3002:
1471:
1287:
976:
3186:. More general fields do not always have differentiable functions, but they do always have polynomials, which can be symbolically differentiated. Similarly, more general fields may not have an order, but one has a notion of a root of a polynomial lying in a field.
3323:
2165:
2787:
2591:
1909:
1800:
2006:
1392:
129:
2669:
2473:
3223:
3182:, but the following corollary does: if a real polynomial factors (has all of its roots) over the real numbers, then its derivative does as well. One may call this property of a field
1556:
1091:
that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a
1604:, but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every
1462:
is zero. The theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
1229:
2674:
2478:
3021:
can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with
1805:
1696:
2059:
2808:
1939:
2045:
is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers.
1307:
466:
525:
3647:
962:
2606:
2410:
3572:
3521:
3429:
481:
204:
2175:
Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization.
471:
802:
476:
456:
138:
1478:
If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the
1485:
3652:
3611:
584:
531:
417:
243:
215:
326:
3606:
835:
448:
286:
258:
48:
3366:
3189:
Thus Rolle's theorem shows that the real numbers have Rolle's property. Any algebraically closed field such as the
706:
670:
452:
331:
225:
220:
210:
311:
2053:
605:
170:
3376:
1194:
1628:, but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
1266:
1138:
1088:
919:
711:
600:
3453:
3053:
1112:
955:
884:
845:
729:
665:
589:
1253:, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of
2245:
1258:
1254:
929:
595:
486:
371:
316:
277:
183:
3318:{\displaystyle 3x^{2}-1=3\left(x-{\tfrac {1}{\sqrt {3}}}\right)\left(x+{\tfrac {1}{\sqrt {3}}}\right),}
3085:
satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer
1608:
in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem,
3419:
3371:
3034:
Particularly, this version of the theorem asserts that if a function differentiable enough times has
934:
914:
840:
509:
433:
407:
321:
3601:
3519:
Ballantine, C.; Roberts, J. (January 2002), "A Simple Proof of Rolle's
Theorem for Finite Fields",
3179:
1397:
1238:
1119:
988:
909:
879:
869:
756:
610:
412:
268:
151:
146:
1257:, which at that point in his life he considered to be fallacious. The theorem was first proved by
1095:. It is a point at which the first derivative of the function is zero. The theorem is named after
3582:
3538:
3361:
1914:
1262:
1234:
874:
777:
761:
701:
655:
536:
460:
366:
361:
165:
696:
691:
3568:
3425:
3174:
Rolle's theorem is a property of differentiable functions over the real numbers, which are an
2277:
2202:
1270:
948:
782:
560:
443:
396:
253:
248:
3618:
3530:
3499:
3468:
3217:
1092:
792:
686:
660:
521:
438:
402:
3397:
1237:, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of
3349:
3193:
has Rolle's property. However, the rational numbers do not – for example,
3190:
2217:, and the function must change from increasing to decreasing (or the other way around) at
2049:
1691:
1613:
1123:
992:
924:
797:
751:
746:
633:
546:
491:
3560:
3415:
2312:
1479:
807:
615:
387:
3632:
3038:
roots (so they have the same value, that is 0), then there is an internal point where
2823:
has more points with equal values and greater regularity. Specifically, suppose that
1586:
is zero. This is because that function, although continuous, is not differentiable at
3641:
3175:
3009:. Thus its second derivative (graphed in green) also has a root in the same interval.
1142:
1011:
787:
551:
306:
263:
17:
3628:
3622:
3330:
1250:
1096:
541:
291:
3487:
2603:, the inequality turns around because the denominator is now negative and we get
2016:(in the extended real line). If the right- and left-hand limits agree for every
1636:
The second example illustrates the following generalization of Rolle's theorem:
1458:, Rolle's theorem applies, and indeed, there is a point where the derivative of
1108:
980:
904:
27:
On stationary points between two equal values of a real differentiable function
3445:
3325:
does not. The question of which fields satisfy Rolle's property was raised in
2839:
2792:
Finally, when the above right- and left-hand limits agree (in particular when
1401:
1007:
650:
574:
301:
296:
200:
3504:
3001:
2782:{\displaystyle f'(c^{-}):=\lim _{h\to 0^{-}}{\frac {f(c+h)-f(c)}{h}}\geq 0,}
2586:{\displaystyle f'(c^{+}):=\lim _{h\to 0^{+}}{\frac {f(c+h)-f(c)}{h}}\leq 0,}
1470:
579:
569:
1404:
centered at the origin. This function is continuous on the closed interval
1076:
645:
392:
349:
38:
3542:
3070:, take as the induction hypothesis that the generalization is true for
1286:
2337:). We shall examine the above right- and left-hand limits separately.
1904:{\displaystyle f'(x^{-}):=\lim _{h\to 0^{-}}{\frac {f(x+h)-f(x)}{h}}}
1795:{\displaystyle f'(x^{+}):=\lim _{h\to 0^{+}}{\frac {f(x+h)-f(x)}{h}}}
3534:
3005:
The red curve is the graph of function with 3 roots in the interval
2160:{\displaystyle f'(x^{-})\leq f'(x^{+})\leq f'(y^{-}),\quad x<y.}
975:
974:
2593:
where the limit exists by assumption, it may be minus infinity.
3132:. Hence, the first derivative satisfies the assumptions on the
2048:
This generalized version of the theorem is sufficient to prove
3567:(2nd ed.). New York: Harper & Row. pp. 201–207.
2221:. In particular, if the derivative exists, it must be zero at
2330:(the argument for the minimum is very similar, just consider
2001:{\displaystyle f'(c^{+})\quad {\text{and}}\quad f'(c^{-})}
2819:
We can also generalize Rolle's theorem by requiring that
3063:
is simply the standard version of Rolle's theorem. For
1387:{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}},\quad x\in .}
3294:
3264:
3226:
2677:
2609:
2481:
2413:
2062:
1942:
1808:
1699:
1488:
1310:
1233:
This version of Rolle's theorem is used to prove the
1197:
51:
3633:
http://mizar.org/version/current/html/rolle.html#T2
3317:
2781:
2663:
2585:
2467:
2159:
2000:
1903:
1794:
1550:
1386:
1223:
123:
3529:(1), Mathematical Association of America: 72–74,
2260:. If these are both attained at the endpoints of
2706:
2510:
2311:Suppose then that the maximum is obtained at an
1837:
1728:
124:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}
2664:{\displaystyle {\frac {f(c+h)-f(c)}{h}}\geq 0,}
2468:{\displaystyle {\frac {f(c+h)-f(c)}{h}}\leq 0,}
1265:. The name "Rolle's theorem" was first used by
956:
8:
2248:attains both its maximum and its minimum in
1639:Consider a real-valued, continuous function
3589:. Boston: Ginn and Company. pp. 30–37.
3398:"A brief history of the mean value theorem"
3178:. As such, it does not generalize to other
2796:is differentiable), then the derivative of
3424:. American Mathematical Soc. p. 224.
3143:. By the induction hypothesis, there is a
2858:th derivative exists on the open interval
2178:The idea of the proof is to argue that if
1428:, but not differentiable at the endpoints
963:
949:
829:
735:
639:
515:
355:
189:
29:
3503:
3326:
3293:
3263:
3234:
3225:
2728:
2720:
2709:
2693:
2676:
2610:
2608:
2532:
2524:
2513:
2497:
2480:
2414:
2412:
2132:
2105:
2078:
2061:
1989:
1968:
1958:
1941:
1859:
1851:
1840:
1824:
1807:
1750:
1742:
1731:
1715:
1698:
1512:
1504:
1487:
1345:
1332:
1326:
1309:
1261:in 1823 as a corollary of a proof of the
1196:
84:
61:
56:
50:
3486:Craven, Thomas; Csordas, George (1977),
3000:
2789:where the limit might be plus infinity.
1474:The graph of the absolute value function
1469:
1416:and differentiable in the open interval
1285:
1087:essentially states that any real-valued
3388:
769:
738:
678:
559:
487:Differentiating under the integral sign
425:
379:
276:
235:
192:
37:
1551:{\displaystyle f(x)=|x|,\quad x\in .}
7:
3565:The Calculus, with Analytic Geometry
2815:Generalization to higher derivatives
2020:, then they agree in particular for
1249:Although the theorem is named after
3396:Besenyei, A. (September 17, 2012).
2052:when the one-sided derivatives are
33:Part of a series of articles about
25:
3522:The American Mathematical Monthly
3488:"Multiplier sequences for fields"
3452:, translated by Butler, Michael,
2809:Fermat's stationary point theorem
1175:, then there exists at least one
3013:The requirements concerning the
2171:Proof of the generalized version
1936:such that one of the two limits
3619:Rolle's and Mean Value Theorems
3170:Generalizations to other fields
2144:
1973:
1967:
1575:between −1 and 1 for which the
1520:
1356:
1103:Standard version of the theorem
2761:
2755:
2746:
2734:
2713:
2699:
2686:
2643:
2637:
2628:
2616:
2565:
2559:
2550:
2538:
2517:
2503:
2490:
2447:
2441:
2432:
2420:
2138:
2125:
2111:
2098:
2084:
2071:
1995:
1982:
1964:
1951:
1892:
1886:
1877:
1865:
1844:
1830:
1817:
1783:
1777:
1768:
1756:
1735:
1721:
1708:
1542:
1527:
1513:
1505:
1498:
1492:
1378:
1363:
1320:
1314:
1212:
1206:
118:
112:
103:
97:
81:
75:
1:
2807:(Alternatively, we can apply
418:Integral of inverse functions
1920:, then there is some number
3607:Encyclopedia of Mathematics
3348:For a complex version, see
2840:continuously differentiable
836:Calculus on Euclidean space
259:Logarithmic differentiation
3669:
3454:Holt, Rinehart and Winston
3367:Intermediate value theorem
3333:, the answer is that only
3077:. We want to prove it for
2296:is zero at every point in
2024:, hence the derivative of
1269:of Germany in 1834 and by
3648:Theorems in real analysis
2292:and so the derivative of
570:Summand limit (term test)
3421:A History of Mathematics
2054:monotonically increasing
1802:and the left-hand limit
1304:, consider the function
1224:{\displaystyle f'(c)=0.}
254:Implicit differentiation
244:Differentiation notation
171:Inverse function theorem
3345:have Rolle's property.
2970:Then there is a number
2842:on the closed interval
2400:. Therefore, for every
2396:attains its maximum at
2381:is smaller or equal to
1290:A semicircle of radius
1267:Moritz Wilhelm Drobisch
1089:differentiable function
712:Helmholtz decomposition
3505:10.1215/ijm/1256048929
3319:
3220:, but its derivative,
3081:. Assume the function
3054:mathematical induction
3010:
2783:
2665:
2587:
2469:
2203:a maximum or a minimum
2161:
2032:and is equal to zero.
2002:
1905:
1796:
1552:
1475:
1388:
1294:
1225:
1072:
1044:, then there exists a
846:Limit of distributions
666:Directional derivative
327:Faà di Bruno's formula
125:
3320:
3100:in the open interval
3004:
2784:
2666:
2596:Similarly, for every
2588:
2470:
2246:extreme value theorem
2162:
2012:and the other one is
2003:
1924:in the open interval
1906:
1797:
1678:in the open interval
1643:on a closed interval
1616:in the open interval
1553:
1473:
1389:
1289:
1255:differential calculus
1226:
1179:in the open interval
1048:in the open interval
978:
930:Mathematical analysis
841:Generalized functions
526:arithmetico-geometric
372:Leibniz integral rule
126:
3653:Theorems in calculus
3372:Linear interpolation
3224:
2675:
2607:
2479:
2411:
2060:
1940:
1806:
1697:
1597:changes its sign at
1593:. The derivative of
1486:
1308:
1195:
935:Nonstandard analysis
408:Lebesgue integration
278:Rules and identities
49:
18:Rolle's Theorem
3377:Gauss–Lucas theorem
2877:intervals given by
2201:must attain either
606:Cauchy condensation
413:Contour integration
139:Fundamental theorem
66:
3450:The Gamma Function
3362:Mean value theorem
3315:
3305:
3275:
3011:
2779:
2727:
2661:
2583:
2531:
2465:
2205:somewhere between
2157:
1998:
1915:extended real line
1901:
1858:
1792:
1749:
1612:will still have a
1571:, but there is no
1548:
1476:
1384:
1295:
1273:of Italy in 1846.
1263:mean value theorem
1235:mean value theorem
1221:
1073:
778:Partial derivative
707:generalized Stokes
601:Alternating series
482:Reduction formulae
457:tangent half-angle
444:Cylindrical shells
367:Integral transform
362:Lists of integrals
166:Mean value theorem
121:
52:
3587:Advanced Calculus
3492:Illinois J. Math.
3469:Kaplansky, Irving
3304:
3303:
3274:
3273:
3216:factors over the
3155:st derivative of
3139:closed intervals
3093:, there exists a
3017:th derivative of
2990:th derivative of
2768:
2705:
2650:
2572:
2509:
2454:
2232:is continuous on
1971:
1899:
1836:
1790:
1727:
1351:
1271:Giusto Bellavitis
983:-valued function
973:
972:
853:
852:
815:
814:
783:Multiple integral
719:
718:
623:
622:
590:Direct comparison
561:Convergence tests
499:
498:
472:Partial fractions
339:
338:
249:Second derivative
16:(Redirected from
3660:
3615:
3590:
3583:Taylor, Angus E.
3578:
3547:
3545:
3516:
3510:
3508:
3507:
3483:
3477:
3475:
3473:Fields and Rings
3465:
3459:
3457:
3442:
3436:
3435:
3411:
3405:
3404:
3402:
3393:
3344:
3338:
3324:
3322:
3321:
3316:
3311:
3307:
3306:
3299:
3295:
3281:
3277:
3276:
3269:
3265:
3239:
3238:
3215:
3184:Rolle's property
3165:
3161:
3154:
3146:
3142:
3138:
3131:
3117:
3099:
3092:
3088:
3084:
3080:
3076:
3069:
3062:
3043:
3037:
3030:
3026:
3020:
3016:
3008:
2997:
2993:
2989:
2985:
2973:
2965:
2961:
2957:
2932:
2920:
2876:
2869:
2857:
2853:
2837:
2830:
2822:
2803:
2799:
2795:
2788:
2786:
2785:
2780:
2769:
2764:
2729:
2726:
2725:
2724:
2698:
2697:
2685:
2670:
2668:
2667:
2662:
2651:
2646:
2611:
2602:
2592:
2590:
2589:
2584:
2573:
2568:
2533:
2530:
2529:
2528:
2502:
2501:
2489:
2474:
2472:
2471:
2466:
2455:
2450:
2415:
2406:
2399:
2395:
2391:
2380:
2365:
2353:
2343:
2336:
2329:
2317:
2307:
2295:
2291:
2275:
2271:
2259:
2243:
2231:
2224:
2220:
2216:
2212:
2208:
2200:
2196:
2166:
2164:
2163:
2158:
2137:
2136:
2124:
2110:
2109:
2097:
2083:
2082:
2070:
2044:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2005:
2004:
1999:
1994:
1993:
1981:
1972:
1969:
1963:
1962:
1950:
1935:
1923:
1919:
1910:
1908:
1907:
1902:
1900:
1895:
1860:
1857:
1856:
1855:
1829:
1828:
1816:
1801:
1799:
1798:
1793:
1791:
1786:
1751:
1748:
1747:
1746:
1720:
1719:
1707:
1692:right-hand limit
1689:
1677:
1673:
1654:
1642:
1627:
1611:
1607:
1603:
1596:
1592:
1585:
1574:
1570:
1557:
1555:
1554:
1549:
1516:
1508:
1461:
1457:
1438:
1434:
1427:
1415:
1393:
1391:
1390:
1385:
1352:
1350:
1349:
1337:
1336:
1327:
1303:
1293:
1239:Taylor's theorem
1230:
1228:
1227:
1222:
1205:
1190:
1178:
1174:
1155:
1136:
1117:
1093:stationary point
1070:
1059:
1047:
1043:
1024:
1005:
986:
965:
958:
951:
899:
864:
830:
826:
793:Surface integral
736:
732:
640:
636:
596:Limit comparison
516:
512:
403:Riemann integral
356:
352:
312:L'Hôpital's rule
269:Taylor's theorem
190:
186:
130:
128:
127:
122:
74:
65:
60:
30:
21:
3668:
3667:
3663:
3662:
3661:
3659:
3658:
3657:
3638:
3637:
3602:"Rolle theorem"
3600:
3597:
3581:
3575:
3561:Leithold, Louis
3559:
3556:
3554:Further reading
3551:
3550:
3535:10.2307/2695770
3518:
3517:
3513:
3485:
3484:
3480:
3467:
3466:
3462:
3444:
3443:
3439:
3432:
3416:Cajori, Florian
3414:
3412:
3408:
3400:
3395:
3394:
3390:
3385:
3358:
3350:Voorhoeve index
3340:
3334:
3286:
3282:
3256:
3252:
3230:
3222:
3221:
3194:
3191:complex numbers
3172:
3163:
3156:
3148:
3144:
3140:
3133:
3128:
3119:
3114:
3107:
3101:
3098:
3094:
3090:
3086:
3082:
3078:
3071:
3064:
3057:
3052:The proof uses
3050:
3039:
3035:
3028:
3022:
3018:
3014:
3007:[−3, 2]
3006:
2995:
2991:
2987:
2975:
2971:
2963:
2959:
2954:
2943:
2934:
2922:
2918:
2911:
2905:
2898:
2891:
2884:
2878:
2874:
2859:
2855:
2843:
2832:
2828:
2820:
2817:
2801:
2797:
2793:
2730:
2716:
2689:
2678:
2673:
2672:
2612:
2605:
2604:
2597:
2534:
2520:
2493:
2482:
2477:
2476:
2416:
2409:
2408:
2401:
2397:
2393:
2382:
2367:
2355:
2345:
2341:
2331:
2319:
2315:
2297:
2293:
2281:
2273:
2261:
2249:
2233:
2229:
2228:By assumption,
2222:
2218:
2214:
2210:
2206:
2198:
2179:
2173:
2128:
2117:
2101:
2090:
2074:
2063:
2058:
2057:
2042:
2038:
2029:
2025:
2021:
2017:
2013:
2009:
1985:
1974:
1954:
1943:
1938:
1937:
1925:
1921:
1918:[−∞, ∞]
1917:
1861:
1847:
1820:
1809:
1804:
1803:
1752:
1738:
1711:
1700:
1695:
1694:
1679:
1675:
1674:. If for every
1656:
1644:
1640:
1634:
1617:
1614:critical number
1609:
1605:
1598:
1594:
1587:
1576:
1572:
1561:
1484:
1483:
1468:
1459:
1440:
1436:
1429:
1417:
1405:
1341:
1328:
1306:
1305:
1298:
1291:
1284:
1279:
1247:
1198:
1193:
1192:
1180:
1176:
1157:
1145:
1126:
1124:closed interval
1115:
1105:
1081:Rolle's theorem
1061:
1049:
1045:
1026:
1014:
995:
993:closed interval
984:
969:
940:
939:
925:Integration Bee
900:
897:
890:
889:
865:
862:
855:
854:
827:
824:
817:
816:
798:Volume integral
733:
728:
721:
720:
637:
632:
625:
624:
594:
513:
508:
501:
500:
492:Risch algorithm
467:Euler's formula
353:
348:
341:
340:
322:General Leibniz
205:generalizations
187:
182:
175:
161:Rolle's theorem
156:
131:
67:
47:
46:
28:
23:
22:
15:
12:
11:
5:
3666:
3664:
3656:
3655:
3650:
3640:
3639:
3636:
3635:
3626:
3616:
3596:
3595:External links
3593:
3592:
3591:
3579:
3573:
3555:
3552:
3549:
3548:
3511:
3498:(4): 801–817,
3478:
3460:
3456:, pp. 3–4
3437:
3430:
3406:
3387:
3386:
3384:
3381:
3380:
3379:
3374:
3369:
3364:
3357:
3354:
3327:Kaplansky 1972
3314:
3310:
3302:
3298:
3292:
3289:
3285:
3280:
3272:
3268:
3262:
3259:
3255:
3251:
3248:
3245:
3242:
3237:
3233:
3229:
3171:
3168:
3147:such that the
3126:
3112:
3105:
3096:
3049:
3046:
2986:such that the
2968:
2967:
2952:
2941:
2916:
2909:
2903:
2896:
2889:
2882:
2871:
2816:
2813:
2804:must be zero.
2778:
2775:
2772:
2767:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2723:
2719:
2715:
2712:
2708:
2704:
2701:
2696:
2692:
2688:
2684:
2681:
2660:
2657:
2654:
2649:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2582:
2579:
2576:
2571:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2527:
2523:
2519:
2516:
2512:
2508:
2505:
2500:
2496:
2492:
2488:
2485:
2464:
2461:
2458:
2453:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2313:interior point
2172:
2169:
2168:
2167:
2156:
2153:
2150:
2147:
2143:
2140:
2135:
2131:
2127:
2123:
2120:
2116:
2113:
2108:
2104:
2100:
2096:
2093:
2089:
2086:
2081:
2077:
2073:
2069:
2066:
2046:
2037:
2034:
1997:
1992:
1988:
1984:
1980:
1977:
1966:
1961:
1957:
1953:
1949:
1946:
1898:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1854:
1850:
1846:
1843:
1839:
1835:
1832:
1827:
1823:
1819:
1815:
1812:
1789:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1745:
1741:
1737:
1734:
1730:
1726:
1723:
1718:
1714:
1710:
1706:
1703:
1633:
1632:Generalization
1630:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1519:
1515:
1511:
1507:
1503:
1500:
1497:
1494:
1491:
1480:absolute value
1467:
1466:Second example
1464:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1355:
1348:
1344:
1340:
1335:
1331:
1325:
1322:
1319:
1316:
1313:
1283:
1280:
1278:
1275:
1246:
1243:
1220:
1217:
1214:
1211:
1208:
1204:
1201:
1139:differentiable
1104:
1101:
1008:differentiable
971:
970:
968:
967:
960:
953:
945:
942:
941:
938:
937:
932:
927:
922:
920:List of topics
917:
912:
907:
901:
896:
895:
892:
891:
888:
887:
882:
877:
872:
866:
861:
860:
857:
856:
851:
850:
849:
848:
843:
838:
828:
823:
822:
819:
818:
813:
812:
811:
810:
805:
800:
795:
790:
785:
780:
772:
771:
767:
766:
765:
764:
759:
754:
749:
741:
740:
734:
727:
726:
723:
722:
717:
716:
715:
714:
709:
704:
699:
694:
689:
681:
680:
676:
675:
674:
673:
668:
663:
658:
653:
648:
638:
631:
630:
627:
626:
621:
620:
619:
618:
613:
608:
603:
598:
592:
587:
582:
577:
572:
564:
563:
557:
556:
555:
554:
549:
544:
539:
534:
529:
514:
507:
506:
503:
502:
497:
496:
495:
494:
489:
484:
479:
477:Changing order
474:
469:
464:
446:
441:
436:
428:
427:
426:Integration by
423:
422:
421:
420:
415:
410:
405:
400:
390:
388:Antiderivative
382:
381:
377:
376:
375:
374:
369:
364:
354:
347:
346:
343:
342:
337:
336:
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
281:
280:
274:
273:
272:
271:
266:
261:
256:
251:
246:
238:
237:
233:
232:
231:
230:
229:
228:
223:
218:
208:
195:
194:
188:
181:
180:
177:
176:
174:
173:
168:
163:
157:
155:
154:
149:
143:
142:
141:
133:
132:
120:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
83:
80:
77:
73:
70:
64:
59:
55:
45:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3665:
3654:
3651:
3649:
3646:
3645:
3643:
3634:
3630:
3627:
3624:
3620:
3617:
3613:
3609:
3608:
3603:
3599:
3598:
3594:
3588:
3584:
3580:
3576:
3574:0-06-043959-9
3570:
3566:
3562:
3558:
3557:
3553:
3544:
3540:
3536:
3532:
3528:
3524:
3523:
3515:
3512:
3506:
3501:
3497:
3493:
3489:
3482:
3479:
3474:
3470:
3464:
3461:
3455:
3451:
3447:
3441:
3438:
3433:
3431:9780821821022
3427:
3423:
3422:
3417:
3410:
3407:
3399:
3392:
3389:
3382:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3359:
3355:
3353:
3351:
3346:
3343:
3337:
3332:
3331:finite fields
3328:
3312:
3308:
3300:
3296:
3290:
3287:
3283:
3278:
3270:
3266:
3260:
3257:
3253:
3249:
3246:
3243:
3240:
3235:
3231:
3227:
3219:
3213:
3209:
3205:
3201:
3197:
3192:
3187:
3185:
3181:
3177:
3176:ordered field
3169:
3167:
3159:
3152:
3136:
3129:
3122:
3115:
3108:
3074:
3067:
3060:
3055:
3047:
3045:
3042:
3032:
3025:
3003:
2999:
2983:
2979:
2955:
2948:
2944:
2937:
2930:
2926:
2919:
2912:
2902:
2895:
2888:
2881:
2872:
2867:
2863:
2851:
2847:
2841:
2835:
2827:the function
2826:
2825:
2824:
2814:
2812:
2810:
2805:
2790:
2776:
2773:
2770:
2765:
2758:
2752:
2749:
2743:
2740:
2737:
2731:
2721:
2717:
2710:
2702:
2694:
2690:
2682:
2679:
2658:
2655:
2652:
2647:
2640:
2634:
2631:
2625:
2622:
2619:
2613:
2600:
2594:
2580:
2577:
2574:
2569:
2562:
2556:
2553:
2547:
2544:
2541:
2535:
2525:
2521:
2514:
2506:
2498:
2494:
2486:
2483:
2462:
2459:
2456:
2451:
2444:
2438:
2435:
2429:
2426:
2423:
2417:
2404:
2389:
2385:
2378:
2374:
2370:
2363:
2359:
2352:
2348:
2338:
2335:
2327:
2323:
2314:
2309:
2305:
2301:
2289:
2285:
2279:
2269:
2265:
2257:
2253:
2247:
2244:, and by the
2241:
2237:
2226:
2204:
2194:
2190:
2186:
2182:
2176:
2170:
2154:
2151:
2148:
2145:
2141:
2133:
2129:
2121:
2118:
2114:
2106:
2102:
2094:
2091:
2087:
2079:
2075:
2067:
2064:
2055:
2051:
2047:
2040:
2039:
2035:
2033:
1990:
1986:
1978:
1975:
1959:
1955:
1947:
1944:
1933:
1929:
1916:
1913:exist in the
1911:
1896:
1889:
1883:
1880:
1874:
1871:
1868:
1862:
1852:
1848:
1841:
1833:
1825:
1821:
1813:
1810:
1787:
1780:
1774:
1771:
1765:
1762:
1759:
1753:
1743:
1739:
1732:
1724:
1716:
1712:
1704:
1701:
1693:
1687:
1683:
1671:
1667:
1663:
1659:
1652:
1648:
1637:
1631:
1629:
1625:
1621:
1615:
1601:
1590:
1583:
1579:
1568:
1564:
1558:
1545:
1539:
1536:
1533:
1530:
1524:
1521:
1517:
1509:
1501:
1495:
1489:
1481:
1472:
1465:
1463:
1455:
1451:
1447:
1443:
1433:
1425:
1421:
1413:
1409:
1403:
1400:is the upper
1399:
1394:
1381:
1375:
1372:
1369:
1366:
1360:
1357:
1353:
1346:
1342:
1338:
1333:
1329:
1323:
1317:
1311:
1301:
1297:For a radius
1288:
1282:First example
1281:
1276:
1274:
1272:
1268:
1264:
1260:
1256:
1252:
1244:
1242:
1240:
1236:
1231:
1218:
1215:
1209:
1202:
1199:
1188:
1184:
1172:
1168:
1164:
1160:
1153:
1149:
1144:
1143:open interval
1140:
1134:
1130:
1125:
1121:
1114:
1110:
1102:
1100:
1098:
1094:
1090:
1086:
1085:Rolle's lemma
1082:
1078:
1068:
1064:
1057:
1053:
1041:
1037:
1033:
1029:
1022:
1018:
1013:
1012:open interval
1009:
1003:
999:
994:
990:
982:
977:
966:
961:
959:
954:
952:
947:
946:
944:
943:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
902:
894:
893:
886:
883:
881:
878:
876:
873:
871:
868:
867:
859:
858:
847:
844:
842:
839:
837:
834:
833:
832:
831:
821:
820:
809:
806:
804:
801:
799:
796:
794:
791:
789:
788:Line integral
786:
784:
781:
779:
776:
775:
774:
773:
768:
763:
760:
758:
755:
753:
750:
748:
745:
744:
743:
742:
737:
731:
730:Multivariable
725:
724:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
684:
683:
682:
677:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
643:
642:
641:
635:
629:
628:
617:
614:
612:
609:
607:
604:
602:
599:
597:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
567:
566:
565:
562:
558:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
527:
523:
520:
519:
518:
517:
511:
505:
504:
493:
490:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
462:
458:
454:
453:trigonometric
450:
447:
445:
442:
440:
437:
435:
432:
431:
430:
429:
424:
419:
416:
414:
411:
409:
406:
404:
401:
398:
394:
391:
389:
386:
385:
384:
383:
378:
373:
370:
368:
365:
363:
360:
359:
358:
357:
351:
345:
344:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
283:
282:
279:
275:
270:
267:
265:
264:Related rates
262:
260:
257:
255:
252:
250:
247:
245:
242:
241:
240:
239:
234:
227:
224:
222:
221:of a function
219:
217:
216:infinitesimal
214:
213:
212:
209:
206:
202:
199:
198:
197:
196:
191:
185:
179:
178:
172:
169:
167:
164:
162:
159:
158:
153:
150:
148:
145:
144:
140:
137:
136:
135:
134:
115:
109:
106:
100:
94:
91:
88:
85:
78:
71:
68:
62:
57:
53:
44:
43:
40:
36:
32:
31:
19:
3629:Mizar system
3623:cut-the-knot
3605:
3586:
3564:
3526:
3520:
3514:
3495:
3491:
3481:
3472:
3463:
3449:
3440:
3420:
3409:
3391:
3347:
3341:
3335:
3211:
3207:
3203:
3199:
3195:
3188:
3183:
3173:
3157:
3150:
3134:
3124:
3120:
3110:
3103:
3072:
3065:
3058:
3051:
3040:
3033:
3027:in place of
3023:
3012:
2981:
2977:
2969:
2950:
2946:
2939:
2935:
2928:
2924:
2914:
2907:
2900:
2893:
2886:
2879:
2865:
2861:
2849:
2845:
2833:
2818:
2806:
2791:
2598:
2595:
2402:
2387:
2383:
2376:
2372:
2368:
2366:, the value
2361:
2357:
2350:
2346:
2339:
2333:
2325:
2321:
2310:
2303:
2299:
2287:
2283:
2267:
2263:
2255:
2251:
2239:
2235:
2227:
2192:
2188:
2184:
2180:
2177:
2174:
1931:
1927:
1912:
1685:
1681:
1669:
1665:
1661:
1657:
1650:
1646:
1638:
1635:
1623:
1619:
1599:
1588:
1581:
1577:
1566:
1562:
1559:
1477:
1453:
1449:
1445:
1441:
1431:
1423:
1419:
1411:
1407:
1395:
1299:
1296:
1251:Michel Rolle
1248:
1232:
1186:
1182:
1170:
1166:
1162:
1158:
1151:
1147:
1132:
1128:
1122:on a proper
1106:
1097:Michel Rolle
1084:
1080:
1074:
1066:
1062:
1055:
1051:
1039:
1035:
1031:
1027:
1020:
1016:
1001:
997:
449:Substitution
211:Differential
184:Differential
160:
3446:Artin, Emil
3056:. The case
2811:directly.)
2340:For a real
905:Precalculus
898:Miscellanea
863:Specialized
770:Definitions
537:Alternating
380:Definitions
193:Definitions
3642:Categories
3383:References
3118:such that
3089:from 1 to
3044:vanishes.
2962:from 1 to
2958:for every
2933:such that
2873:there are
2344:such that
2028:exists at
1402:semicircle
1191:such that
1120:continuous
1060:such that
989:continuous
885:Variations
880:Stochastic
870:Fractional
739:Formalisms
702:Divergence
671:Identities
651:Divergence
201:Derivative
152:Continuity
3612:EMS Press
3448:(1964) ,
3261:−
3241:−
3218:rationals
3166:is zero.
2998:is zero.
2771:≥
2750:−
2722:−
2714:→
2695:−
2653:≥
2632:−
2575:≤
2554:−
2518:→
2457:≤
2436:−
2213:, say at
2134:−
2115:≤
2088:≤
2080:−
2050:convexity
1991:−
1881:−
1853:−
1845:→
1826:−
1772:−
1736:→
1531:−
1525:∈
1482:function
1367:−
1361:∈
1339:−
875:Malliavin
762:Geometric
661:Laplacian
611:Dirichlet
522:Geometric
107:−
54:∫
3585:(1955).
3563:(1972).
3471:(1972),
3418:(1999).
3356:See also
3158:f
3121:f
3041:f
3024:f
2947:f
2936:f
2854:and the
2683:′
2487:′
2392:because
2384:f
2369:f
2334:f
2278:constant
2189:f
2181:f
2122:′
2095:′
2068:′
1979:′
1948:′
1814:′
1705:′
1666:f
1658:f
1578:f
1567:f
1563:f
1450:f
1442:f
1439:. Since
1277:Examples
1203:′
1167:f
1159:f
1113:function
1111:-valued
1077:calculus
1063:f
1036:f
1028:f
915:Glossary
825:Advanced
803:Jacobian
757:Exterior
687:Gradient
679:Theorems
646:Gradient
585:Integral
547:Binomial
532:Harmonic
397:improper
393:Integral
350:Integral
332:Reynolds
307:Quotient
236:Concepts
72:′
39:Calculus
3631:proof:
3614:, 2001
3543:2695770
2272:, then
2197:, then
2036:Remarks
1565:(−1) =
1245:History
1141:on the
1131:,
1010:on the
910:History
808:Hessian
697:Stokes'
692:Green's
524: (
451: (
395: (
317:Inverse
292:Product
203: (
3571:
3541:
3428:
3329:. For
3180:fields
3068:> 1
2906:≤ ⋯ ≤
2838:times
2671:hence
2601:< 0
2475:hence
2405:> 0
2354:is in
1406:[−
1302:> 0
1259:Cauchy
1156:, and
1107:If a
1025:, and
752:Tensor
747:Matrix
634:Vector
552:Taylor
510:Series
147:Limits
3539:JSTOR
3401:(PDF)
3210:− 1)(
3141:, …,
3130:) = 0
3048:Proof
2931:]
2923:[
2913:<
2899:<
2885:<
2870:, and
2852:]
2844:[
2364:]
2356:[
2290:]
2282:[
2270:]
2262:[
2258:]
2250:[
2242:]
2234:[
1655:with
1653:]
1645:[
1560:Then
1414:]
1398:graph
1135:]
1127:[
1069:) = 0
1004:]
996:[
991:on a
979:If a
575:Ratio
542:Power
461:Euler
439:Discs
434:Parts
302:Power
297:Chain
226:total
3569:ISBN
3426:ISBN
3413:See
3339:and
3214:+ 1)
3153:− 1)
2945:) =
2209:and
2187:) =
2149:<
1690:the
1664:) =
1448:) =
1435:and
1396:Its
1165:) =
1109:real
1034:) =
981:real
656:Curl
616:Abel
580:Root
3621:at
3531:doi
3527:109
3500:doi
3162:at
3137:− 1
3075:− 1
3061:= 1
2994:at
2974:in
2921:in
2836:− 1
2831:is
2800:at
2707:lim
2511:lim
2318:of
2280:on
2276:is
2041:If
2014:≤ 0
2010:≥ 0
2008:is
1970:and
1838:lim
1729:lim
1602:= 0
1591:= 0
1569:(1)
1118:is
1083:or
1075:In
987:is
287:Sum
3644::
3610:,
3604:,
3537:,
3525:,
3496:21
3494:,
3490:,
3352:.
3202:=
3198:−
3123:′(
3109:,
3031:.
2980:,
2927:,
2892:≤
2864:,
2848:,
2703::=
2507::=
2407:,
2375:+
2360:,
2349:+
2324:,
2308:.
2302:,
2286:,
2266:,
2254:,
2238:,
2225:.
2056::
1930:,
1834::=
1725::=
1684:,
1649:,
1622:,
1580:′(
1444:(−
1422:,
1418:(−
1410:,
1241:.
1219:0.
1185:,
1150:,
1137:,
1099:.
1079:,
1065:′(
1054:,
1019:,
1006:,
1000:,
459:,
455:,
3625:.
3577:.
3546:.
3533::
3509:.
3502::
3476:.
3458:.
3434:.
3403:.
3342:F
3336:F
3313:,
3309:)
3301:3
3297:1
3291:+
3288:x
3284:(
3279:)
3271:3
3267:1
3258:x
3254:(
3250:3
3247:=
3244:1
3236:2
3232:x
3228:3
3212:x
3208:x
3206:(
3204:x
3200:x
3196:x
3164:c
3160:′
3151:n
3149:(
3145:c
3135:n
3127:k
3125:c
3116:)
3113:k
3111:b
3106:k
3104:a
3102:(
3097:k
3095:c
3091:n
3087:k
3083:f
3079:n
3073:n
3066:n
3059:n
3036:n
3029:f
3019:f
3015:n
2996:c
2992:f
2988:n
2984:)
2982:b
2978:a
2976:(
2972:c
2966:.
2964:n
2960:k
2956:)
2953:k
2951:b
2949:(
2942:k
2940:a
2938:(
2929:b
2925:a
2917:n
2915:b
2910:n
2908:a
2904:2
2901:b
2897:2
2894:a
2890:1
2887:b
2883:1
2880:a
2875:n
2868:)
2866:b
2862:a
2860:(
2856:n
2850:b
2846:a
2834:n
2829:f
2821:f
2802:c
2798:f
2794:f
2777:,
2774:0
2766:h
2762:)
2759:c
2756:(
2753:f
2747:)
2744:h
2741:+
2738:c
2735:(
2732:f
2718:0
2711:h
2700:)
2691:c
2687:(
2680:f
2659:,
2656:0
2648:h
2644:)
2641:c
2638:(
2635:f
2629:)
2626:h
2623:+
2620:c
2617:(
2614:f
2599:h
2581:,
2578:0
2570:h
2566:)
2563:c
2560:(
2557:f
2551:)
2548:h
2545:+
2542:c
2539:(
2536:f
2526:+
2522:0
2515:h
2504:)
2499:+
2495:c
2491:(
2484:f
2463:,
2460:0
2452:h
2448:)
2445:c
2442:(
2439:f
2433:)
2430:h
2427:+
2424:c
2421:(
2418:f
2403:h
2398:c
2394:f
2390:)
2388:c
2386:(
2379:)
2377:h
2373:c
2371:(
2362:b
2358:a
2351:h
2347:c
2342:h
2332:−
2328:)
2326:b
2322:a
2320:(
2316:c
2306:)
2304:b
2300:a
2298:(
2294:f
2288:b
2284:a
2274:f
2268:b
2264:a
2256:b
2252:a
2240:b
2236:a
2230:f
2223:c
2219:c
2215:c
2211:b
2207:a
2199:f
2195:)
2193:b
2191:(
2185:a
2183:(
2155:.
2152:y
2146:x
2142:,
2139:)
2130:y
2126:(
2119:f
2112:)
2107:+
2103:x
2099:(
2092:f
2085:)
2076:x
2072:(
2065:f
2043:f
2030:c
2026:f
2022:c
2018:x
1996:)
1987:c
1983:(
1976:f
1965:)
1960:+
1956:c
1952:(
1945:f
1934:)
1932:b
1928:a
1926:(
1922:c
1897:h
1893:)
1890:x
1887:(
1884:f
1878:)
1875:h
1872:+
1869:x
1866:(
1863:f
1849:0
1842:h
1831:)
1822:x
1818:(
1811:f
1788:h
1784:)
1781:x
1778:(
1775:f
1769:)
1766:h
1763:+
1760:x
1757:(
1754:f
1744:+
1740:0
1733:h
1722:)
1717:+
1713:x
1709:(
1702:f
1688:)
1686:b
1682:a
1680:(
1676:x
1672:)
1670:b
1668:(
1662:a
1660:(
1651:b
1647:a
1641:f
1626:)
1624:b
1620:a
1618:(
1610:f
1606:x
1600:x
1595:f
1589:x
1584:)
1582:c
1573:c
1546:.
1543:]
1540:1
1537:,
1534:1
1528:[
1522:x
1518:,
1514:|
1510:x
1506:|
1502:=
1499:)
1496:x
1493:(
1490:f
1460:f
1456:)
1454:r
1452:(
1446:r
1437:r
1432:r
1430:−
1426:)
1424:r
1420:r
1412:r
1408:r
1382:.
1379:]
1376:r
1373:,
1370:r
1364:[
1358:x
1354:,
1347:2
1343:x
1334:2
1330:r
1324:=
1321:)
1318:x
1315:(
1312:f
1300:r
1292:r
1216:=
1213:)
1210:c
1207:(
1200:f
1189:)
1187:b
1183:a
1181:(
1177:c
1173:)
1171:b
1169:(
1163:a
1161:(
1154:)
1152:b
1148:a
1146:(
1133:b
1129:a
1116:f
1071:.
1067:c
1058:)
1056:b
1052:a
1050:(
1046:c
1042:)
1040:b
1038:(
1032:a
1030:(
1023:)
1021:b
1017:a
1015:(
1002:b
998:a
985:f
964:e
957:t
950:v
528:)
463:)
399:)
207:)
119:)
116:a
113:(
110:f
104:)
101:b
98:(
95:f
92:=
89:t
86:d
82:)
79:t
76:(
69:f
63:b
58:a
20:)
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