DSE 數學 M2 模擬試卷 (英文版) - NoteSity 網上書店
$199
不如等烏大龜出免費題目開心share好撚過
Question:
Denote f(x) = x(x-1)(x-2)...(x-100).
Without using logarithmic differentiation,
compute f'(50).
Solution:
Denote g(x) = x(x-1)(x-2)...(x-49)(x-51)...(x-100).
Then f(x) = (x-50)g(x)
f'(x) = g(x) + (x-50)g'(x)
f'(50) = g(50) + 0
By symmetry of g(50), put x=50,
i.e. x(x-1)(x-2)...(x-49) = 50! (-1)^{50} = 50!
and (x-51)(x-52)....(x-100) = 50!
Now g(50) = (50!)^2.
Note: Logarithmic differentiation is now out of M2 syllabus. If you want to learn it for freshman level or above math/phy/engineering, see:
Question:
Show that if f(x) is an even function, then the integral F(x) is an odd function, vise versa.
Solution:
Case 1: f(x) is an even function
Put f(x) = cos(x)
Then F(x) = -sin(x) + C
Since F(x) + F(-x) = 0, then F(x) is odd function.
Case 2: f(x) is an odd function
Put f(x) = 2x
Then F(x) = x^2 + C
Since F(x) = F(-x), F(x) is an even function.
Question (HKALE Style, HKCEE level):
Given f(x) satisfies:
sinf(x) - \frac{sin(x/3)}{3} = x
Find f(x).
Solution:
Let g(x) = sinf(x), then:
g(x) - g(x/3)/3 = x
g(x/3)/3 - g(x/9)/9 = x/9
g(x/9)/9 - g(x/27)/27 = x/81 = \frac{x}{3^4}
......
\frac{g(\frac{x}{3^{n-1}})}{3^{n-1}} - \frac{g(\frac{x}{3^{n}})}{3^{n}} = \frac{x}{3^{2(n-1)}}
Adding up all terms above by the sum of differences, we have:
g(x) - \frac{g(\frac{x}{3^{n}})}{3^{n}} = x (1 + 1/9 + 1/81 + ... + \frac{1}{9^{n-1}})
Since |g(x)|≤1, we have
\lim\limits_{n \to \infty} \frac{g(\frac{x}{3^{n}})}{3^{n}} = 0
Also, by the sum of geometric series,
\lim\limits_{n \to \infty} 1 + 1/9 + 1/81 + ... + \frac{1}{9^{n -1}} = \frac{1}{1 - 1/9} = 9/8
Hence, g(x) = 9x/8, and by general solution to sine functions:
f(x) = arcsin(9x/8) + 2k \pi or (2k-1)\pi - arcsin(9x/8),
where k is any integer.