2條Partial differentiation問題

2012-11-12 3:41 am

1. The concentration of salt in a fluid at (x,y,z)is given by F(x,y,z)=4x^2+3y^4+2x^2z^2 mg/cm^3. You are at the point(1,1,1) .

(a) In which direction should you move if you want the concentration to increase the fastest?

(Give your answer as a vector.)

(b) You start to move in the direction you found in part (a) at a speed of cm/sec. How fast is the concentration changing?
rate of change =

2. For g(x,y)= (x^2+5y+15)^1/2, find the best linear approximation of g(x,y) for (x,y) near (1,4) .

更新1:

謝謝你的解答,自由自在請問題1(b) 為何可以這樣計算 1. 點解partial x/partial t=v X (12/(12^2+12^2+4^2)^1/2) 2. 咁用得 partial differentiation,請問除了t之外仲有咩係x,y,z既變量? 謝謝!

回答 (1)

用戶9112

2012-11-12 5:37 am

✔ 最佳答案

in (1)(b) you did not write down the speed???

2012-11-11 21:37:29 補充：
http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-1114.png

圖片參考：http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-1114.png

2012-11-12 13:11:07 補充：
The direction of greatest rate of change is 12i + 12j +4k
The unit vector in that direction is 3/√19 i + 3/√19 j + 1/√19 k
Then if the speed is v the velocity is 3v/√19 i + 3v/√19 j + v/√19 k
dx/dt is the veolcity in x direction so it is 3v/√19

2012-11-12 13:13:42 補充：
F only change with x, y and z
x,y and z are the position
then why the position change? because it is moving as indicated in the . so x,y,z changes with t

2012-11-12 13:15:52 補充：
the question does not give any hint as to why the position change so we can only assume t is the only variable

2012-11-12 13:16:07 補充：
Suppose we have another scenario that the position will be when the light intensity change then the light intensity become a variable. You can imagine a lot of factors as well, but as far as this question is concerned, only speed is mentioned...

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