推 ivylan:謝謝!!! 111.251.121.48 07/08 16:24
※ 引述《ivylan (藍藍)》之銘言：
: 4.lim (2x)^x=?
: x→0
: 想確認一下答案是不是1
取ln後
xln2x = xln2 + xlnx
這兩項極限都有值
第一項0
第二項
lim xlnx
= lim (lnx / 1/x)
x->0
= lim (1/x)/(-1/x^2)
= 0
取exp就是1 沒錯
x^x^........更複雜的情況我有發過文
: → →
: 8.(a)if f(x,y,z) = (x^2+y^2+z^2)^(1/2) , then▽*(▽f) = ?
f = r
∂f/∂x = 2x/2r = x/r
→ → →
▽f = (x/r,y/r,z/r) = (1/r)R r = |R| R : 位置向量
▽*(▽f) = [(∂/∂x)^2 + (∂/∂y)^2 + (∂/∂z)^2]f
(∂/∂x)(∂/∂x)f = (∂/∂x)(x/r) = [r - x(x/r)]/r^2 = (y^2+z^2)/r^3
同理(∂/∂y)^2f = (x^2 + z^2)/r^3
(∂/∂z)^2f = (x^2 + y^2)/r^3
=> ▽*(▽f) = 2(x^2 + y^2 + z^2)/r^3 = 2r^2/r^3 = 2/r = 2/√[x^2+y^2+z^2]
: → → →
: (b)if F(x,y,z) = <x^3+y , y^3+x , z^3+y > , then ▽(▽*F) = ?
▽*F = 3x^2 + 3y^2 + 3z^2 = 3r^2 r^2 = x^2 + y^2 + z^2
▽(3r^2) = 3 * 2r (x/r , y/r, z/r)
= 6 (x, y, z)
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