試題及答案微積分試卷

浙江大學(xué)2006–2007學(xué)年秋冬學(xué)期《微積分I》課程期末考試試卷（A卷 一、求導(dǎo)數(shù)或微分（每小題6分）dy(1yxsin浙江大學(xué)2006–2007學(xué)年秋冬學(xué)期《微積分I》課程期末考試試卷（A卷 一、求導(dǎo)數(shù)或微分（每小題6分）dy(1yxsin4x(arctan2x)3ln2d2d d2tt(2)設(shè)x ds,y sin(t-s)ds，求ts2 dx.d00(3)yy(x是由方程exy2xxy10x的可導(dǎo)函數(shù)，求.x二、求積分（每小題6分6(4) x6xx2dx0x(5)arctanedxe2 d xx.(6)11三、求極限（每小題6分12cosx1(7)求3x03limxa11f(af(a0f(a)(x f三、求極限（每小題6分12cosx1(7)求3x03limxa11f(af(a0f(a)(x f(x)f(a)(8)1(9)設(shè)u(11)(12)(1n)n，求limunnnnnn四、選擇題（每小題5分，每小題所給4個選項中只1個是符合要求的，請將所選那項的代碼填入【】中）.3x(10)設(shè)0tarcsin(t)dt, 1)dt.則當(dāng)x0時t620(A)與是同階但不等價無窮小(C)的高階無窮小(B)與是等價無窮小(D)的高階無窮小【】(11)設(shè)級數(shù)ann(A)(a a )必收斂)必收斂 n nnn(a2na2n1(a2na2n1必收斂必收斂【】nnex,xx(12)f(xxF(x) f(tdtF(xx0x(A)極限不存在(C)連續(xù)但不可導(dǎo)(B)極限存在，但不連續(xù)(D可導(dǎo)【】(13)yf(x為連續(xù)函數(shù)，除點xaf(x二階可導(dǎo)yf(x的圖形如圖，yf(A)有一個拐點，一個極小值點，一個極大值點有二個拐點，一個極小值點，一個極大值點有一個拐點，一個極小值點，二個極大值點有一個拐點，二個極小值點，一個極大值點cbaox【2五、（本題滿分9分(14)設(shè)曲線五、（本題滿分9分(14)設(shè)曲線yax2(xa0y1x2AOAya；a的值使V(a)六、（本題滿分9分(15)將函數(shù)f(xxarctanx1ln(1x2x0處展開成泰勒級數(shù)（2數(shù)）并指明成立范圍七、（本題滿分8分(16)x0xf(x)(x4)e2(x2)ex23sin （1）y (4cos4xlnxsin4)1xtt2（2）x ds,y sinudu200dy(esint)t222dysin2t2tesint2 (sintcost)222e2dddd2 2ee2dxt2dt2（3）exy(dxdy)2dxxdyydx0xydyx0(t3)9t2dt 9t2dt27sin （1）y (4cos4xlnxsin4)1xtt2（2）x ds,y sinudu200dy(esint)t222dysin2t2tesint2 (sintcost)222e2dddd2 2ee2dxt2dt2（3）exy(dxdy)2dxxdyydx0xydyx0(t3)9t2dt 9t2dt27633 x9(x3)2dx200arctanexdxtarctantdt1arctant t2 e2dt2(1t2t21arctant11)d1arctantarctant12(2tt1tt2t11arctane 2e2 d x1t2dt2arctan1txx0102cosx12cosxxln2cosx11lim3x（7）3x3xx0x3x0x33lim1cosxlim1ln1cosx1222x3x36f(x)f(a)f(a)(xf(a)(xa)[f(x)f(8)lim limf(a)(x f(x)f(a)f(x)f(a)f(a)(xxxaf(x)f(a)f(a)(x1[ff(a)(xa)2[f(x)f(a)(xxx f(x)f(a)f(a)1[f(洛必達法則2(xx1ln(11)ln(12)ln(1n（9）lnnnnnn1nixlimlnunlim 1 ln(1xln(1dx2ln210n1n04elimlnen e2ln21limn41223xarcsinx 23311 x0 x3x0 x2 x1te x1xeF(x)eF(0)11F(0)F0x2e edt tdt1 ,0xt 0F(0 F1223xarcsinx 23311 x0 x3x0 x2 x1te x1xeF(x)eF(0)11F(0)F0x2e edt tdt1 ,0xt 0F(0 F(0)0F(0)不存在交點：A( ,a)yay1aA212(Ⅰ)V(a)Vxxax) 21a(1a)01ox0,0aa3(4f(a)(af(a)a0,aV(a(1(1（15）f(x)arctanx,f(0)0,f(0)f(x) (1)nx2n,1x1n2nxxf(x) f(x)dxx2n1x1，f(x) f,00nf(x)x2n1x,n0(2n1)(2nxxf(x)( 1)e2(x1)ex,f(0)0（16）xx2xe2)0,(xx1f(x) e2xexx44x0,f(x)f(0)0