Quod Erat Demonstrandum

2010/07/22

暑期無聊閱讀

Filed under: Additional / Applied Mathematics,HKALE,HKCEE — johnmayhk @ 9:02 上午
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學校圖書館檢來一本舊教科書,名為 Applied Numerical Analysis。

在解方程一章,以下例引起動機。

If you worked for a mining company the following might be a typical problem:

There are two intersecting mine shafts that meet at an angle of 123^o, as shown in the figure above. The straight shaft has a width of 7 feet, while the entrance shaft is 9 feet wide. What is the longest ladder that can be negotiate the turn? You can neglect the thickness of the ladder members, and assume it is not tipped as it is maneuvered around the corner. Your solution should provide for the general case in which the angle, A, is a variable, as well as the widths of the shafts.

修附加數的同學,或許你也做過類似習題,比如考慮角 A 的大小是 90 度的情況。

教科書隨即道:只要解出以下超越方程的 C 值:

\frac{9\cos (\pi - 123^o - C)}{\sin^2(\pi - 123^o - C)} - \frac{7\cos C}{\sin^2 C} = 0

代入下式

l = \frac{9}{\sin (\pi - 123^o - C)} + \frac{7}{\sin C}

便可計算梯子的最大長度。

無聊的授課員可能指出:寫"\pi - 123^o"不妥當。但重點是如何獲得上述結論,修 M2 的同學,這可以是大家的習題。

當然,這不過是引起動機之例,在那一課的習題中,可以找到很多其他的「實例」,讓人感到非線性方程在「生活」中俯拾皆是,見:

雖然問題的結構和初中的"change the subject"分別不太大。

書中也有一些第一次見(我孤陋寡聞嘛…)的寫法,比如作者把四階導數寫成

f^{iv}(x)

6 則迴響 »

  1. 個fourth derivative咁寫….第一眼睇落以為complex variable果類野= =.
    John sir你有冇睇過日本人拍果D綜藝節目,有套叫恐佈數學,好似係北野武有份,係講一D現實生活中既數學

    迴響 由 Justin — 2010/07/22 @ 10:00 上午 | 回覆

    • 冇睇過呀,原來是 NOW TV 100 台的節目,留意下先。

      迴響 由 johnmayhk — 2010/07/22 @ 8:20 下午 | 回覆

  2. john ng,有條題想問一下你。
    consider the system of linear equations
    X+3Y+2Z=2
    Z+KY+Z=2
    X+11Y+KZ=2
    show that the system of linear equations(E) must have a unique solution.

    i have got that when K=7 or K=-2, E has no real solution.
    then can we still say that E must have a unique solution.

    迴響 由 jaychan00 — 2010/07/22 @ 5:33 下午 | 回覆

    • 第 2 條式是

      x + ky + z = 2

      嗎?

      如果是,

      則對應係數的行列式是

      k^2 – 5k + 14
      = (k – 2.5)^2 + 7.75

      對於任何實數 k,上式一定非零,

      故題目的方程組必有唯一解。

      但如果第 2 條式真的是

      z + ky + z = 2

      ky + 2z = 2

      則對應係數的行列式是

      k^2 – 2k – 16

      可以是零,即方程組不一定有唯一解。

      迴響 由 johnmayhk — 2010/07/22 @ 8:14 下午 | 回覆

  3. 恩=。= 系 x +ky +z = 2.=.=忘記了completing square這招絕招。

    迴響 由 jaychan00 — 2010/07/22 @ 8:24 下午 | 回覆

  4. thank you john sir

    迴響 由 jaychan00 — 2010/07/22 @ 8:24 下午 | 回覆


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