Re: [微積] 3題考古題
※ 引述《h29479341 (FLY)》之銘言：
: x-sin^(-1)x
: 1. lim -----------------
: x->0 (sinx)^(3)
因為 [arcsin(x)]' = (1 - x^2)^{-1/2}, arcsin(0) = 0
x ∞ -1/2 n
=> arcsin(x) = ∫[Σ ( )(-t^2) ]dt
0 n=0 n
x
= ∫(1 + t^2/2 + ...)dt
0
= x + x^3/6 + ...
(sin x)^3 = (x - x^3/6 + ...)^3 = x^3 +/- ...
x - arcsin(x) x - (x + x^3/6 + O(x^5))
lim --------------- = -------------------------- = -1/6
x→0 sin^3(x) x^3 + O(x^5)
: 2. lim (e^x+1)^(1/x)
: x->∞
edit:
1
(1 + e^x)^{1/x} = e (1 + ---)^{1/x}
e^x
x→∞時右邊顯然趨近 1
(以下做法比較麻煩可忽略)
1/x ln(1 + e^x) ln(1 + e^x)
lim e = exp( lim ----------- )
x→∞ x→∞ x
e^x/(1+e^x)
= exp( lim ----------- )
x→∞ 1
= e
: 3. If tan^(-1)(y/x)= ln根號(x^2+3y^2) find dy/dx=?
F(x, y(x)) = arctan(y/x) - 1/2 ln(x^2 + 3y^2) = 0
1 y 1 2x
----------- (- ---) - --- ----------
1 + (y/x)^2 x^2 2 x^2 + 3y^2
dy/dx = -Fx/Fy = - ------------------------------------
1 1 1 6y
----------- --- - --- ----------
1 + (y/x)^2 x 2 x^2 + 3y^2
y 1 2x
--------- + --- ----------
x^2 + y^2 2 x^2 + 3y^2
= ----------------------------
x 1 6y
--------- - --- ----------
x^2 + y^2 2 x^2 + 3y^2
2y(x^2 + 3y^2) + 2x(x^2 + y^2)
= --------------------------------
2x(x^2 + 3y^2) - 6y(x^2 + y^2)
3y^3 + y^2x + yx^2 + x^3
= ---------------------------
-3y^3 + 3y^2x - 3yx^2 + x^3
: 其中第1.3題 有反三角函數唷
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.217.33.242
※ 編輯: suhorng 來自: 61.217.33.242 (02/05 00:07)
→
02/05 23:39, , 1F
02/05 23:39, 1F
推
02/06 01:41, , 2F
02/06 01:41, 2F
→
02/06 20:35, , 3F
02/06 20:35, 3F
討論串 (同標題文章)