數學分析-復旦-dvd8附送

⒈⒉用定義證明，函數y

x03x3xfx0⑴limf(x0x)f(x0) ⑵limf(x)f(x0)⑶

xf(x0h)h

f(x0?y2x23x1(1,2)(2,1)f(x為(,x0f(1sinx)3f(1sinx)8x(x)其中xx0xyf(x在(1,f(1個焦點（4.25。1cos⑴y|sinx ⑵1cos⑶ye|x| ⑷y|ln(x1)||x|1asin1,(a x x2 x⑴y x

⑵yax xxex x ⑷yex2

x⑶yax2 x

xfxx0|fx)|x0fx在[a,b]f(a)

f(b0f(af(b0fx在(afx在有限區(qū)間(ab

fxfx)

f(x)f(x)g(x)f(x)xg(xf(0)g(0) ⒈用定義證明(cosx)sinx⑴(cscx)cotxcscx ⑵(cotx)csc2x 1x⑶(arccosx) ⑷(arccotx)11xx2⑸(ch1x) ⑹(th1x)(cth1x)x21xx⑴f(x)3sinxlnx ⑵f(x)xcosxx23x⑶f(x)(x27x5)sinx ⑷f(x)x2(3tanx2secx)x⑸f(x)exsinx4cosxx

3 ⑹f(x)

2sinxx33x⑺f(x) ⑻f(x)xsinx2lnxxxcos x⑼f(x)

x3cotx ⑽f(x)ln

xsinxcos；xsinxcos⑾f(x)(exlog3x)arcsinx ⑿f(x)(cscx3lnx)x2shxxsec⒀f(x)xcscx ylnx在(e,1)

xsinf(x) arctan當ayxylogaxlimyxnyax2bxcS1x,y)|過xy)可以作該拋物線的兩條切線S2xy)|過xy)只可以作該拋物線的一條切線S3xy)|過xy)不能作該拋物線的切線}，fxxx0處可導，gxxx0處不可導,證明c1fxc2gxxx0處fxgxxx0處都不可導,c1fxc2gxxx0處一定fx)gxxx0設fij(x)（i,j f11 f12

f11

f12

f1nd f21

f2n

k

f

f

f

fn1

n2

fnn ⒈求下列函數的導數：⑴y(2x2x1)2 ⑵ye2xsin3xlnlnx11

⑷y x⑸ysinx3 ⑹y xx⑺y ln(x x1) ⑻yarcsin(ex2x⑼y 21 ⑽y ln

x2 (2x2sinx1x1x

1ln2

⑿y 1cscx232x2343x31cscx232x2343x3a2xa2a2xa2x⒉求下列函數的導數：⑴ylnsinx ⑵yln(cscxcotx)a2x2x⑶a2x2x

a2

x ⑷yln(x x2x2aa；x2a⑸yx2a2

a2ln(x x2a2)⒊fx3x；⑴f ) ⑵fl3x；n f( ⑷arctanff(⑸f(f(ex2 ⑹sin(f(sinx)) ⑺ff(x)

f(f(⑴yxx⑶ycosxx

1y3 y3 ⑷ylnx(2x1)⑸y⑺ysinxx

1x1 ⑹y1x1⒌對下列隱函數 ⑴yxarctany ⑵yxey1xcos sinyx ⑷xyln(y1)xcos⑸ex2yxy20 ⑹tan(xy)xy0⑺2ysinxxlny0 ⑻x3y33axy0xat2⑴ybt3xt2sin⑶yt2cosxacos3⑸yasin3xt1

x1t2⑵ytt3xaet⑷ybetxsh⑹ych⑺y

tt

x⑻y

1t1txe2tcos2⑼ye2tsin2

xln(1t2⑽ytarctan 2tt 2tt求曲線x ，y 上與t1對應的點處的切線和法線方程1t 1t

確定y為x的函數，其中t為參變量，求 tsinyy

xa(costtsinya(sinttcos

t設函數ugxxx0yf(u在uu0gx0處連續(xù)。請舉例說明，在以下情況中，復合函數yfgxxx0處并非一定不可導：⑴ugxx0yf(uu0⑵ugxx0yf(u在u0⑶ugxx0yf(uu0f(ug(uh(uh(u)1ux f(u)g(u)h(u) ⑵f(u)g(u)⑶h(u)g(u) ⑷logh(u)g(u)1f2(u)h21f2(u)h2h(u) ⒈求下列函數的高階導數：⑴yx32x2x1,求y ⑵yx4lnx，求yxlnx⑶y

11

y xx⑸ysinx3，求y、y ⑹yx3 x⑺yx2e3xy⑻yex2arcsinxy⑼yx3cos2xy(80⑽y(2x21)shxy(99nyn⑴ysin2x ⑵y2xlnxe⑶yx ⑷y

；x25x⑸yexcosx ⑹ysin4xcos4x

x2f(x)x2

xx1⑴[f(x2)] ⑵f x⑶[f(lnx)] ⑷[lnf(x)]⑸[f(ex)] ⑹[f(arctanx)]⑴yarctanx（y(1x21⑵yarcsinx（xy(1x2yd2dx2⑴ex2yx2y0 ⑵tan(xy)xy0⑶2ysinxxlny0 ⑷x3y33axy0d2dx2xat2 xatcos ⑴ybt3 ⑵yatsin xt(1sint xaet ⑶ytcos ⑷ybet x⑸y

1t⑹1t

xsinycos

y⑴d2x ⑵d3 3(y)2yydy ( (⑴y3xtanx,求d2y ⑵yx4ex，求d4y11xx2⑶y ，求d2y ⑷y ，求dx2⑸yxsin3xd3yln⑺