高等数学(同济大学第五版)答案解析.pdf

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1、д乬811߸ᅮϟ߫ᑇ䴶⚍䲚Ёાѯᰃᓔ䲚ǃ䯁䲚ǃऎඳǃ᳝⬠䲚ǃ᮴⬠䲚˛ᑊߚ߿ᣛߎᅗӀⱘ㘮⚍᠔៤ⱘ⚍䲚(⿄Ўᇐ䲚)੠䖍⬠(1){(xy)

2、x0y0}2㾷ᓔ䲚᮴⬠䲚ᇐ䲚ЎR䖍⬠Ў{(xy)

3、x0៪y0}22(2){(xy)

4、1xy4}22㾷᮶䴲ᓔ䲚জ䴲䯁䲚᳝⬠䲚ᇐ䲚Ў{(xy)

5、1xy4}2222䖍⬠Ў{(xy)

6、xy1៪xy4}2(3){(xy)

7、yx}22㾷ᓔ䲚ऎඳ᮴⬠䲚ᇐ䲚Ў{(xy)

8、yx}䖍⬠Ў{(xy)

9、yx}222

10、2(4){(xy)

11、x(y1)1}{(xy)

12、x(y2)4}㾷䯁䲚᳝⬠䲚ᇐ䲚Ϣ䲚ড়ᴀ䑿Ⳍৠ2222䖍⬠Ў{(xy)

13、x(y1)1}{(xy)

14、x(y2)4}22x2Ꮖⶹߑ᭄f(x,y)xyxytan䆩∖f(txty).y22tx㾷f(tx,ty)(tx)(ty)(tx)(ty)(tan)tyt2Mx2y2xytanxJHt2f(x,y)KLyI3䆩䆕ߑ᭄F(xy)lnxlny⒵䎇݇㋏ᓣF(xyuv)F(xu)F(xv

15、)F(yu)F(yv)䆕ᯢF(xyuv)ln((xy)ln(uv)(lnxlny)(lnulnv)lnxlnulnxlnvlnylnulnylnvF(xu)F(xv)F(yu)F(yv)wuv4Ꮖⶹߑ᭄f(uvw)uw䆩∖f(xyxyxy)xy(xy)(xy)㾷f(xyxyxy)(xy)(xy)xy2x(xy)(xy)5∖ϟ߫৘ߑ᭄ⱘᅮНඳ2(1)zln(y2x1)2㾷㽕Փߑ᭄᳝ᛣНᖙ乏y

16、2x1>02ᬙߑ᭄ⱘᅮНඳЎD{(xy)

17、y2x10}11(2)zxyxy㾷㽕Փߑ᭄᳝ᛣНᖙ乏xy>0xy>0ᬙߑ᭄ⱘᅮНඳЎD{(xy)

18、xy0xy>0}(3)zxy2㾷㽕Փߑ᭄᳝ᛣНᖙ乏y0xy0ेxyѢᰃ᳝x0Ϩxy2ᬙߑ᭄ᅮНඳЎD={(xy)

19、x0y0xy}x(4)zln(yx)1x2y222㾷㽕Փߑ᭄᳝ᛣНᖙ乏yx>0x01xy022ᬙߑ᭄ⱘᅮНඳЎD={(xy)

20、yx0x

21、0x+y1}uR2x2y2z21(Rr0)(5)x2y2z2r222222222㾷㽕Փߑ᭄᳝ᛣНᖙ乏Rxyz0Ϩx+y+zr022222ᬙߑ᭄ⱘᅮНඳЎD={(xyz)

22、rx+y+zR}z(6)uarccosx2y222z222㾷㽕Փߑ᭄᳝ᛣНᖙ乏x+y0Ϩ

23、

24、1ेzx+yx2y222222ᬙߑ᭄ᅮНඳЎD={(xyz)

25、zx+yx+y0}6∖ϟ߫৘ᵕ䰤1xy(1)lim22(x,y)(0,1)xy1xy10㾷li

26、m1(x,y)(0,1)x2y201ln(xey)(2)lim(x,y)(1,0)x2y2ln(xey)ln(1e0)㾷limln2(x,y)(1,0)x2y212022xy4(3)lim(x,y)(0,0)xy2xy4(2xy4)(2xy4)㾷limlim(x,y)(0,0)xy(x,y)(0,0)xy(2xy4)11lim(x,y)(0,0)(2xy4)4xy(4)lim(x,y)(0,0)xy11xyxy(xy11)㾷limli

27、m(x,y)(0,0)xy11(x,y)(0,0)(xy11)(xy11)xy(xy11)limlimxy11)2(x,y)(0,0)xy(x,y)(0,0)sin(xy)(5)lim(x,y)(2,0)ysin(xy)sinxy㾷limlimx122(x,y)(2,0)y(x,y)(2,0)xy1cos(x2y2)(6)lim22x2y2(x,y)(0,0)(xy)e1cos(x2y2)1cos(x2y2)1㾷limlimlim22x2y222x2y

28、2(x,y)(0,0)(xy)e(x,y)(0,0)xy(x,y)(0,0)e1costsintlimlim0t0tt017䆕ᯢϟ߫ᵕ䰤ϡᄬ೼xy(1)lim(x,y)(0,0)xy䆕ᯢབᵰࡼ⚍p(xy)⊓y0䍟৥(00)xyx߭limlim1(x,y)(0,0)xyx0xy0བᵰࡼ⚍p(xy)⊓x0䍟৥(00)xyy