Re: [微分] 2題求救...

[Honor1984 (希望願望成真)]

※ 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

--
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◆ From: 122.124.102.144
推 shadowlin:請問grad是指切線嗎?那A*B不是有XY項嗎?
→ shadowlin:dot又是什麼意思呢?對不起程度不好> <
→ shadowlin:為什麼知道要用x/a+y/b=1呢?是怎麼想到?
→ shadowlin:和為什麼知道要用(a,-b)去乘?
→ Honor1984:dot內積 grad法向量方向導數 垂直切線
→ Honor1984:x/a + y/b = 1截距式 因為a,b>0所以此切
→ Honor1984:線之向量平行(a,-b) 為何a,b>0 看grad
推 shadowlin:謝謝!!