Quod Erat Demonstrandum

2010/04/26

一對直線

Filed under: Additional / Applied Mathematics,Pure Mathematics — johnmayhk @ 4:09 下午
Tags:

表示一對直線(A pair of straight lines)的方程可以是

(a_1x+ b_1y + c_1)(a_2x+ b_2y + c_2) = 0

化簡後得二次式 Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

但有沒有「別樣」的方程,亦可表示一對直線(起碼,一對平行線)?

中四的歐同學今天偶爾問起一例,偶爾答答。

看看以下方程:

(x + y)\log(x + y) = x + y - 1

同學試找其 \frac{dy}{dx},從而估計一下它的圖像。

用 Winplot 輸入 log(x+y)*(x+y)-x-y+1=0,得圖像為兩條平行線。

始終感覺有點怪,我可能想錯了什麼…

[OT] 午飯,學校門口有人派傳單,宣傳一個數學網站。其中一個口號是:「勢必揭起網上學習旋風」。上去看看,數學東西是有的,但感覺也是宣傳多於一切。假假地在下也有十多年所謂「網上教學」經驗,從來看不到「網上學習」(esp. math. stuff)在本地會產生什麼「旋風」,荒涼地「營運」或(半)休眠才是常態。

4 則迴響 »

  1. Ha, long time not leaving comment here~

    By simple calculus it can be shown that the equation t ln t – t + 1 = 0 has only one real solution, namely t = 1, even after allowing ln t to have complex value when t<0. So there should be one straight line x+y=1 plotted. I guess it is numerical error and the two equations plotted in Winplot is very close?

    迴響 由 馬高迪克 — 2010/04/27 @ 1:58 上午 | 回覆

    • Thank you 馬高迪克! (ot: why change your ‘name’ into Chinese?)

      Yes, xlnx – x + 1 = 0 has exactly one root (x = 1), but what I’d used in the Winplot was ‘log’ (instead of ‘ln’) and the fact that xlogx – x + 1 = 0 has a root 7.292411085… (other than 1)
      resulting the graph of a pair of straight lines.

      And it is still very strange to me…

      P.S. Sorry, I can’t remember, are you staying postdoc now? Thank you again for leaving a message, have a nice day!

      迴響 由 johnmayhk — 2010/04/27 @ 9:05 上午 | 回覆

  2. Actually I forget what name I usually use here. My MSN is MarcoDick, though. Sometimes I find the Chinese translation 馬高迪克 more 型, like 迪克與牛仔.

    I am not that old, now being first year PhD student in NYU Computer Science; yet Mathematics is always in my heart.

    迴響 由 馬高迪克 — 2010/04/27 @ 9:14 上午 | 回覆

    • Thank you 迪克博士 for having the heart of math and willing to share! ^_^

      迴響 由 johnmayhk — 2010/04/27 @ 9:21 上午 | 回覆


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