考研数学三知识点总结

高数三角函数变换cos(A−B)=cosAcosB+sinAsinBcos(A+B)=cosAcosB+sinAsinBsin( A−B)=sinAcosB−cosAsinBsin( A+B)=sinAcosB+cosAsinBsinAcosB=12 [sin ( A+B)+sin( A−B)]sinx cosx =12 sin2xsinAsinB= 12 [cos(A− B)−cos( A+B)]sin2x =12(1−cos2x )cosAcosB=12 [cos( A−B)+cos( A+B)]cos2 x =12 (1+cos2x )cos2x =1−tan 2x1+tan 2xsin2x =2tanx1+tan2xarcsinx +arccosx =π2arctanx +arccotx =π2arctanx +arctan 1x =π2圆柱体积 V =π r 2h圆锥体积 V =13 π r 2h球体积 V =43 π r3椭圆面积 S=π ab抛物线y 2=2p x交点坐标 ( p2 ,0 )准线 x =− p2点到直线距离 ∣ax0 +by0 +c∣√a2+b2第一类间断点：包括可去间断点和跳跃间断点。可去间断点：间断点处左右极限存在但不等于该点函数值。f (x0 +0 )= f (x0 −0 )≠ f ( x0 )跳跃间断点：间断点处左右极限存在但不相等。f (x0 +0 )≠ f ( x0 −0 )第二类间断点：间断点处左右极限至少有一个是∞重要极限limx →0sinxx=1limx →∞(1+1x )x =elimx →0(1+x )1x =ex 趋向于 0 时的等价无穷小sinx ∼x tanx ∼x arcsinx ∼ x arctanx ∼ x 1−cosx ∼ 12 x 2ln(1+x )∼x loga(x +1)∼ xlna e x−1∼x a x−1∼x lnan√1+x −1∼ xn (1+bx )a−1∼abx导数公式(a x)'=ax lna (loga x )'= 1x lna(tanx )'=sec2 x (cotx )'=−csc 2 x (secx )'=secx tanx (cscx )'=−cscx cotx(arcsinx )'=1√1−x 2 (arccosx )'=−1√1−x 2 (arctanx )'=11+x 2 (arccotx )'=−11+x 2[sin (ax +b)](n)=ansin(ax +b+n2 π )[cos(ax +b)](n)=ancos(ax +b+n2 π )(1ax +b )(n)=(−1)nann!(ax +b)n+1[ln(ax +b)](n)=(−1)n−1(n−1)!an(ax +b)n积分公式∫dx√x 2±a2=ln∣x +√x 2±a2∣+C∫dx√a2−x 2=arcsin xa+C∫dxx 2−a2=12 ln∣x −ax +a∣+C∫dxx 2+a2= 1a arctan xa+C∫dxa2 x 2+b2= 1ab arctan axb +c∫ secx dx =ln∣secx +tanx∣+c∫cscx dx =ln∣cscx −cotx∣+c∫√a2− x 2dx = a22 arcsin x2+ x2 √a2− x 2+c∫√x 2±a2dx = x2 √x 2±a2±a22 ln∣x +√x 2±a2∣+c∫0π2 sin n x dx =∫0π2 cosn x dx =(n−1)!!n!!π2 (n为偶数)∫0π2 sin n x dx =∫0π2 cos...