高等数学上复习要点

高等数学（1）期末复习提相

第一章函数f(x)中符号f xf(xf(xf(xyxf(x)f(x)f(x称为奇函数，奇函数的图形关于原点对称。yyxya

(为实数(a0,a④对数函数：yloga (a0,asinxcosxtanxcotyearctan2(1yeuuv2varctanww1x。分解后的函数前三个都是基本第二 极限与连(a2xk(a2xka)(a2xka2xk xk(a2xk x2ax (xx)(xx 1x x x x axnaxn1 x

n

n

n0xx0bxmb10

bm1x

nlimsinx lim(11)

e （或lim(1xx

elimsin(x)(x)011lim(1 f(x) （或lim1g(x

ef( f

g(xsin limsinlimsinxlim1 1 1x0sin x03sin 3limsin 2 2 ) )2 )lim(x2)

lim

lim

xx x11 x(11) lim[(11)x x xx0f(xxx0xx0f(x第三 导数与微f(xxx0

f(x0x)f(x0x

f(x)f(x0xf(xxx0f(x0y

f(x上点(x0,f(x0yf(x在点(x0,f(x0yf(x0)(xx0)f(x0在x0点不一x3xx3xlny1ln(x1)1ln(x

y

yy

2(xxx3x

3(xx6(x2xxy dy(t) d(uv)dud(uv)vdu

v

)vdu

(vdyyxdxyuuxdx第四章导数的应用 若在区间(a,bf(x)0f(x在区间(a,b上单调增加；若在区间(abf(x)0f(x在区间(abf(xxx0f(x00f(xxx0的左右由正变负（f(x00），xx0f(xf(xxx0的左右由负变正（f(x00），xx0f(x若在区间(a,bf(x)0f(x在区间(a,b上是凹函数；若在区间(a,bf(x)0f(x在区间(ab

f(xa

f(x)ayayf(x

f(xxx0yf(xf(x在区间[abf(xf(x第五 不定积(f(x)g(x))dxf(x)dxcf(x)dxcf (c[k1f1(x)k2f2(x)]dxk1f1(x)dxk2f2d f(x)dx

fF(x)dxF(x)及df(x)dxfdF(x)F(x)在利用分部积分时，如何选择uv将排在后面的函数选做v，具体地说：若被积函数是x3e2x，则将e2x选做v；若被积函数是x2arctanx，则将x2选做v。 a(k1f1(x)k2f2(x))dxk1af1(x)dxk2af2 af(x)dxaf(x)dxcf

af(x)dxbfaaf(x)dxa若G(x) f(t)dt，G(x)f p1p1 10xpp1p1yf(xyg(xxa,xbSbSaf(x)g(x)对于对称区间(a,a)a当f(x)为奇函数时 f(x)dx 当f(x)为偶函数时 －af(x)dx20f(x)dx2－af第七 无穷级 若an和bn收敛，则(anbn (anbn)an

若anc为常数，则can can 级数收敛的必要条件：若an收敛，则liman0 几何级数aqnq1q1 npn

p1p1⒌知道函数的泰勒级数和马克劳林级数，记住ex,sinx,cosx和ln(1x anbn(n1,2,，那么

第八 常微分方yp(xyq(x