微積分初步期末綜合整理

1、微積分初步2021年7月一、填空題(每題 4分，此題共20分)、， 11.函數(shù)f (x)ln(x 2)(2, 1)( 1,4.假設(shè)函數(shù) f(x) xsin? tk,0,在x00處連續(xù)，那么X的定義域是2.假設(shè)lim竺竺 2，那么k 2.xkx3.曲線ex在點(diǎn)(0, 1)處的切線方程是23.24.25.曲線y 、x在點(diǎn)(1,1)處的切線方程是 yd e x dx微分方程(y )3d4.dxn (x21)dx 0.26.假設(shè)函數(shù)f(x)5.微分方程yy,y(0)1的特解為y4xy(4)3 xsin xk,5y sin X的階數(shù)為4 .1,0,在x00處連續(xù)，那么1ln(x 2)(2, 1) ( 1

2、,2.6.函數(shù)f (x)2x 的定義域是27.28.(sinx) dx微分方程(y )7.假設(shè)函數(shù)f(x)x：2,k,在x 0處連續(xù)，那么k02. 29.函數(shù)f (x)si nx c.4xy y5 s in:1的定義域是(,5).5 xx的階數(shù)為8.曲線y1.x在點(diǎn)(1, 1)處的斜率是.230.limx.1xsinx9. 2xdx2Xc .ln 231.f(x)2x，那么 f (x) = 2x(ln 2)2 .F (x) c，那么10.微分方程y x21 .y2x滿足初始條件y(0) 1的特解為14 x2的定義域是ln(x 1)(1,0) (0,2.11.函數(shù) f (x)12.函數(shù)yX 2口

3、的間斷點(diǎn)是=x 1 .1 13.函數(shù)y3(x21)2的單調(diào)增加區(qū)間是1,).f (2x 3)dx-F233.微分方程xy(y)34.函數(shù)f (x)1ln(xsin 2x35.lim2.x 0X36.弋卜:/假設(shè) y = x (x -1)( Xf(x)dx32.37.(2x 3) c.-2)( x - 3)，那么的定義域是(2,1)( 1,).2) 假設(shè)d e x dx4 sin x ex y的階數(shù)是3.y (0) = -6 .14.右 f (x)dx-H-*sin2x c，那么 f (x) =2cos2x .2e x dx .15.16.17.微分方程(y )3 函數(shù)f (x 2) 卄 sin

4、 6x右limx 0 sin kx4xy x2 4x2，那么k18.曲線f (x) ex19.5.y sinx的階數(shù)為3.7，那么 f (x).38.39.微分方程y(2,3)(3,y, y(0)1的特解為y)3.40.1在(0,2)處的切線斜率是1.41.42.27 (1+ln3 )e 2 c1假設(shè)一是f (x)的一個(gè)原函數(shù)，那么f (x)X(y)4 ln y y 函數(shù)f (x 1)sin 2x為3階微分方程.x2 2x 7，那么 f (x)2 6.微分方程(y )344.函數(shù) f (x 2)43.4xy(4) x2 4x7y sin x的階數(shù)為4.5，那么 f (x)x2145設(shè)函數(shù)f(x

5、).2xsinx1,k,k = -1 .在x= 0處連續(xù)，那么46曲線f(x) ex 1在(0,2)點(diǎn)的斜率是1.1 347(5x3 3x 2)dx 4.48微分方程xy (y )2y40的階數(shù)是3處連續(xù).A . 0B. 1 C . 23.以下結(jié)論中(C. f (x)在x x0處不連續(xù)，那么一定在x0處不可導(dǎo).)49 .函數(shù)f (x)ln(x 2)的定義域是(2,3)(3,).正確.A.f (x)在x x0處連續(xù)，那么一定在 x0處可微.50. limxsinx"2函數(shù)的極值點(diǎn)一定發(fā)生在其駐點(diǎn)上 .f(X)在x x0處不連續(xù)，那么一定在 x0處不可導(dǎo).51.f(x)f (3) = 2

6、7(1 In 3).52.2dex =eD .函數(shù)的極值點(diǎn)一定發(fā)生在不可導(dǎo)點(diǎn)上.1,_4.以下等式中正確的選項(xiàng)是(D. dxd(2、. X). xa . sin xdx d(cosx)153.函數(shù)f(x) 的定義域是(-2 , 2)V4 x254 假設(shè) sin xdx-cosx+cb. ln xdx d)xc. axdx d(ax)(y )3 4xy1D. dx xy5sin x的階數(shù)為(d(2 ._ x)55 .曲線在任意一點(diǎn)處的切線斜率為、x，且曲線過(guò)點(diǎn)(1 , 1),那么32-1曲線方程為yx23356.由定積分的幾何意義知,xa2 x2dx2 a0457.微分方程y3y0的通解為y c

7、e 3x58.函數(shù)f (x)15 x的疋義域是(1,0)(0,5)ln(x1)A. 2 ;B. 3 ;C.4D. 56.設(shè)f(x1) x2 2x 3，那么 f (x)(D2 X4)A. x21B . X22c. x24D . X247.假設(shè)函數(shù)f(x)在點(diǎn)X0處可導(dǎo)，那么(B limX X0f(x)A，但Af(Xo)是錯(cuò)誤的.A函數(shù)f(X)在點(diǎn)X0處有定義B.lim f (x)X XA，但Af (Xo)C函數(shù)f(x)在點(diǎn)X0處連續(xù)D.函數(shù)f(x)在點(diǎn)X0處可5.微分方程B. 3 ;)微59.設(shè) f (x 1)22x 1,那么 f (x)2 22 x2x360.函數(shù)y的間斷點(diǎn)是=-1 .x 16

8、1.f (x)ln x，2那么 f (x)-362.曲線f (x).X1在(0,1)點(diǎn)的斜率是1263.假設(shè) f(x)dxcos2x c,貝U f (x)-4cos2x .64.微分方程xy(y)0的階數(shù)是 265.微分方程yy且y(0)1的特解是yex8.函數(shù)yx24x6在區(qū)間(4,4)是(A.先減后增)、單項(xiàng)選擇題(每題4分，此題共20分)1.設(shè)函數(shù)y xsinx，那么該函數(shù)是(A.偶函數(shù)).A.偶函數(shù)B.奇函數(shù) C.非奇非偶函數(shù) D 既奇又偶函數(shù)2.當(dāng) k (C. 2)時(shí)，函數(shù) f (x)x2 2,k,0，在x0A.先減后增C.單調(diào)減少9.假設(shè) f (x).先增后減.單調(diào)增加. x(x

9、0)，那么f (x)dx(B.A.x c)x21 2x210.微分方程C.A. 111.設(shè)函數(shù)yB.D.3(y)34xy5yyx . x c23 3x x2 c2sin x的階數(shù)為(c. 3 )B. 2 C. 310 x 10xD. 5，那么該函數(shù)是(B.偶函數(shù)).A.奇函數(shù) B.偶函數(shù) C.非奇非偶函數(shù)D 既奇又偶函數(shù)12.當(dāng)x0時(shí)，以下變量中為無(wú)窮小量的是(c. ln(1 x)b.沁 c . ln(1 x)xd-4x13.設(shè) y lg2 x，那么 dy(d.dx)A.單調(diào)增加B.單調(diào)減少1a.dx2xB . dx xxln 10ln10c . - dxx1x ln10dx14.在切線斜率為

10、2x的積分曲線族中，通過(guò)點(diǎn)(1, 4)的曲線為(C. y =x2 + 3 ).2 2A. y x 1 B . y x 222C. y = x + 3 D . y = x + 4以下等式成立的是A .x3 dxd3x)ln3A .x .d3xdx_2、3 dxB2d(1 x )ln 31xC .dx.xd . xD1ln xdx d(-)xC.先增后減D .先減后增18.函數(shù)y x22x 7在區(qū)間2,2是C.先減后增15.微分方程yy 1的通解是A. yCex1)A. yxCx 1Ce 1 ； b. y e ；c. yx C ; Dy x2 C2e x ex16.函數(shù)y的圖形關(guān)于A.坐標(biāo)原點(diǎn)對(duì)稱(chēng)

11、.2A.坐標(biāo)原點(diǎn)B. x軸C . y軸D. yxx“e2, x0亠 17.當(dāng) k (D. 3)時(shí)，函數(shù)f (x)在x 0處k,x0連續(xù).a. o b. 1 c . 2 d . 325.以下微分方程中為可別離變量方程的是 B.魚(yú) xy ydxA dyB dyA.x y ;b.xy y;dxdxC. 3xy sin x ；D.凹 x(y x)dxdx26.設(shè) f (x1) x21,那么f (x)(C x(x 2)A x(x1)B .2 xC . x(x2)D (x2)(x 1)27.假設(shè)函數(shù)f x在點(diǎn)X0處可導(dǎo)，那么B. lim f x A，但x x0A f x。是錯(cuò)誤的.a.函數(shù)f x在點(diǎn)xo處

12、有定義B . lim f x A，但A f x0x xoC 函數(shù)f x在點(diǎn)xo處連續(xù) D 函數(shù)f x在點(diǎn)xo處可微28.滿足方程fX 0的點(diǎn)一定是函數(shù) fx的C.駐點(diǎn)A.極值點(diǎn)B .最值點(diǎn)C .駐點(diǎn)D .間斷點(diǎn)C .先減后增D .先增后減19.以下等式成立的是(A.plf (x)dx f(x) dxdA .dxf (x)dxf(x)B.f (x)dx f (x)A.單調(diào)減少B 單調(diào)增加29. xf (x)dx (A. xf (x)A. xf (x) f (x) C B.C. x2 f (x) c230.設(shè) f (x 1) x21f (x) C)xf (x) cD.(x 1) f (x) C那么

13、 f (x)(A. x(x 2)C .d f (x)dxf(x)D .df (x0 f (x)20.微分方程yy 1的通解為B ycex 1)A. y cex 1 -B.y cex 1；- 12D.c. y -x c;y x c2e設(shè)函數(shù)y -xx21.e，那么該函數(shù)是B .偶函數(shù).2A.奇函數(shù)B.偶函數(shù)C.非奇非偶函數(shù)D .既奇又偶函數(shù)22.f (x)sin x1,當(dāng)C.x0時(shí)，f x為無(wú)窮小x旦 量.A .xB .xC .x 0 D . x 123.函數(shù)y x 12在區(qū)間2,2是D.先減后增2A. x(x 2) B x C x(x 2) d (x 2)(x1)e x ex31.設(shè)函數(shù)y，那

14、么該函數(shù)是(B.偶函數(shù)).2A.奇函數(shù)B .偶函數(shù)C .非奇非偶函數(shù)D .既奇又偶函數(shù)32.函數(shù) f (x)x 32的間斷點(diǎn)是A.x 1, x 2)x 3x2A . x 1,x2B x 3C . x 1,x2,x3D.無(wú)間斷點(diǎn)33.以下結(jié)論中C. f x在xx0處不連續(xù)，那么一定在X。處不可導(dǎo)正確.A f x在x x0處連續(xù)，那么一定在 x0處可微.B .函數(shù)的極值點(diǎn)一定發(fā)生在其駐點(diǎn)上 .C f x在x x0處不連續(xù)，那么一定在 x0處不可導(dǎo).D.函數(shù)的極值點(diǎn)一定發(fā)生在不可導(dǎo)點(diǎn)上.34.如果等式1f (x)exdx那么 fXA. 1x35.以下微分方程中,B.12xD.C.D.sin xxxy

15、e方程.2A . yxC . y xy36.設(shè)函數(shù)ycos yln yxeA.奇函數(shù)連續(xù).B.偶函數(shù)2時(shí),38.以下函數(shù)在指定區(qū)間A . sinx B39.以下等式正確的選項(xiàng)是A 3xdxd3xln 340-yyyxsin x-y sin x，那么該函數(shù)是A.xye奇函數(shù)C.非奇非偶函數(shù)函數(shù)f (x)x22,.既奇又偶函數(shù)A.k,3xdxC dxC.、xd、x以下無(wú)窮積分收斂的是B.sinxdx b1dx x41.以下函數(shù)中為奇函數(shù)是上單調(diào)減少的是2xd3xIn 3dx1 x2D.d(1x2)1ln xdx d()xe 2xdx).2xdx1xdx(D.ln(x+ 12).A.xsinx B.

16、lnx C.x+2 D.ln(x+ 12)42. 當(dāng)k= C.2時(shí)，函數(shù)fx=在x=0處連續(xù)。A.0 B.1 C.2 D.e+143. 函數(shù)y= 2 +1在區(qū)間-2 , 2是B .先單調(diào)下降再單調(diào)上升A.單調(diào)下降B .先單調(diào)下降再單調(diào)上升C.先單調(diào)上升再單調(diào)下降D .單調(diào)上升44 .在切線斜率為2X的積分曲線族中，通過(guò)點(diǎn)1, 4的曲線為A . y= 23 。A . y= 23 B.y= 24 C.y= 22 D.y= 2 145.微分方程yy, y(0)1的特解為C.ye )A .y 0.5 2b . y ec. yeD.yex 146.函數(shù)y1ln x的定義域?yàn)閐.x0且x4).x4A .x

17、 0 B .x 4 c .x 0且x1D.x0且x 447.函數(shù)f (x)ln x 在 xe處的切線方程是(C.y1x 1).e11 ,11A.yx b.yx 1C. yx1D.y -x e 1eeee1、D. 2x12x是線性微分148 .以下等式中正確的選項(xiàng)是D. 一 dx d2.、x,xln xdxa . sinxdx d(cosx) b.c. axdx d(ax)-J49 .以下等式成立的是(A. 一dx-JA . 一 f (x)dx f (x) B . f (x)dx f (x) dxxd. 1 dx d(2 . x)f (x)dx f (x)c. d f (x)dx f (x)D.

18、df(x)f(x)50 .以下微分方程中為可別離變量方程的是B.dydxxy y)A.業(yè) x y ； dxB.屯dxxyy ；-dy.C.xy sin x ；dxD.dydxx(y x)51 .以下函數(shù)中為奇函數(shù)是D In(x1 x2)A.xsi nxB.In x c.x2 xd. In(x .1 x2)52 .當(dāng)k c . 2時(shí)，函數(shù)e' f(x)x 1,k,xx0在x 0處0連續(xù)A. 0B . 1C.2D.e153 .函數(shù)y x 1在區(qū)間2,2是B.先單調(diào)下降再單調(diào)上升A.單調(diào)下降B.先單調(diào)下降再單調(diào)上升C.先單調(diào)上升再單調(diào)下降D .單調(diào)上升A . y =x2 + 3B2.y =

19、x + 4C.y2 x2 D2 .y x 155 .微分方程yy, y(0)1的特解為c. yexA . y0.5x2Bx.y eC.yx eD.yex154 .在切線斜率為2x的積分曲線族中，通過(guò)點(diǎn)1, 4的曲線為A. y =x2 + 3 .56.假設(shè)函數(shù)f ( x)2x那么呵f (x)(A.4.計(jì)算定積分In xdxD 不存在解:e2In xdxxlne21e xdxx2e2e2157 當(dāng) k= (A. 1)時(shí),函數(shù)f(x)1,k0，在x0連續(xù).A .58.f(x)竺,當(dāng)xD.0時(shí)，fX為無(wú)窮小量.A.B.C.1 D.0x 1解:原式lim (xx 1 (x6.設(shè)y2x e解:yc 2x2

20、edy(2e7.計(jì)算不定積分5.計(jì)算極限lim0處解:1x21x2 5x 4 1)(x 1) 4)(x 1) cosx，求 dy.sin x2xsin x)dxxcosxdxlim -x 1 x 4159.(1'cos x2 .e .sinx 2x )dxD.0)3xcosxdx = xsinx sinxdx xsinxcosxA.0 B.1 C. 2 D. 4338.計(jì)算定積分e3-dxx60.設(shè)f x是連續(xù)的奇函數(shù)，那么定積分f (x)dx( D. 0)解:3e0a . 2 f (x)dx-a0af(x)dx-aa0 f(x)dx1x 、 1 ln xe311 d(1 lnx)一

21、1 ln xdxe3三、計(jì)算題此題共 44分，每題11分i.計(jì)算極限limx 22 x2 x6x 83x 2解：原式 lim (x 4)(x2)x 2 (x 2)(x1)3x 272xi計(jì)算極限limx 22 1 ln x解：xm2x 3x2 計(jì)算極限linx 224 1X Xm2H X(X 2)(X 1)(x 2)(x 2)2lim x 2 x 3x2解：原式=xm 3 (X 3)(x一1)-(x 3)(x 3)32.設(shè)yln3x cos x，求 dy .解: y13cos2 x( sinx)xdy丄23sin xcos x)dxx3.計(jì)算不定積分(2x 1)10dx解:(2x 1)10dx

22、 =1 101112 (2x 1) d(2x 1)尹 1) cx2 4x9.計(jì)算極限lim令一空x 1 x 5x 4 lim a 5)(x 1) x 1 (x.市e解：原式10.設(shè) y解: ydy11.解:4)( x 1)In x，求 dy.1 1e'x 1x 1 xx 1Ji計(jì)算不定積分1 cos-dx =x1)dxx1cosTdxx1 1 cos- dx x112.計(jì)算定積分xexdx01解:xexdx0解:xe 013.計(jì)算極限limx 1limx 1exdx2小x 2xx2114.設(shè) y ln x cosex，求 dy.解:1sin xeX15.計(jì)算不定積分牙dxX解:2xy

23、2e解:dy ( 2e2x16.計(jì)算定積分1eln xdx27.計(jì)算不定積分解:23 ? x2 )dx2sin、x ,x X解:17.計(jì)算極限limx 2x2 6x 8x24解:28.18.設(shè) y x .、XIn cosx，求 dy.sin . xdx = 2 sin .' xd x1 計(jì)算定積分2xeXdx02cos x解:解:12xexdx 2xex01x e 0dx2e 2e解:x1xdxi19.計(jì)算不定積分x29.計(jì)算極限limx 3解：原式1ex1ex c30.設(shè) y2 xcosxdx解:22 x cos xdxx sin x22 sin xdx cos X000220.計(jì)算

24、定積分解:6x80202X21. 計(jì)算極限lim 231.解:x 4 x 5x 4解：原式 lim (X 4)(x2)x 4 (x 4)(x 1)lim2x 4 xx2 2x 15x29lim(X 5)(X 3) x 3(x 3)(x 3)x . x ln cosx, 求 dy.12-( sinx) cosx12 tan x弓x2計(jì)算不定積分dytan x)dx(1 2x)9dx(1 2x)9dx =9(1-2x)9d(12x)20 (12x)1032.計(jì)算定積分1xe0xdx解:y 2X ln23cos3xdy(2X ln2 3cos3x)dx23.計(jì)算不定積分xcosxdx解:xcosxd

25、x =xsin xe151 n x24.計(jì)算定積分dx1X解:sinxdx22. 設(shè) y 2X sin3x，求 dy.xsinx cosx解:1xe xdx033.計(jì)算極限解：原式34 .設(shè) yxexdxe 1 5ln x , dx1解:xe(11 5ln x)d(1 5lnx)110(15lnx)235.10(3625.計(jì)算極限limx 21) 2X2 3x 2x24解:解：原式 limx 226.設(shè) y e 2x(x 1)(x 2)(x 2)(x 2)X X , 求 dy.3xx24lim (x 1)(x2)x 2(x 2)(x2)3sin 5x cos x，求25cos5x 3cos02

26、y . x( sin x)25cos5x 3sinxcos x計(jì)算不定積分(1 x) dxx' x(1 x)2dx=、 x(1x)2d(1- x)勺1x)3 C計(jì)算極限lim亡占x 3 x29解:原式lim© 3)(x 1)2x 3 (x38.解:3)( x-，求x13)339.計(jì)算不定積分2t(x 111 x_i2exdxe；d 1e d x1ex c40.計(jì)算定積分7 xcosxdx0解:：xcosxdx=xsi nx02si nxdx解：y25 計(jì)算不定積分解"xx)27 計(jì)算不定積分解:cosx 02 =212 0sin x 3 x2 zcosx 2dx 2

27、 (1x)2d(1cosxxdxx)2(1x)3 c3cos xdx x8 計(jì)算不定積分5解：(1 2x) dx2 cos xd x(12x)5dx1 (12x)5d(122x)1存12x)6 c41 .解:42.解:計(jì)算極限x lim x 2 x43.解:44.解:m2H X3x 2x 6lim(Xx 21)(x 2)3)(x 2)9 計(jì)算不定積分x一巳dxx5 e1設(shè) yx2ex，求 y- 2 - 1 y 2xex x2ex( p)x1ex (2x計(jì)算不定積分(2x 1)10dx計(jì)算定積分1exdx03 .設(shè)y解：y解:10.1)解:計(jì)算定積分(2x1)10dx-(2x 1)10d(2x2

28、1 exdx0111)13.x xeex11exdx01,求yx解:-dxxx sinxdx 021 .sin2 2計(jì)算定積分I12、x 112xx2 3x 22x解:-51exd(5-sin xdx0 2ex)1021-xsin xdx2 5 ex ccos<dx12 xe012xexdx014 計(jì)算定積分e2In xdx1xlnxdx12 xexdx02 2-ee2In xdxe21e2x-dxxx2xe2ex12 exdx0-2ee2 1e2 1辭-linx3x 2lim (X解.x 12x1 zxx 2(x3.設(shè)y2xcos3x, 求< y 。解:y2xcos2x(sin x)34.設(shè)yx2In cosx，求y .2.計(jì)算極限2x2limx 11)(x 2)2)e315.解:計(jì)算定積分dx1 x 1 lnxdxX、-1 ln x1 1 lnxd(1 啦2而xe3