請參考附件
填充第 12 題
官方答案有誤，應是 16/3
114 明倫高中
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114 明倫高中
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Re: 114 明倫高中
第 9 題
(7 + x)^(1/3)/x = lim[1/(3n - 1)][f(x^2/n) + f(2x^2/n) + ... + f(nx^2/n)]
= lim[n/(3n - 1)](1/n)Σf(x^2 * (k/n))
= (1/3)∫f(x^2t)dt (從 0 積到 1)
兩邊同乘以 3x^2
3x(7 + x)^(1/3) = ∫f(x^2t)x^2dt = ∫f(x^2t)d(x^2t) (從 0 積到 1)
令 u = x^2t，F(u) = ∫f(u)du
3x(7 + x)^(1/3) = ∫f(u)du (從 0 積到 x^2) = F(x^2) - F(0)
兩邊同時微分
(4x + 21)/(7 + x)^(2/3) = 2xf(x^2)
x = 1 代入
可得
2f(1) = 25/4
f(1) = 25/8
(7 + x)^(1/3)/x = lim[1/(3n - 1)][f(x^2/n) + f(2x^2/n) + ... + f(nx^2/n)]
= lim[n/(3n - 1)](1/n)Σf(x^2 * (k/n))
= (1/3)∫f(x^2t)dt (從 0 積到 1)
兩邊同乘以 3x^2
3x(7 + x)^(1/3) = ∫f(x^2t)x^2dt = ∫f(x^2t)d(x^2t) (從 0 積到 1)
令 u = x^2t，F(u) = ∫f(u)du
3x(7 + x)^(1/3) = ∫f(u)du (從 0 積到 x^2) = F(x^2) - F(0)
兩邊同時微分
(4x + 21)/(7 + x)^(2/3) = 2xf(x^2)
x = 1 代入
可得
2f(1) = 25/4
f(1) = 25/8