✔ 最佳答案
1. 求偏導數, 很基本的問題!
f_y(x,y) = 2x^2+2xy-3y^2.
2. C 以參數式表現:
C = {(x,y,z) = (t,-3t,2t): 0≦t≦1}
故 ds = √[1^2+(-3)^2+2^2] dt = (√14)dt
而 x+y^2-2z = t+(-3t)^2-2(2t) = 9t^2-3t
故原積分變成 √14(9t^2-3t) 對 t 在 [0,1] 的積分, 結果是
√14(3-3/2) = (3/2)√14.
3. 2.2^2 = 4.84
√4.9 = √(4.84+0.06)
x = a = 4.84, dx = 0.06 ==> d(√x) = [1/(2√x)]dx = 0.06/4.4 = .014
故 √4.9 ≒ 2.2+.014 = 2.213636... ≒ 2.214
切線近似之誤差:
|f(x) - f(a) - f'(a)(x-a)| = |f"(t)|(x-a)^2/2, for some t 在 (a,x) 之內.
f(x) = √x ==> f"(x) = -1/(4x^{3/2})
故 |f(x) - f(a) - f'(a)(x-a)| ≦(.06^2/2)/(4*2.2^3) < .00005
故 √4.9≒2.214 其誤差在 .00036+.00005 < .0005 之內.
Check: √4.9 = 2.21359 ≒ 2.214
2011-03-29 14:24:25 補充:
d(√x) = [1/(2√x)]dx = 0.06/4.4 = .013636...
√4.9 ≒ 2.2+.013636... = 2.213636... ≒ 2.214