(a) The function y depends on three independent variables w, s and d as follows:y(w, s, d)= ws^3 / d^4Find the percentage increase in y when w increase by 2%, s decreases by 3% and d increases by 1%.
(b)A right-angled triangle having sides x, y and z is shown below:
圖片參考：http://i926.photobucket.com/albums/ad110/alantse30/154f7636.png
If x is increasing at a rate of 2cm /s while y is decreasing at 3cm/s, calculate the rate at which z is changing. Given that x = 5cm and y =3cm at the present moment.
數學Partial Differentiation(20點)
2010-12-13 7:40 am
回答 (2)
2010-12-13 8:30 am
✔ 最佳答案
a) 圖片參考：http://i1191.photobucket.com/albums/z467/robert1973/Crazypardiff1.jpg
Hence the percentage increase is -3%
b) Using Chain Rule:
圖片參考：http://i1191.photobucket.com/albums/z467/robert1973/Crazypardiff2.jpg
2010-12-13 00:35:01 補充：
Part (a)
Last line should be = -0.03 ws^3/d^4
參考: 原創答案
2010-12-13 8:38 am
y = ws^3 /d^4
線性解法
w + 2%, s + 3 %, d + 1%
=(2 + 3 * 3 - 4 * 1) %
= 7 %
實際值 = 1.02 * 1.03^3 / 1.01^4 = 1.0710909522633878760471046041808
% increase = 7.10909522633878760471046041808%
x2 = y2 + z2
dx/dt = 2, dy/dt = 3
z2 = x2 - y2
z2 = 25 - 9
z = 4
另
d(z2)/dt = d(x2)/dt - d(y2)/dt
d(z2)/dz * dz/dt = d(x2)/dx * dx/dt - d(y2)/dy * dy/dt
2z * dz/dt = 2x * dx/dt - 2y * dy/dt
代入x, y, z, dy/dt, dx/dt
8 dz/dt = 10 * 5 - 6 * 3
dz/dt = 32 / 8 = 4
2010-12-15 00:51:32 補充：
messed up + and - in part 1, should be
(2 - 3*3 - 4*1) = -11%
線性解法
w + 2%, s + 3 %, d + 1%
=(2 + 3 * 3 - 4 * 1) %
= 7 %
實際值 = 1.02 * 1.03^3 / 1.01^4 = 1.0710909522633878760471046041808
% increase = 7.10909522633878760471046041808%
x2 = y2 + z2
dx/dt = 2, dy/dt = 3
z2 = x2 - y2
z2 = 25 - 9
z = 4
另
d(z2)/dt = d(x2)/dt - d(y2)/dt
d(z2)/dz * dz/dt = d(x2)/dx * dx/dt - d(y2)/dy * dy/dt
2z * dz/dt = 2x * dx/dt - 2y * dy/dt
代入x, y, z, dy/dt, dx/dt
8 dz/dt = 10 * 5 - 6 * 3
dz/dt = 32 / 8 = 4
2010-12-15 00:51:32 補充：
messed up + and - in part 1, should be
(2 - 3*3 - 4*1) = -11%
收錄日期: 2021-04-13 17:43:14
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20101212000051KK01573