高等数学微积分课件 多元变量的函数

1、 2 ( , )2f x yxxyy 2 (0,0)0(0)(0)022f 2 (1,2)1(1)(2)229f 2 (2,1)2(2)(1)127f Example 1, page 536 22 ( , )f x yxy Example 2, page 536 2 ( , )g x y xy x y y = x Example 2, page 536 22 ( , )1h x yxy x y 11 1 1 Example 2, page 536 1113 300240 4888 an d pxyqxy Applied Example 3, page 537 ( , )R x yxpyq 11

2、13 300240 4888 xxyyxy 22 131 300240 484 xyxyxy Applied Example 3, page 537 10002000 11 3000 48 xy 13 2400 88 xy 0y 0 x Thus, the graph of the domain is: x y 2000 1000 11 3000 48 xy 13 2400 88 xy D Applied Example 3, page 537 y x Example 5, page 540 4 2 2 4 4 2 2 4 y x y x c Example 5, page 540 2 1 1

3、2 4 3 2 1 0 1 2 y x Example 6, page 540 0 (, )( , ) lim h ff xh yf x y xh 0 ( ,)( , ) lim k ff x ykf x y yk f x What does mean? ( , ) f f x b x slope of f x What does mean? What does ean? m f y ( , )slope of f f c y y What does ean? m f y 223 ( , )f x yxxyy 223 ( , )fyxyyxx 2 2 f x yx Example 1, pag

4、e 546 223 ( , )f x yxxyy 223 ( , )fyyyxxx 2 23x f yy y Example 1, page 546 223 ( , )f x yxxyy 2 (1,2) 2(1)22 f x 2 (1,2) 2(1)(2)3(2)8 f y Example 1, page 546 22 ( , ) xy w x y xy 22 ( , ) x y w y x yx 2 22 2 2 ()(2 ) () wxxx xx yyy y 22 222 () () y yx xy Example 2, page 547 22 ( , ) xy w x y xy 22 (

5、 , ) xy w yx yx 22 222 ()(2 ) () xxx x wyyy yy 22 222 () () x xy xy Example 2, page 547 22 5 ( , )()g s tsstt 22 5 ( , )()g s tstst 224 5()(2) g ssttts s 224 5(2)()st sstt Example 2, page 547 22 5 ( , )()g s tsstt 22 5 ( , )()g s tstst 422 5()(2 )ss g ttst t 224 5(2)()ts sstt Example 2, page 547 22

6、( , ) uv h u ve 22 ( , ) uv hevu 22 2 uv h eu u 22 2 uv ue Example 2, page 547 22 ( , ) uv h u ve 22 ( , ) uv hveu 22 ( 2 ) vu h ev u 22 2 uv ve Example 2, page 547 ( , , )ln yz wf x y zxyzxexy , fff xyz Example 3, page 549 ( , , )ln yz wf x y zxyzxexy ( ,ln, ) yz y zyzewyf xxxx ln y x z yzefy Examp

7、le 3, page 549 ( , , )ln yz wf x y zxyzxexy ( , , )ln yz wfyyxzxxezyx y y z x xzxzfe y Example 3, page 549 ( , , )ln yz wf x y zxyzxexy ( ,)ln, zy x yxywfzyzexx zy z fxyxye Example 3, page 549 Applied Example 4, page 550 1/3 1/31/3 2 3020 3 x y fxy x 2/3 2/32/3 1 3010 3 y x fxy y Applied Example 4,

8、page 550 1/3 273 (125,27)202012 1255 x f 2/3 7 9 12525 (125,27)101027 279 y f Applied Example 4, page 550 Applied Example 4, page 550 2 2 () xxx f ff xx 2 () xyx f ff y xy 2 2 () yyy f ff yy 2 () yxy f ff x yx f x y f x f y x y 2 ff xyx y 2 2 ff yyy x y 2 2 ff xxx 2 ff yxy x 22 ff y xx y When both a

9、re continuous 3222 ( , )33f x yxx yxyy 22 363xxyy 3222 (33) x fxx yxyy x 22 (363) xx fxxyy x 66xy 22 (363) xy fxxyy y 66xy 6()xy6()yx Example 6, page 552 3222 ( , )33f x yxx yxyy 2 362xxyy 3222 (33) y fxx yxyy y 2 ( 362 ) yx fxxyy x 66xy 2 ( 362 ) yy fxxyy y 62x 6()yx2(31)x Example 6, page 552 2 ( ,

10、 ) xy f x ye 2 2xy y e 2 () xy x fe x 2 2 () xy xx fy e x 2 4xy y e 2 2 () xy xy fy e y 22 3 22 xyxy yexy e 2 2 2(1) xy yexy Example 7, page 553 2 ( , ) xy f x ye 2 2 xy xye 2 () xy y fe y 2 (2) xy yx fxye x 22 3 22 xyxy yexy e 2 (2) xy yy fxye y 22 2(2)(2) xyxy xexyxy e 2 2 2(12) xy xexy 2 2 2(1) x

11、y yexy Example 7, page 553 ( , )0 f a b x ( , )0 f a b y ( , )0 f a b x ( , )0 f a b y ( , )0 f a b x ( , )0 f a b y ( , )0( , )0an d ff a ba b xy 22 ( , )f x yxy Example 1, page 561 22 ( , )f x yxy Example 1, page 561 22 ( , )f x yxy Example 1, page 561 22 ( , )344484f x yxxyyxy 644488 and xy fxyfx

12、y Example 2, page 562 22 ( , )344484f x yxxyyxy Example 2, page 562 322 ( , )412362f x yyxyy 2 2122436 and xy fxfyy 2 201224360 nd axyy Example 3, page 562 322 ( , )412362f x yyxyy Example 3, page 562 322 ( , )412362f x yyxyy Example 3, page 562 22 131 ( , )300240 484 R x yxyxyxy ( , )1801405000C x

13、yxy Applied Example 3, page 563 22 22 ( , )( , )( , ) 131 300240(1801405000) 484 131 1201005000 484 P x yR x yC x y xyxyxyxy xyxyxy Applied Example 3, page 563 22 131 ( , )1201005000 484 P x yxyxyxy 1131 12001000 2444 and xy PxyPyx Applied Example 3, page 563 2 131315 ( , ) 24481616 D x y 131 244 xx

14、yyxy PPP Applied Example 3, page 563 22 131 (208,64)(208)(64)(208)(64) 484 120(208) 100(64) 5000 $10,680 P 22 131 ( , )1201005000 484 P x yxyxyxy Applied Example 3, page 563 22222 12345 ddddd 111222333 ()()(), , , and so on.df xydf xydf xy 22222 12345 ddddd 222 112233 22 4455 () () () () () f xyf xy

15、f xy f xyf xy 222 112233 22 4455 mxbymxbymxby mxbymxby 222 112233 22 4455 ( , )()()() ()() f m bmxbymxbymxby mxbymxby 111222333 444555 2()2()2() 2()2() f mxby xmxbyxmxby x m mxbyxmxby x 222 111 122223333 22 44445555 2 mxbxy xmxbxy xmxbxy x mxbxy xmxbxy x 22222 1234512345 1 122334455 2()() () xxxxxmx

16、xxxx b y xy xy xy xy x 222 112233 22 4455 ( , )()()() ()() f m bmxbymxbymxby mxbymxby 112233 4455 2()2()2() 2()2() f mxbymxbymxby b mxbymxby 1234512345 2()5()xxxxx mbyyyyy 222 112233 22 4455 ( , )()()() ()() f m bmxbymxbymxby mxbymxby 1234512345 ()5xxxxx mbyyyyy 00an d ff mb 22222 1234512345 1 12233

17、4455 ()() xxxxxmxxxxx b y xy xy xy xy x 123123 (.). nn xxxx mnbyyyy 2222 123123 1 12233 (.)(.) . nn nn xxxxmxxxx b y xy xy xy x Example 1, page 570 2222 123123 1 12233 (.)(.) . nn nn xxxxmxxxx b y xy xy xy x 22222 (12345 )(12345) (1)(1)(3)(2)(4)(3)(3)(4)(6)(5) mb 551561mb Example 1, page 570 15517mb

18、 123123 (.)5. nn xxxx mbyyyy (12345)513436mb Example 1, page 570 15517mb551561mb Example 1, page 570 Applied Example 3, page 572 x p x2 xp 2.238.04.8483.6 5.436.029.16194.4 7.034.549.00241.5 11.530.0132.25345.0 14.628.5213.16416.1 40.7167.0428.411280.6 540.716740.7428.411280.6 an dbmbm Applied Examp

19、le 3, page 572 0.8139.99an dmb ( )0.8139.99(049.37) pf xxx Applied Example 3, page 572 2 ( ) ( 0.8139.99) 0.8139.99 R xxp xx xx 2 2 ( ) 0.8139.99(425) 0.8135.9925 P xRC xxx xx Applied Example 3, page 572 ( )1.6235.99P xx 0.81(22.22)39.9921.99p Applied Example 3, page 572 222 ( )2(1)321h xxxxx Exampl

20、e 1, page 580 y xExample 1, page 580 ( )622(31)h xxx 1 3 1 3 1 3 1 3 Example 1, page 580 1 2 , 3 3 22 1 2122 ,2 3 3333 f 2 3 1 2 , 3 3 2 3 Example 1, page 580 ( , , )( , )( , )F x yf x yg x y 22 ( , , )( , )( , ) (2)(1) F x yf x yg x y xyxy Example 2, page 582 22 2(1)Fxyxy 402010 xy FxFyFxy 11 42 an

21、d xy 114 10 423 or Example 2, page 582 141142 ()() 433233 yieldsand xy 411 342 xy intoand 1 3 x 2 3 y 1 2 , 3 3 Example 2, page 582 Applied Example 5, page 582 Applied Example 5, page 582 22 ( , , )( , )( , )4(43600)F x yf x yg x yxyxy 22 420 480 436000 x y Fyx Fxy Fxy Applied Example 5, page 582 2y x 22 2 48040 or y xyxy x 22 420 480 436000 x y Fyx Fxy Fxy Applied Example 5, page 582 22 4436000yy 45015 2y 22(15 2)30 2xy 30 215 2an dxy 22 420 4