2015考研數(shù)學(xué)高等數(shù)學(xué)重難點(diǎn)題型精講講義

1、x2 2xx2 1ex 的漸近線(xiàn)x2 2xx2 1ex 的漸近線(xiàn)10f(xx3x x1f(x sin1ln(1x)xx2 4 xa4x2e2xdx ，求a . xa4x2e2xdx ，求a .x xaa cos dt 5，求a ，b .2】已知ex t02000，求 ， .3】已知(nn 515x sin4f(xdt g(x) (115x sin4f(xdt g(x) (1t)t dt x0f(xg(xt00（A）高階無(wú)窮小 （B）低階無(wú)窮小 （C）5ABC使ex(1BxCx21Axo(x32 31】求極限n 6nnnn n nnnn n n n n nn12xdx3】求極限n 1】確定方程2

2、x 1x2的實(shí)根的個(gè)數(shù)74 3【例2】證明方程4arctan4 3【例2】證明方程4arctanx x 30恰有兩個(gè)實(shí)根方程ex ax2有幾個(gè)實(shí)根f(a f(b0f(af(b1f(x在(,(ab)82】已知f(x在0,上連續(xù)，dxx02】已知f(x在0,上連續(xù)，dxx0 f (xcosxdx 0 10,1上可導(dǎo)，f(1) xf20證明：存在0,1f(f(04f(xg(x在abg(x0f (a) f (b) g(a) g(b)0f() f g（2）在(ab內(nèi)至少存在一點(diǎn).9fff(x在0,1上二階可導(dǎo)，且xxfff(x在0,1上二階可導(dǎo)，且xx 5f(x是0,1上可導(dǎo)f(00, f(11，證明存

3、在兩個(gè)不同的點(diǎn)1,2(0,1112f1311證明：存在0, , ( 227f(x在ab(ab內(nèi)可導(dǎo)(0abf(a f(7f(x在ab(ab內(nèi)可導(dǎo)(0abf(a f(b f(xlim f(xk8xaf(xlim f(x09f(xh f(xhf(x(01)f (xf (x01證明：2f (x0)(xf (x f (x f(x )(xx )2 00002f (x) h2 f (xhxf(xh f(x f(x)h2f (xh f (xhf(x(0 f (x)h2 hf(xh f(x)h2f (x)h2 hf(xhf(x)h2 f (x)h2 o(f (x)h2 hf(xhf(x)h2 f (x)h2

4、o(h2hf(xh) flim 所以 f f (x)21即210f(x在abf (x0 x1x2為abf(x1) f(x2)fx1 x2 2211f(x在0,1上二階可導(dǎo)，且maxxf(x 1 f 140f(0) f(1)1x0 x1x1x0 x1x2ln(1xx2 11證明f (x)dx f 300bbbf(x)g(x)dx f g 22 aabf(x) 在abf (bbbf(x)g(x)dx f g 22 aabf(x) 在abf (xf (x)dx ，abb sin1x【例 1】計(jì)算不定積分 1cos(x1) (xf232f(x，x3(xf xe (1ex4f(x xe (1ex4f(x

5、 lnt dt f(x f1x1 1x21x2【例6】已知 tf (2xt)dt arctanx , f(1)1，f (x)dx22012yy2 7】計(jì)算二重積分ydxdyDx2,y0,y2yy2 7】計(jì)算二重積分ydxdyDx2,y0,y2,xD2x42【例8】交換積分次序 dydy，并計(jì)算22129】計(jì)算二重積分cos(xydxdy，其D(x,y0 x,0 y Df(xy)dxdy a，其D(xy0 x1,0 y1DD1f(xf (x f(x2f(x0f (x f(x（1）fx（2）求y f (x 2012x2y2t12x2y2tfx y )dxdyf (t22f(t在0,f(te4t 3

6、】y(xy3y2y2ex，圖形在(0,1)yx2x1在(0,1)y y(x4xdyx2y)dx0y y(xx1,x2y0y y(x22【例5】z f(e sin y)，f(u)具有二階連續(xù)導(dǎo)數(shù)22【例5】z f(e sin y)，f(u)具有二階連續(xù)導(dǎo)數(shù)e z，求f(u)x2 xx6f(x在0,f(0)1f(x f(xf(t)dt 0（1）f（2）ex f(xx1】z f(xyyg(xf(uvg(xx1 2. yyy0)Z yZ 1Zyy0)Z yZ 1Z2】以u(píng)x2 yv xy2zyz3z z(xy由方程F(x,y)0所確定，證 zxyx4】求原點(diǎn)到由方程5x26xy5y2 85x24y2

7、4上求一點(diǎn)，到2x5x24y2 4上求一點(diǎn)，到2x3y603S(x( x2243S(x( x224246n（1）（2）（4）n(nnnn(n2n1(1)n(n2 n6】求(1)n(n2 n6】求17f(xx2 5x8f(xln(1x2x21 9f(x1 10f(x12x2 10f(x12x2 1】計(jì)算曲線(xiàn)積分 e sinyb(xy)dxe cosyax)dyLxxL沿著y 2axx2 到原點(diǎn)的曲線(xiàn)xdyLx2 y2 14x L 起點(diǎn)為(a,b)，終點(diǎn)為(c,d)，曲線(xiàn)積分I 11 y2 f(xy)dxy2 f(xy)1dyxL （2）若abcdI 【例4】設(shè)S為橢球z 1的上半部分，點(diǎn)P(x,

8、y,z)S ，為S 在點(diǎn)P2(x, y,SdS【例 5】計(jì)算曲面積分2x3dydz2y3dzdx3(z【例 5】計(jì)算曲面積分2x3dydz2y3dzdx3(z2 1)dxdy ，其中z 1x2 y2xdydz ydzdx 222，其中2x 2y 3(x2 y2 z22數(shù)學(xué)高等數(shù)學(xué)重難點(diǎn)題型講義例1】2】1213】4】5】76】7】8】1)f9f(0)0, f數(shù)學(xué)高等數(shù)學(xué)重難點(diǎn)題型講義例1】2】1213】4】5】76】7】8】1)f9f(0)0, f(0)0, f (0)4,x10】3 1】a0或2】a1b4 12000，求，3】已知.(nn 1216,B,C 232322】 (2 233】1】

9、3【例3】a0時(shí)，無(wú)根；0a時(shí)，1個(gè)根；a 時(shí)，2個(gè)根；a時(shí)，3個(gè)根4443】1】3【例3】a0時(shí)，無(wú)根；0a時(shí)，1個(gè)根；a 時(shí)，2個(gè)根；a時(shí)，3個(gè)根4449f(xh f(xhf(x(01)f (xf (x01證明：2(x0)(xf公式f(x) f (x ) f(x )(xx )2 f(xhx00002(xh2 ff (xh f(x f(x)h2(xh2 ff(xh) f (x)hf(x 2(xh2 fhf(xh)f(x)h2 f (x)h2 o(h2limhf(xf(x)hlim f (xlim f (x) 即lim 1221】extan x22】arctan32 3】ln4f(x1】extan x22】arctan32 3】ln4f(x f1) 1(ln33427】44(29】10】1（1）f(xex（2）22f(t4t2【例 3】4yx5f(uC1euC2eu16（1）f(xex(1 2 y25】(8