Quod Erat Demonstrandum

2010/10/18

某插值法習題

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 2:56 下午
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一道普通的題:

f(x) = \sin(\frac{\pi}{2}x)

設多項式 p(x) 使 p(0) = f(0), p(1) = f(1), p(2) = f(2)

估計絕對誤差 |f(x) - p(x)| 在區間 (0.5,1) 上的上限。 (more…)

廣告

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. (more…)

2008/11/09

Initial guess

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 9:36 下午
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數值法解方程的題目,通常會提供 initial guess。但有時就算沒有提供,我們也可以約略估計,比如解

x - \frac{x^3}{9} + \frac{x^5}{11} - \frac{x^{15}}{2008} = 0.123

同學可以估計其中一個實數解,大概是什麼嗎?(先問:為何知道實數解一定存在?) (more…)

2008/10/22

Integrate polynomial of degree less than 4

Filed under: Additional / Applied Mathematics — johnmayhk @ 8:30 下午
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To find the definite integral of a polynomial of degree less than 4, we can use the following formula.

\int_a^b p(x)dx = \frac{b - a}{6}[p(a) + 4p(\frac{a+b}{2}) + p(b)]

e.g.

\int_2^4 (x-2)(x-4)(x-7)dx = \frac{4-2}{6}[0 + 4(3-2)(3-4)(3-7) + 0] = \frac{16}{3}

Nothing special, it is just something about Simpson’s rule.

2008/09/29

Just a question of applied math. from a F.7 boy

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 6:13 下午
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Just discuss an easy AL Applied Mathematics (II) Question with students.

Let f(x), g(x), h(x) be twice differentiable functions such that f(x) = g^2(x) + x^3h(x).

(a) Let p(x) = \frac{f(x)}{g(x)}, where g(0) \ne 0. Show that p(0) = g(0), p'(0) = g'(0), p''(0) = g''(0).

(b) Using (a), or otherwise, find Taylor’s expansion of the function \frac{2x^4 - 3x^3 + x + 4}{\sqrt{x + 4}} about x = 0, up to the term in x^2. (more…)

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