考研数学公式大全(高清版).doc
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1、愿将过往寄存于星空,抬头皆是耀眼的感动! 高等数学公式导数公式:1(arcsin x) =(tgx) = sec x21 x2(ctgx) = csc x21(arccos x) = (secx) = secxtgx1 x2(cscx) = cscxctgx1+(arctgx) =(ax) = axlna1 x2111+ x(log x) =(arcctgx) = a2xlna基本积分表:tgxdx= ln cos x +Cdx= sec2xdx = tgx + Ccosdx2xctgxdx = ln sin x +C= csc2xdx = ctgx + Csec xdx = ln sec x
2、 +tgx +Ccsc xdx = ln csc x ctgx +C2sin xsec xtgxdx = sec x + Ccsc xctgxdx = csc x + Cdx+ xdx1x= arctg +Caxa2222aax aaxaxdx =+ C1ln ashxdx = chx + C=ln+C a2a x + adx xdx1a + xln+Cchxdx = shx + C222a a xdxx= ln(x + x2 a ) + C2= arcsin +C2 a2a2x2ax22n 1I = sinnxdx = cosnxdx =In2nn00xa2xxa222+ a a x222d
3、x =dx =dx =xxa222+ a222+ln(x + x2+ a a2) + C+ C2x2x22aa2 a xln x + x2222xarcsin + C2a三角函数的有理式积分:2u ,cos x = 1u22x2dusin x =,u = tg ,dx =1+ u21+ u21+ u2 一些初等函数:两个重要极限:xx exsin x双曲正弦: shx = elim=12x0x+ ex1x双曲余弦:chx = elim(1+ ) = e = 2.718281828459045.2xxshxchxexx e+ exx双曲正切:thx =earshx = ln(x + x2+1)1
4、)archx = ln(x + x211+ x1 xarthx = ln2三角函数公式:诱导公式:函数sincostgctg角 A-sin cos -tg -ctgcos sin ctg tgcos -sin -ctg -tgsin -cos -tg -ctg-sin -cos tg ctg-cos -sin ctg tg-cos sin -ctg -tg-sin cos -tg -ctgsin cos tg ctg90-90+180-180+270-270+360-360+和差角公式:和差化积公式: + 2 2 sin( ) = sin cos cos sin cos( ) = cos co
5、s m sin sin sin + sin = 2sincossin + sin sin = 2 coscos + cos = 2 coscos cos = 2 sintg tg1m tg tgtg( ) =22 + 2 cosctg ctg m1ctg ctg2 2ctg( ) = + sin2 倍角公式:sin 2 = 2 sin cossin 3 = 3sin 4sincos 3 = 4 cos 3cos3tg tg3cos 2 = 2 cos2 1=1 2 sin2 = cos2 sin23ctg2 12ctg2tgctg2 =3tg3 =13tg 2tg2 =1tg 2半角公式:si
6、n = 21 cos1+ coscos = 2221 cos 1 cossin1+ cos1+ cos 1+ cossin1 costg = =ctg = =21+ cossin21 cossinabc正弦定理:= 2R余弦定理:c2 a2 b2 2abcosC=+sin A sin B sinC反三角函数性质:arcsin x = arccosxarctgx = arcctgx22高阶导数公式莱布尼兹(Leibniz)公式:n(uv)(n)=Cnku(nk) (k)vk=0n(n 1)n(n 1)L(n k +1)k!=u(n)v nu(n 1)v+u(n2)v+L+u(nk)v(k) +L
7、+ uv(n)2!中值定理与导数应用:拉格朗日中值定理:f b f a b a)( ) ( ) = ()(f( ) ( )()ff b f a柯西中值定理:=( ) ( ) F()F b F a当F(x) = x时,柯西中值定理就是拉格朗日中值定理。曲率:弧微分公式: = + 其中 ds1 y dx, y = tg2s平均曲率: =K. 从 点到 点,切线斜率的倾角变化量; :: MMs MM弧长。y+ (1 y=dM点的曲率:K = lim=.sdss0)2 3直线:K = 0;1半径为a的圆:K = .a 定积分的近似计算:bb a矩形法:f (x) (y + y +L+ y )0 1 n
8、1nabb a 1梯形法:f (x) (y + y ) + y +L+ y 0n1n1n2abb a抛物线法:f (x) (y + y ) + 2(y + y +L+ y ) + 4(y + y +L+ y )0n24n213n13na定积分应用相关公式:功:W = F s水压力:F = p Am m引力:F = k12,k为引力系数r2b1b a函数的平均值:y =f (x)dxab1b a均方根:2f (t)dta空间解析几何和向量代数:空间2点的距离:d = M M = (x x )2+ (y y )2+ (z z )212212121向量在轴上的投影:Pr j AB = AB cos,
9、是AB与u轴的夹角。uv1vvv1vPr j (a + a ) = Pr ja + Pr javu22vva b = a b cos = a b + a b + a b ,是一个数量,xxyyzza b + a b + a bz z两向量之间的夹角:cos=xxyyax2 ay az+2+2bx +by +bz222ijkvvvvvvvv vc = a b = a ay a , c = a b sin.例:线速度:v = wr.xzbx by bzax ay az向量的混合积:abc = (a b)c = b by b = a b c cos,为锐角时,vvvv vvvvvxzcx cy cz
10、代表平行六面体的体积。 平面的方程:v1、点法式:A(x x ) + B(y y ) +C(z z ) = 0,其中n =A,B,C,M (x , y ,z )00000002、一般方程:Ax + By +Cz + D = 0xyz3、截距世方程:+ + =1a b cAx + By +Cz + D平面外任意一点到该平面的距离:d =000A2+ B2+ C2x = x + mtx x0my y0z z00v空间直线的方程:= t,其中s =m,n, p;参数方程:y = y + nt0npz = z + pt0二次曲面:x2y22z221、椭球面: +=1a2bcx2y22、抛物面: +=
11、z(, p,q同号)2p 2q3、双曲面:x2yby2222zcz2222单叶双曲面: +=1ax22双叶双曲面: =(1 马鞍面)a2bc多元函数微分法及应用zzuxudx + dy + dzy zu全微分:dz = xdx + dydu =y全微分的近似计算:z dz = f (x, y)x + f (x, y)yxy多元复合函数的求导法:dz z u z vdt u t v tz = f 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,uxuyvxvydu =dx + dyd
12、v = dx + dy隐函数的求导公式:dyFxFyd2yFxFx dy隐函数F(x, y) = 0, = ,=( ) ( )dxdx2x Fyy Fy dxzxFxFzyzFy隐函数F(x, y,z) = 0, = , = Fz F Fu vG GF(x, y,u,v) = 0隐函数方程组:(F,G)(u,v)FuFv =J=G(x, y,u,v) 0=Gu Gvu vuxuy1 (F,G)= J (x,v)v1 (F,G)J (u,x)1 (F,G) = xv1 (F,G)= J (y,v) = yJ (u, y)微分法在几何上的应用:x = (t)空间曲线x x0 y y0 z z0处的
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