Re: [微積] 幾題微積分
f(x) = x^2
f(x+△x) - f(x)
f'(x) = lim -----------------
△x→0 △x
x^2 + 2x△x + (△x)^2 - x^2
= lim -----------------------------
△x→0 △x
2x△x + (△x)^2
= lim -----------------
△x→0 △x
= lim 2x + △x
△x→0
= 2x
: 2. 第5題
: http://i.imgur.com/KblB1XG.jpg
f = x^2 + 2y^2 - 3z^2
▽f = (2x, 4y, -6z)
Dv(f) = ▽f dot v_hat = (1/√3)(2x + 4y - 6z)
Dv(f)|(1,1,1) = 0
or
let x = y = z = t
f(t) = t^2 + 2t^2 - 3t^2 = 0
df/dt = 0
: 3.
: http://i.imgur.com/uHlgefH.jpg
f(x,y) = arcsin(xy)
yi + xj
▽f = ----------------
√(1 - (xy)^2)
▽f(1,0) = j
D(f) (min) | (1,0) = -|▽f| = -1
: 4. 第3題
: http://i.imgur.com/9fOQycm.jpg
3.
1
-----------
x(a + bx)
1 1
= - --------------
b x(x + (a/b))
1 b 1 1
= - -(- - ----------)
b a x x + (a/b)
1
∫----------- dx
x(a + bx)
1 dx dx
= -(∫--- -∫-----------)
a x x + (a/b)
= (1/a)(ln|x| - ln|x + a/b|)
5.
x = exp(-t)*cost
y = exp(-t)*sint
dx/dt = exp(-t)*(-cost - sint)
dy/dt = exp(-t)*(-sint + cost)
d length = √[(dx/dt)^2 + (dy/dt)^2] dt
= √[exp(-t)^2 *2] dt
= √2 * exp(-t) dt
length = | ∫d length |
= | √2 ∫exp(-t) dt |
= | -√2 exp(-t) |
= √2 exp(-t) (帶上下界才是答案)
: 題目有點多，不好意思!!
: 謝謝大家幫忙，感恩
: -----
: Sent from JPTT on my Asus ASUS_Z017DA.
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※ 編輯: aromaQ626 (218.161.55.240), 06/11/2017 18:23:27
※ 編輯: aromaQ626 (218.161.55.240), 06/11/2017 18:26:08
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