作者Honor1984 (希望願望成真)
看板trans_math
標題Re: [微分] 2題求救...
時間Sat Nov 1 15:09:29 2008
※ 引述《victor7935 (victor)》之銘言：
: 1.
: Show that
: the hyperbola x^2-y^2=5
grad A: 2x i -2y j
: and
: the ellipse 4x^2+9y^2=72
: are orthogonal.
grad B: 8x i + 18y j
二方程式聯立
13x^2 = 117
=> x^2 = 9
grad A dot grad B = 16x^2 - 36y^2
= 16x^2 - 36(x^2-5)
= -20x^2 + 180
= 0
所以在交會點處orthogonal
: 2.
: Show that the sum of the x- and y-intercepts of
: any line tangent to the graph of
: (1/2) (1/2) (1/2)
: x + y = c
: is constant and equal to c.
x, y ≧ 0
graph的grad 1/2√x i + 1/2√y j
x/a + y/b = 1 a , b > 0
(a , -b) dot (1/2√x , 1/2√y) = 0
a^2 b^2
---- = -----
x y
令x = a^2k
y = b^2k
k > 0
ak + bk = 1 => k = 1 / (a + b)
a √k + b √k = √c = √(a + b)
=> a + b = c const
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 122.124.102.144
1F：推 shadowlin:請問grad是指切線嗎?那A*B不是有XY項嗎? 122.125.165.33 11/01 15:45
2F：→ shadowlin:dot又是什麼意思呢?對不起程度不好> < 122.125.165.33 11/01 15:57
3F：→ shadowlin:為什麼知道要用x/a+y/b=1呢?是怎麼想到? 122.125.165.33 11/01 16:15
4F：→ shadowlin:和為什麼知道要用(a,-b)去乘? 122.125.165.33 11/01 16:18
5F：→ Honor1984:dot內積 grad法向量方向導數 垂直切線122.124.102.144 11/01 16:19
6F：→ Honor1984:x/a + y/b = 1截距式 因為a,b>0所以此切122.124.102.144 11/01 16:20
7F：→ Honor1984:線之向量平行(a,-b) 為何a,b>0 看grad122.124.102.144 11/01 16:21
8F：推 shadowlin:謝謝!! 122.125.165.33 11/01 17:07