Re: [考古] 交大88第八題

[GSXSP (Gloria)]

※ 引述《johnnyzsefb (AJ)》之銘言：
: ８．A heat-seeking particle is located at the point (2,-3) on a metal
:     plate whose temperature at (x,y) is T(x,y)=20-4x^2-y^2.
:     Find the path of the particle as it continuously moves in the direction
:     of maximum temperature increase.
▽T = (-8x , -2y)

Let path be  (t , f(t))

direction: (1 , f'(t))  =  ▽T  = (-8t , -2f(t)) = -8t(1 , f(t)/4t)
                                                 = (1 , f(t)/4t)

 f'(t) = f(t)/4t    ...解微方

4tf'(t) = f(t)

Let f(t) = K*t^r 代回原式

可以解到 r   然後再利用 (2,-3) 解 K

應該就解完了 ##

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→ GSXSP:補一下  t要選範圍
推 johnnyzsefb:水喔!~方法真讚XD
→ johnnyzsefb:但是選範圍啥意思? 要說t為實數嗎?
→ johnnyzsefb:path (t,t^(1/4)-3-2^(1/4)) t為大於0
→ johnnyzsefb:之實數?
→ GSXSP:(2,3)起點 一個方向增加 一個方向減少
→ GSXSP:(2,-3)
→ GSXSP:還有我上面有一個等號稍欠考慮
→ GSXSP:-8t提出來..所以t<0才可以 然而我們要從t=0
→ GSXSP:開始 所以是從t=2 跑到t=0
→ GSXSP:抱歉 上面手誤 是從t=2開始
→ GSXSP:然後可以看看(0,0)的確是最大值
→ GSXSP:講的有點混亂@@ 自行體會一下吧XD