:as well as endpoints
2208:
2185:
2156:
2123:
2097:
2069:
2040:
2013:
1983:
1946:
1923:
1758:
1735:
1690:
1657:
1600:
1561:
1520:
1497:
1477:
1457:
1417:/3, local minimum at
1208:
1109:
1081:first derivative test
1062:extreme value theorem
977:if there exists some
897:
781:
746:
726:
693:
659:
495:
422:
385:
349:
303:
139:, such as the set of
70:mathematical analysis
51:
4342:Miscellaneous topics
4282:hyperbolic functions
4267:irrational functions
4145:Exponential function
3998:Sequences and series
3764:Integration by parts
3170:; Weir, Maurice D.;
2989:Infimum and supremum
2967:domain of a function
2903:
2864:greatest lower bound
2739:greatest lower bound
2668:
2504:
2382:
2361:
2338:
2309:
2305:and the results are
2263:
2240:
2219:
2197:
2168:
2133:
2106:
2086:
2067:{\displaystyle x=50}
2052:
2038:{\displaystyle x=50}
2023:
1993:
1957:
1935:
1770:
1747:
1700:
1667:
1610:
1571:
1532:
1506:
1486:
1466:
1446:
1188:
790:
755:
735:
702:
668:
629:
431:
394:
361:
320:
274:
4329:List of derivatives
4165:History of calculus
4080:Cauchy condensation
3977:Exterior derivative
3934:Lagrange multiplier
3670:Maximum and minimum
3501:Limit of a sequence
3489:Limit of a function
3436:Graph of a function
3416:Continuous function
3228:Weisstein, Eric W.
3203:Weisstein, Eric W.
2638:In relation to sets
2458:partial derivatives
2377:feet of fencing is
2370:{\displaystyle 200}
2249:{\displaystyle 100}
2078:. Now retrieve the
1455:{\displaystyle 200}
984:such that, for all
575:local minimum point
4262:rational functions
4229:Method of Fluxions
4075:Alternating series
3972:Differential forms
3954:Partial derivative
3914:Divergence theorem
3796:Quadratic integral
3564:Leibniz's notation
3554:Mean value theorem
3539:Partial derivative
3484:Indeterminate form
3277:2017-02-15 at the
2944:
2909:
2693:. Furthermore, if
2683:
2597:
2450:
2442:
2434:
2400:
2367:
2344:
2324:
2293:
2246:
2225:
2206:{\displaystyle 50}
2203:
2180:
2163:that implies that
2151:
2118:
2092:
2064:
2035:
2008:
1978:
1941:
1918:
1916:
1753:
1730:
1685:
1652:
1595:
1556:
1518:{\displaystyle xy}
1515:
1502:is the width, and
1492:
1472:
1452:
1371:Global maximum at
1290:Global minimum at
1203:
1143:Maxima and minima
1135:
892:
776:
741:
721:
688:
654:
518:is said to have a
490:
417:
380:
344:
298:
66:
4422:
4421:
4348:Complex calculus
4337:
4336:
4218:Law of Continuity
4150:Natural logarithm
4135:Bernoulli numbers
4126:Special functions
4085:Direct comparison
3949:Multiple integral
3823:Integral equation
3719:Integral calculus
3650:Stationary points
3624:Other techniques
3569:Newton's notation
3534:Second derivative
3426:Finite difference
3189:978-0-321-58876-0
3178:(12th ed.).
3168:Thomas, George B.
3153:978-0-547-16702-2
3121:978-0-495-01166-8
3009:Mex (mathematics)
2868:least upper bound
2789:is an element of
2719:least upper bound
2493:to prove this by
2347:{\displaystyle 0}
2228:{\displaystyle 0}
2095:{\displaystyle x}
1944:{\displaystyle 0}
1861:
1818:
1790:
1756:{\displaystyle x}
1650:
1626:
1495:{\displaystyle y}
1475:{\displaystyle x}
1440:
1439:
1406:Local maximum at
1276:stationary points
1264:second derivative
1258:First derivative
1201:
744:{\displaystyle f}
619:topological space
538:> 0 such that
16:(Redirected from
4457:
4352:Contour integral
4250:
4100:Limit comparison
4009:Types of series
3968:Advanced topics
3959:Surface integral
3803:Trapezoidal rule
3742:Basic properties
3737:Riemann integral
3685:Taylor's theorem
3411:Concave function
3406:Binomial theorem
3383:
3376:
3369:
3360:
3355:
3347:
3315:
3298:and the output (
3288:
3282:
3268:
3262:
3261:
3253:
3244:
3243:
3241:
3240:
3225:
3219:
3218:
3216:
3215:
3200:
3194:
3193:
3164:
3158:
3157:
3142:(9th ed.).
3132:
3126:
3125:
3110:(6th ed.).
3109:
3096:
3079:
3073:
3072:
3067:
3061:
3042:
3041:
3035:
2957:output value is
2918:
2916:
2915:
2910:
2874:, respectively.
2767:greatest element
2692:
2690:
2689:
2684:
2651:greatest element
2606:
2604:
2603:
2598:
2593:
2572:
2571:
2550:
2549:
2537:
2536:
2476:throughout. The
2411:
2409:
2407:
2406:
2401:
2376:
2374:
2373:
2368:
2353:
2351:
2350:
2345:
2333:
2331:
2330:
2325:
2304:
2302:
2300:
2299:
2294:
2257:
2255:
2253:
2252:
2247:
2234:
2232:
2231:
2226:
2214:
2212:
2210:
2209:
2204:
2191:
2189:
2187:
2186:
2181:
2162:
2160:
2158:
2157:
2152:
2127:
2125:
2124:
2119:
2101:
2099:
2098:
2093:
2073:
2071:
2070:
2065:
2044:
2042:
2041:
2036:
2017:
2015:
2014:
2009:
1987:
1985:
1984:
1979:
1950:
1948:
1947:
1942:
1927:
1925:
1924:
1919:
1917:
1895:
1891:
1887:
1886:
1885:
1862:
1860:
1849:
1841:
1819:
1817:
1806:
1791:
1789:
1778:
1762:
1760:
1759:
1754:
1739:
1737:
1736:
1731:
1694:
1692:
1691:
1686:
1661:
1659:
1658:
1653:
1651:
1646:
1632:
1627:
1622:
1614:
1604:
1602:
1601:
1596:
1565:
1563:
1562:
1557:
1524:
1522:
1521:
1516:
1501:
1499:
1498:
1493:
1481:
1479:
1478:
1473:
1461:
1459:
1458:
1453:
1427:
1426:
1416:
1415:
1368:
1353:
1329:
1325:
1321:
1317:
1313:
1212:
1210:
1209:
1204:
1202:
1200:
1192:
1177:inflection point
1175:= 0, this is an
1137:
1132:
1120:
1119:
1118:
1073:Fermat's theorem
1028:
1009:
992:within distance
983:
975:
974:
964:
945:
926:
925:
914:
913:
901:
899:
898:
893:
870:
869:
840:
839:
821:
820:
785:
783:
782:
777:
750:
748:
747:
742:
730:
728:
727:
722:
714:
713:
697:
695:
694:
689:
687:
663:
661:
660:
655:
650:
649:
605:within distance
565:within distance
556:
499:
497:
496:
491:
468:
467:
426:
424:
423:
418:
413:
389:
387:
386:
381:
373:
372:
353:
351:
350:
345:
314:
313:
307:
305:
304:
299:
268:
267:
253:
210:
114:Pierre de Fermat
21:
4465:
4464:
4460:
4459:
4458:
4456:
4455:
4454:
4425:
4424:
4423:
4418:
4414:Steinmetz solid
4399:Integration Bee
4333:
4315:
4241:
4183:Colin Maclaurin
4159:
4127:
4121:
3993:
3987:Tensor calculus
3964:Volume integral
3900:
3875:Basic theorems
3838:Vector calculus
3832:
3713:
3680:Newton's method
3515:
3494:One-sided limit
3470:
3451:Rolle's theorem
3441:Linear function
3392:
3387:
3341:
3324:
3319:
3318:
3289:
3285:
3279:Wayback Machine
3269:
3265:
3256:Garrett, Paul.
3255:
3254:
3247:
3238:
3236:
3227:
3226:
3222:
3213:
3211:
3202:
3201:
3197:
3190:
3166:
3165:
3161:
3154:
3134:
3133:
3129:
3122:
3098:
3097:
3093:
3088:
3083:
3082:
3070:
3069:
3068:
3064:
3039:
3038:
3036:
3032:
3027:
2984:Derivative test
2980:
2975:
2974:
2925:
2924:
2923: = 0.
2901:
2900:
2888:
2880:
2845:natural numbers
2822:totally ordered
2780:maximal element
2762:minimal element
2735:minimal element
2666:
2665:
2662:maximal element
2640:
2624:
2563:
2541:
2528:
2502:
2501:
2491:Rolle's theorem
2423:
2417:
2380:
2379:
2378:
2359:
2358:
2336:
2335:
2307:
2306:
2261:
2260:
2259:
2238:
2237:
2236:
2217:
2216:
2195:
2194:
2193:
2166:
2165:
2164:
2131:
2130:
2129:
2104:
2103:
2084:
2083:
2050:
2049:
2021:
2020:
1991:
1990:
1955:
1954:
1933:
1932:
1915:
1914:
1893:
1892:
1877:
1867:
1863:
1853:
1839:
1838:
1810:
1798:
1782:
1768:
1767:
1745:
1744:
1698:
1697:
1665:
1664:
1633:
1615:
1608:
1607:
1569:
1568:
1530:
1529:
1504:
1503:
1484:
1483:
1482:is the length,
1464:
1463:
1444:
1443:
1424:
1422:
1413:
1411:
1362:
1351:
1327:
1323:
1319:
1315:
1311:
1213:
1186:
1185:
1122:
1114:
1112:
1111:
1104:
1077:critical points
1054:
1011:
1001:
978:
972:
971:
947:
937:
923:
922:
912:strict extremum
911:
910:
861:
831:
812:
788:
787:
753:
752:
733:
732:
705:
700:
699:
666:
665:
641:
627:
626:
539:
459:
429:
428:
392:
391:
364:
359:
358:
318:
317:
311:
310:
272:
271:
265:
264:
236:
193:
160:
46:
35:
28:
23:
22:
15:
12:
11:
5:
4463:
4461:
4453:
4452:
4447:
4442:
4437:
4427:
4426:
4420:
4419:
4417:
4416:
4411:
4406:
4401:
4396:
4394:Gabriel's horn
4391:
4386:
4385:
4384:
4379:
4374:
4369:
4364:
4356:
4355:
4354:
4345:
4343:
4339:
4338:
4335:
4334:
4332:
4331:
4326:
4324:List of limits
4320:
4317:
4316:
4314:
4313:
4312:
4311:
4306:
4301:
4291:
4290:
4289:
4279:
4274:
4269:
4264:
4258:
4256:
4247:
4243:
4242:
4240:
4239:
4232:
4225:
4223:Leonhard Euler
4220:
4215:
4210:
4205:
4200:
4195:
4190:
4185:
4180:
4175:
4169:
4167:
4161:
4160:
4158:
4157:
4152:
4147:
4142:
4137:
4131:
4129:
4123:
4122:
4120:
4119:
4118:
4117:
4112:
4107:
4102:
4097:
4092:
4087:
4082:
4077:
4072:
4064:
4063:
4062:
4057:
4056:
4055:
4050:
4040:
4035:
4030:
4025:
4020:
4015:
4007:
4001:
3999:
3995:
3994:
3992:
3991:
3990:
3989:
3984:
3979:
3974:
3966:
3961:
3956:
3951:
3946:
3941:
3936:
3931:
3926:
3924:Hessian matrix
3921:
3916:
3910:
3908:
3902:
3901:
3899:
3898:
3897:
3896:
3891:
3886:
3881:
3879:Line integrals
3873:
3872:
3871:
3866:
3861:
3856:
3851:
3842:
3840:
3834:
3833:
3831:
3830:
3825:
3820:
3819:
3818:
3813:
3805:
3800:
3799:
3798:
3788:
3787:
3786:
3781:
3776:
3766:
3761:
3760:
3759:
3749:
3744:
3739:
3734:
3729:
3727:Antiderivative
3723:
3721:
3715:
3714:
3712:
3711:
3710:
3709:
3704:
3699:
3689:
3688:
3687:
3682:
3674:
3673:
3672:
3667:
3662:
3657:
3647:
3646:
3645:
3640:
3635:
3630:
3622:
3621:
3620:
3615:
3614:
3613:
3603:
3598:
3593:
3588:
3583:
3573:
3572:
3571:
3566:
3556:
3551:
3546:
3541:
3536:
3531:
3525:
3523:
3517:
3516:
3514:
3513:
3508:
3503:
3498:
3497:
3496:
3486:
3480:
3478:
3472:
3471:
3469:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3423:
3418:
3413:
3408:
3402:
3400:
3394:
3393:
3388:
3386:
3385:
3378:
3371:
3363:
3357:
3356:
3339:
3334:
3323:
3322:External links
3320:
3317:
3316:
3312:critical value
3308:critical point
3283:
3263:
3245:
3220:
3195:
3188:
3180:Addison-Wesley
3159:
3152:
3127:
3120:
3100:Stewart, James
3090:
3089:
3087:
3084:
3081:
3080:
3062:
3029:
3028:
3026:
3023:
3022:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2986:
2979:
2976:
2908:
2889:
2881:
2879:
2876:
2774:(poset) is an
2682:
2679:
2676:
2673:
2639:
2636:
2623:
2620:
2608:
2607:
2596:
2592:
2588:
2585:
2582:
2579:
2575:
2570:
2566:
2562:
2559:
2556:
2553:
2548:
2544:
2540:
2535:
2531:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2474:differentiable
2419:Main article:
2416:
2413:
2399:
2396:
2393:
2390:
2387:
2366:
2354:respectively.
2343:
2323:
2320:
2317:
2314:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2245:
2224:
2202:
2179:
2176:
2173:
2150:
2147:
2144:
2141:
2138:
2117:
2114:
2111:
2091:
2076:critical point
2063:
2060:
2057:
2046:
2045:
2034:
2031:
2028:
2018:
2007:
2004:
2001:
1998:
1988:
1977:
1974:
1971:
1968:
1965:
1962:
1940:
1929:
1928:
1913:
1910:
1907:
1904:
1901:
1898:
1896:
1894:
1890:
1884:
1880:
1876:
1873:
1870:
1866:
1859:
1856:
1852:
1847:
1844:
1842:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1816:
1813:
1809:
1804:
1801:
1799:
1797:
1794:
1788:
1785:
1781:
1776:
1775:
1752:
1741:
1740:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1695:
1684:
1681:
1678:
1675:
1672:
1662:
1649:
1645:
1642:
1639:
1636:
1630:
1625:
1621:
1618:
1605:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1566:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1514:
1511:
1491:
1471:
1451:
1438:
1437:
1404:
1389:
1388:
1369:
1347:
1346:
1343:
1332:
1331:
1308:
1300:
1299:
1288:
1280:
1279:
1256:
1246:
1245:
1234:
1228:
1227:
1214:
1199:
1195:
1184:
1181:
1180:
1165:
1159:
1158:
1151:
1145:
1144:
1141:
1103:
1100:
1053:
1050:
903:
902:
891:
888:
885:
882:
879:
876:
873:
868:
864:
860:
857:
853:
849:
846:
843:
838:
834:
830:
827:
824:
819:
815:
810:
807:
804:
801:
798:
795:
775:
772:
769:
766:
763:
760:
740:
720:
717:
712:
708:
686:
682:
679:
676:
673:
653:
648:
644:
640:
637:
634:
506:If the domain
501:
500:
489:
486:
483:
480:
477:
474:
471:
466:
462:
458:
455:
451:
448:
445:
442:
439:
436:
416:
412:
408:
405:
402:
399:
379:
376:
371:
367:
343:
340:
337:
334:
331:
328:
325:
297:
294:
291:
288:
285:
282:
279:
162:A real-valued
159:
156:
123:As defined in
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4462:
4451:
4448:
4446:
4443:
4441:
4438:
4436:
4433:
4432:
4430:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4359:
4357:
4353:
4350:
4349:
4347:
4346:
4344:
4340:
4330:
4327:
4325:
4322:
4321:
4318:
4310:
4307:
4305:
4302:
4300:
4297:
4296:
4295:
4292:
4288:
4285:
4284:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4259:
4257:
4255:
4251:
4248:
4244:
4238:
4237:
4233:
4231:
4230:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4198:Infinitesimal
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4170:
4168:
4166:
4162:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4133:
4132:
4130:
4124:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4067:
4065:
4061:
4058:
4054:
4051:
4049:
4046:
4045:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4010:
4008:
4006:
4003:
4002:
4000:
3996:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3969:
3967:
3965:
3962:
3960:
3957:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3939:Line integral
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3911:
3909:
3907:
3903:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3876:
3874:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3846:
3844:
3843:
3841:
3839:
3835:
3829:
3826:
3824:
3821:
3817:
3814:
3812:
3811:Washer method
3809:
3808:
3806:
3804:
3801:
3797:
3794:
3793:
3792:
3789:
3785:
3782:
3780:
3777:
3775:
3774:trigonometric
3772:
3771:
3770:
3767:
3765:
3762:
3758:
3755:
3754:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3724:
3722:
3720:
3716:
3708:
3705:
3703:
3700:
3698:
3695:
3694:
3693:
3690:
3686:
3683:
3681:
3678:
3677:
3675:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3652:
3651:
3648:
3644:
3643:Related rates
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3625:
3623:
3619:
3616:
3612:
3609:
3608:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3578:
3577:
3574:
3570:
3567:
3565:
3562:
3561:
3560:
3557:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3526:
3524:
3522:
3518:
3512:
3509:
3507:
3504:
3502:
3499:
3495:
3492:
3491:
3490:
3487:
3485:
3482:
3481:
3479:
3477:
3473:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3403:
3401:
3399:
3395:
3391:
3384:
3379:
3377:
3372:
3370:
3365:
3364:
3361:
3353:
3352:
3346:
3340:
3338:
3335:
3333:
3329:
3326:
3325:
3321:
3313:
3309:
3305:
3301:
3297:
3293:
3287:
3284:
3280:
3276:
3273:
3267:
3264:
3259:
3252:
3250:
3246:
3235:
3231:
3224:
3221:
3210:
3206:
3199:
3196:
3191:
3185:
3181:
3177:
3173:
3169:
3163:
3160:
3155:
3149:
3145:
3141:
3137:
3131:
3128:
3123:
3117:
3113:
3108:
3107:
3101:
3095:
3092:
3085:
3077:
3066:
3063:
3059:
3055:
3051:
3047:
3043:
3034:
3031:
3024:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2981:
2977:
2972:
2968:
2964:
2960:
2956:
2952:
2948:
2941:
2937:
2933:
2929:
2922:
2906:
2898:
2893:
2886:
2877:
2875:
2873:
2869:
2865:
2861:
2857:
2854:
2850:
2846:
2841:
2839:
2838:
2833:
2832:
2827:
2823:
2818:
2816:
2812:
2808:
2804:
2800:
2796:
2793:such that if
2792:
2788:
2784:
2781:
2777:
2773:
2769:
2768:
2763:
2759:
2758:least element
2755:
2754:partial order
2750:
2748:
2744:
2740:
2736:
2732:
2731:least element
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2677:
2663:
2659:
2655:
2652:
2648:
2645:
2637:
2635:
2633:
2629:
2621:
2619:
2617:
2613:
2594:
2586:
2583:
2580:
2577:
2573:
2568:
2560:
2557:
2554:
2546:
2542:
2538:
2533:
2529:
2525:
2519:
2516:
2513:
2507:
2500:
2499:
2498:
2496:
2495:contradiction
2492:
2488:
2484:
2479:
2475:
2472:must also be
2471:
2467:
2463:
2459:
2455:
2446:
2438:
2431:
2430:Peano surface
2427:
2422:
2414:
2412:
2397:
2394:
2391:
2388:
2385:
2364:
2355:
2341:
2321:
2318:
2315:
2312:
2287:
2284:
2281:
2275:
2272:
2269:
2266:
2243:
2222:
2200:
2177:
2174:
2171:
2148:
2145:
2142:
2139:
2136:
2115:
2112:
2109:
2089:
2081:
2077:
2061:
2058:
2055:
2048:reveals that
2032:
2029:
2026:
2019:
2005:
2002:
1999:
1996:
1989:
1975:
1972:
1969:
1966:
1963:
1960:
1953:
1952:
1951:
1938:
1911:
1908:
1905:
1902:
1899:
1897:
1888:
1882:
1878:
1874:
1871:
1868:
1864:
1857:
1854:
1850:
1845:
1843:
1832:
1829:
1826:
1820:
1814:
1811:
1807:
1802:
1800:
1795:
1792:
1786:
1783:
1779:
1766:
1765:
1764:
1750:
1724:
1721:
1718:
1712:
1709:
1706:
1703:
1696:
1682:
1679:
1676:
1673:
1670:
1663:
1647:
1643:
1640:
1637:
1634:
1628:
1623:
1619:
1616:
1606:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1567:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1528:
1527:
1526:
1525:is the area:
1512:
1509:
1489:
1469:
1449:
1435:
1431:
1420:
1409:
1405:
1402:
1398:
1394:
1391:
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1360:
1356:
1349:
1348:
1344:
1342:
1338:
1334:
1333:
1309:
1306:
1302:
1301:
1297:
1293:
1289:
1286:
1282:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1255:
1251:
1248:
1247:
1243:
1239:
1235:
1233:
1230:
1229:
1225:
1224:
1219:
1215:
1197:
1193:
1183:
1182:
1178:
1174:
1170:
1166:
1164:
1161:
1160:
1156:
1152:
1150:
1147:
1146:
1142:
1139:
1138:
1131:
1130:
1125:
1117:
1108:
1101:
1099:
1097:
1092:
1090:
1086:
1082:
1078:
1074:
1070:
1065:
1063:
1059:
1051:
1049:
1047:
1043:
1039:
1035:
1030:
1026:
1022:
1018:
1014:
1008:
1004:
999:
995:
991:
987:
981:
976:
968:
962:
958:
954:
950:
944:
940:
935:
931:
927:
919:
915:
906:
889:
883:
877:
874:
866:
862:
855:
847:
844:
836:
832:
828:
825:
817:
813:
805:
802:
799:
770:
767:
764:
738:
718:
715:
710:
706:
677:
674:
671:
646:
642:
638:
635:
624:
623:
622:
620:
616:
612:
608:
604:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
554:
550:
546:
542:
537:
533:
530:at the point
529:
528:maximum point
525:
521:
517:
513:
509:
504:
487:
481:
475:
472:
464:
460:
453:
446:
443:
440:
414:
403:
400:
397:
377:
374:
369:
365:
357:
356:
355:
335:
329:
315:
312:minimum value
289:
283:
269:
266:maximum value
261:
257:
251:
247:
243:
239:
234:
230:
229:minimum point
226:
222:
218:
214:
208:
204:
200:
196:
191:
187:
186:maximum point
183:
179:
175:
172:
169:defined on a
168:
165:
157:
155:
153:
149:
144:
142:
138:
137:infinite sets
134:
130:
126:
121:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
63:
59:
55:
50:
44:
40:
33:
19:
4450:Superlatives
4309:Secant cubed
4234:
4227:
4208:Isaac Newton
4178:Brook Taylor
3845:Derivatives
3816:Shell method
3669:
3544:Differential
3349:
3303:
3299:
3295:
3291:
3286:
3266:
3237:. Retrieved
3233:
3223:
3212:. Retrieved
3208:
3198:
3175:
3162:
3139:
3130:
3105:
3094:
3075:
3065:
3057:
3053:
3049:
3045:
3033:
3019:Saddle point
2939:
2935:
2931:
2927:
2920:
2871:
2867:
2863:
2859:
2855:
2848:
2842:
2836:
2835:
2830:
2829:
2825:
2819:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2779:
2765:
2761:
2757:
2751:
2726:
2722:
2714:
2710:
2706:
2702:
2698:
2694:
2657:
2653:
2646:
2641:
2625:
2615:
2611:
2609:
2482:
2469:
2466:saddle point
2461:
2453:
2451:
2356:
2128:, and since
2074:is our only
2047:
1930:
1742:
1441:
1436: = −4.
1433:
1429:
1418:
1407:
1400:
1396:
1392:
1384:
1380:
1376:
1372:
1364:
1358:
1354:
1340:
1336:
1304:
1295:
1291:
1284:
1271:
1267:
1259:
1253:
1249:
1241:
1237:
1231:
1221:
1217:
1172:
1168:
1162:
1154:
1148:
1127:
1123:
1115:
1093:
1066:
1055:
1046:real numbers
1031:
1024:
1020:
1016:
1012:
1006:
1002:
997:
993:
989:
985:
979:
970:
966:
960:
956:
952:
948:
942:
938:
933:
929:
921:
917:
909:
907:
904:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
552:
548:
544:
540:
535:
531:
527:
523:
519:
515:
512:metric space
507:
505:
502:
309:
263:
259:
255:
249:
245:
241:
237:
232:
228:
224:
220:
216:
212:
206:
202:
198:
194:
189:
185:
181:
177:
173:
166:
161:
145:
141:real numbers
122:
109:
105:
97:
93:
85:
77:
73:
67:
61:
60:, 0.1≤
57:
53:
18:Local minima
4377:of surfaces
4128:and numbers
4090:Dirichlet's
4060:Telescoping
4013:Alternating
3601:L'Hôpital's
3398:Precalculus
3332:Convergence
3144:Brooks/Cole
3136:Larson, Ron
3112:Brooks/Cole
2947:mathematics
2870:of the set
2785:of a poset
2776:upper bound
2644:ordered set
1421: = −1+
1410: = −1−
928:if for all
4429:Categories
4173:Adequality
3859:Divergence
3732:Arc length
3529:Derivative
3239:2020-08-30
3214:2020-08-30
3172:Hass, Joel
3086:References
2628:functional
1171:) is 0 at
1121:occurs at
1034:continuous
1010:, we have
946:, we have
786:such that
597:) for all
158:Definition
148:statistics
125:set theory
118:adequality
64:≤1.1
4372:of curves
4367:Curvature
4254:Integrals
4048:Maclaurin
4028:Geometric
3919:Geometric
3869:Laplacian
3581:linearity
3421:Factorial
3230:"Maximum"
3205:"Minimum"
2963:arguments
2801:(for any
2743:databases
2587:∈
2558:−
2389:×
2285:−
2146:−
2080:endpoints
1970:−
1906:−
1875:−
1830:−
1722:−
1680:−
1638:−
1587:−
1096:piecewise
875:≥
852:⟹
848:ε
803:∈
797:∀
765:ε
762:∃
716:∈
681:→
473:≥
444:∈
438:∀
407:→
375:∈
4435:Calculus
4362:Manifold
4095:Integral
4038:Infinite
4033:Harmonic
4018:Binomial
3864:Gradient
3807:Volumes
3618:Quotient
3559:Notation
3390:Calculus
3306:compare
3275:Archived
3174:(2010).
3140:Calculus
3102:(2008).
3058:minimums
3054:maximums
2978:See also
2971:codomain
2955:function
2866:and the
2824:set, or
2809:), then
2713:), then
1262:− 1 and
1140:Function
1102:Examples
1042:interval
557:for all
524:relative
254:for all
225:absolute
211:for all
182:absolute
164:function
131:are the
110:absolute
98:relative
86:extremum
82:function
4299:inverse
4287:inverse
4213:Fluxion
4023:Fourier
3889:Stokes'
3884:Green's
3606:Product
3466:Tangent
3304:values;
3076:extrema
2951:arg max
2936:arg min
2928:arg min
2885:Arg max
2853:closure
2837:maximum
2831:minimum
2656:, then
1423:√
1412:√
1330:, ....
1113:√
1038:compact
1019:) >
955:) >
698:. Then
514:, then
78:minimum
74:maximum
4382:Tensor
4304:Secant
4070:Abel's
4053:Taylor
3944:Matrix
3894:Gauss'
3476:Limits
3456:Secant
3446:Radian
3296:points
3186:
3150:
3118:
3050:minima
3046:maxima
2907:argmax
2756:, the
2649:has a
2460:as to
1363:0.1 ≤
1335:2 cos(
1274:gives
1052:Search
982:> 0
965:, and
221:global
178:global
176:has a
171:domain
106:global
102:domain
72:, the
4246:Lists
4105:Ratio
4043:Power
3779:Euler
3596:Chain
3586:Power
3461:Slope
3302:) as
3294:) as
3025:Notes
2826:chain
2820:In a
2770:of a
2717:is a
2660:is a
2454:local
2258:into
1367:≤ 1.1
1361:with
1350:cos(3
1298:= 0.
1252:/3 −
1157:= 0.
1087:, or
1000:with
969:is a
936:with
920:is a
617:is a
581:, if
520:local
510:is a
235:, if
192:, if
104:(the
94:local
92:(the
90:range
80:of a
4115:Term
4110:Root
3849:Curl
3310:and
3184:ISBN
3148:ISBN
3116:ISBN
3056:and
3052:(or
3048:and
2897:sinc
2834:and
2737:and
2701:and
2489:and
2398:2500
2334:and
2313:2500
2235:and
2175:<
2113:>
1763:is:
1339:) −
1326:, ±5
1322:, ±3
1314:, ±4
1303:cos(
1240:= 1/
1067:For
845:<
768:>
625:Let
589:) ≤
547:) ≥
522:(or
244:) ≤
223:(or
201:) ≥
180:(or
76:and
41:and
3591:Sum
3330:at
2945:In
2840:.
2805:in
2725:in
2721:of
2672:max
2365:200
2282:100
2244:100
2178:100
2143:100
2006:100
1967:100
1903:100
1869:100
1827:100
1719:100
1677:100
1635:200
1584:200
1554:200
1450:200
1399:− 2
1395:+ 3
1044:of
996:of
988:in
932:in
751:if
609:of
601:in
577:at
569:of
561:in
427:if
324:min
278:max
258:in
231:at
215:in
188:at
146:In
129:set
108:or
96:or
68:In
4431::
3348:.
3248:^
3232:.
3207:.
3182:.
3146:.
3114:.
3074::
3071:PL
3060:).
3044::
3040:PL
2856:Cl
2813:=
2797:≤
2749:.
2733:,
2634:.
2392:50
2386:50
2201:50
2062:50
2033:50
1425:15
1414:15
1357:)/
1244:.
1220:=
1126:=
1083:,
1071:,
1032:A
1005:≠
941:≠
526:)
227:)
184:)
154:.
62:x
56:)/
3382:e
3375:t
3368:v
3314:.
3300:y
3292:x
3270:"
3260:.
3242:.
3217:.
3192:.
3156:.
3124:.
3078:.
2973:.
2940:x
2932:x
2921:x
2887:.
2872:S
2860:S
2858:(
2849:S
2815:b
2811:m
2807:A
2803:b
2799:b
2795:m
2791:A
2787:A
2783:m
2727:T
2723:S
2715:m
2711:T
2707:S
2703:m
2699:T
2695:S
2681:)
2678:S
2675:(
2658:m
2654:m
2647:S
2616:f
2612:f
2595:,
2591:R
2584:y
2581:,
2578:x
2574:,
2569:3
2565:)
2561:x
2555:1
2552:(
2547:2
2543:y
2539:+
2534:2
2530:x
2526:=
2523:)
2520:y
2517:,
2514:x
2511:(
2508:f
2483:f
2470:z
2462:z
2410:.
2395:=
2342:0
2322:,
2319:0
2316:,
2303:,
2291:)
2288:x
2279:(
2276:x
2273:=
2270:y
2267:x
2256:,
2223:0
2213:,
2190:.
2172:x
2161:,
2149:y
2140:=
2137:x
2116:0
2110:x
2090:x
2059:=
2056:x
2030:=
2027:x
2003:=
2000:x
1997:2
1976:x
1973:2
1964:=
1961:0
1939:0
1912:x
1909:2
1900:=
1889:)
1883:2
1879:x
1872:x
1865:(
1858:x
1855:d
1851:d
1846:=
1836:)
1833:x
1824:(
1821:x
1815:x
1812:d
1808:d
1803:=
1796:y
1793:x
1787:x
1784:d
1780:d
1751:x
1728:)
1725:x
1716:(
1713:x
1710:=
1707:y
1704:x
1683:x
1674:=
1671:y
1648:2
1644:x
1641:2
1629:=
1624:2
1620:y
1617:2
1593:x
1590:2
1581:=
1578:y
1575:2
1551:=
1548:y
1545:2
1542:+
1539:x
1536:2
1513:y
1510:x
1490:y
1470:x
1434:x
1430:x
1419:x
1408:x
1401:x
1397:x
1393:x
1385:x
1381:x
1377:x
1373:x
1365:x
1359:x
1355:x
1352:π
1341:x
1337:x
1328:π
1324:π
1320:π
1316:π
1312:π
1307:)
1305:x
1296:x
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