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高等数学公式

导数公式:

(tanx)??sec2x(cotx)???csc2x(secx)??secx?tanx(cscx)???cscx?cotx(ax)??axlna(logax)??基本积分表:

(arcsinx)??11xlna1?x21(arccosx)???1?x21(arctanx)??1?x21(arccotx)???1?x2?tanxdx??lncosx?C?cotxdx?lnsinx?C?secxdx?lnsecx?tanx?C?cscxdx?lncscx?cotx?Cdx1x?arctan?C?a2?x2aadx1x?a?ln?x2?a22ax?a?Cdx1a?x??a2?x22alna?x?Cdxx?a2?x2?arcsina?C?2ndx2?sec?cos2x?xdx?tanx?Cdx2?csc?sin2x?xdx??cotx?C?secx?tanxdx?secx?C?cscx?cotxdx??cscx?Cax?adx?lna?Cx?shxdx?chx?C?chxdx?shx?C?dxx2?a2?ln(x?x2?a2)?C?2In??sinxdx??cosnxdx?00n?1In?2n???

考研

xa222x?adx?x?a?ln(x?x2?a2)?C22xa22222x?adx?x?a?lnx?x2?a2?C22xa2x2222a?xdx?a?x?arcsin?C22a22

三角函数的有理式积分:

2u1?u2x2dusinx?, cosx?, u?tg, dx? 22221?u1?u1?uA.积化和差公式:

sin?cos??1?sin(???)?sin(???)?2cos?sin??1?sin(???)?sin(???)?2cos?cos??1?cos(???)?cos(???)? sin?sin???1?cos(???)?cos?????? 22B.和差化积公式:

①sin??sin??2sin???2222????????????③cos??cos??2cos ④cos??cos???2sin cossin2222bca1.正弦定理:=== 2R (R为三角形外接圆半径)

sinAsinBsinCcos??? ②sin??sin??2cos???sin???

2..余弦定理:a2=b2+c2-2bccosA b2=a2+c2-2accosB c2=a2+b2-2abcosC

b2?c2?a2cosA?

2bc1111abc3.S⊿=a?ha=absinC=bcsinA=acsinB==2R2sinAsinBsinC

22224Ra2sinBsinCb2sinAsinCc2sinAsinB====pr=p(p?a)(p?b)(p?c)

2sinA2sinB2sinC1 (其中p?(a?b?c), r为三角形内切圆半径) 4.诱导公试

2 -? ?-? sin -sin? cos tan cot -ctg? -ctg? +ctg? -ctg? +ctg?

三角函数值等于?的同名三角函数值,前面加上一个把?看作锐角时,原三角函数值的符号;即:函数名不变,符号看象限

+cos? -tg? -tg? +tg? +sin? -cos? -sin? -sin? -cos? ?+? 2?-? 考研 ?+? 2k+cos? -tg? +sin? +cos? +tg?

?2?? ?? sin +cos? +cos? -cos? -cos? cos +sin? -sin? -sin? tan +ctg? -ctg? +ctg? cot +tg? -tg? +tg? -tg?

?23??? 23??? 2+sin? -ctg? 5.和差角公式

①sin(???)?sin?cos??cos?sin? ②cos(???)?cos?cos??sin?sin? ③tg(???)?tg??tg? ④tg??tg??tg(???)(1?tg??tg?)

1?tg??tg?6.二倍角公式:(含万能公式)

①sin2??2sin?cos??222tg? 1?tg2?221?tg2?②cos2??cos??sin??2cos??1?1?2sin??

1?tg2?tg2?1?cos2?2tg?1?cos2?22sin???③tg2?? ④ ⑤cos??1?tg2?21?tg2?2 ?7.半角公式:(符号的选择由所在的象限确定)

2①sin?2???1?cos??1?cos??1?cos? ②sin2? ③cos?? 222221?cos??? ⑤1?cos??2sin2 ⑥1?cos??2cos2 222④cos2?2⑦1?sin??(cos?sin)2?cos?sin222????2

考研

⑧tg?2??1?cos?sin?1?cos???

1?cos?1?cos?sin?高阶导数公式——莱布尼兹(Leibniz)公式:

(uv)(n)k(n?k)(k)??Cnuvk?0nn(n?1)(n?2)n(n?1)?(n?k?1)(n?k)(k)?u(n)v?nu(n?1)v??uv?????uv???uv(n)2!k!中值定理与导数应用:

拉格朗日中值定理:f(b)?f(a)?f?(?)(b?a)f(b)?f(a)f?(?)柯西中值定理:?F(b)?F(a)F?(?)多元函数微分法及应用

当F(x)?x时,柯西中值定理就是拉格朗日中值定理。全微分:dz??z?z?u?u?udx?dy   du?dx?dy?dz?x?y?x?y?z全微分的近似计算:?z?dz?fx(x,y)?x?fy(x,y)?y多元复合函数的求导法:dz?z?u?z?vz?f[u(t),v(t)]   ???? dt?u?t?v?t?z?z?u?z?vz?f[u(x,y),v(x,y)]   ? ????x?u?x?v?x当u?u(x,y),v?v(x,y)时,?u?u?v?vdu?dx?dy   dv?dx?dy ?x?y?x?y隐函数的求导公式:FxFFdydyd2y??隐函数F(x,y)?0,  ??,  2?(?x)+(?x)?dxFy?xFy?yFydxdxFyF?z?z隐函数F(x,y,z)?0, ??x,  ???xFz?yFz

多元函数的极值及其求法:

考研

设fx(x0,y0)?fy(x0,y0)?0,令:fxx(x0,y0)?A, fxy(x0,y0)?B, fyy(x0,y0)?C??A?0,(x0,y0)为极大值2AC?B?0时,???A?0,(x0,y0)为极小值??2则:值?AC?B?0时,      无极?AC?B2?0时,       不确定???

常数项级数:

1?qn等比数列:1?q?q???q?1?q(n?1)n 等差数列:1?2?3???n?2111调和级数:1?????是发散的23n2n?1级数审敛法:

1、正项级数的审敛法——根植审敛法(柯西判别法):???1时,级数收敛?设:??limnun,则???1时,级数发散n?????1时,不确定?2、比值审敛法:???1时,级数收敛U?设:??limn?1,则???1时,级数发散n??Un???1时,不确定?3、定义法:sn?u1?u2???un;limsn存在,则收敛;否则发散。n??

交错级数u1?u2?u3?u4??(或?u1?u2?u3??,un?0)的审敛法——莱布尼兹定理:? ?un?un?1如果交错级数满足,那么级数收敛且其和s?u,其余项r的绝对值r?u。?limu?01nnn?1n??n??绝对收敛与条件收敛:

(1)u1?u2???un??,其中un为任意实数;(2)u1?u2?u3???un??如果(2)发散,而(1)收敛,则称(1)为条件收敛级数。调和级数:?1(?1)n发散,而?收敛;nn1  级数:?n2收敛;p?1时发散1  p级数:?np  p?1时收敛

如果(2)收敛,则(1)肯定收敛,且称为绝对收敛级数;考研