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Chapter 7.7 (Hint) 5. Hint: (a)(b) ᐭ Midpoint Rule Ȥ Simpson’s RuleḄȄǷ஺ (c) ᑖ ωǷ᧲, ᵨᑖZᑖ஺ 23. Hint: (a)(f) ᵨ8ᡠᦟḄAᶍ˴3f

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Chapter 7.7 (Hint)

5. Hint: (a)(b) 代入 Midpoint Rule 和 Simpson’s Rule的公式求近似值。 (c) 計算積分 實際值時, 使用分部積分。

23. Hint: (a)(f) 用第四章所教的方法約略畫出f′′(x)和f(4)(x)的圖形, 並找出極值可能發 生的點。 或是用 Maple, Mathematica, 或是 Matlab 等軟體, 直接畫圖來找 upper bound。

(j) 假設|ES| < 0.0001, 計算這個不等式, 找出n的下界。

45. Hint: 將[a, b]分割成n個 subintervals: [xi−1, xi], 1 ≤ i ≤ n, 並証明在一個 subinter- val 內, 恆有T[xi

−1,xi]<Rxi

xi−1

f(x) dx < M[xi

−1,xi]。 46. Hint: 考慮取n = 2使用 Simpson’s Rule計算Rb

a f(x) dx的值, 証明S2 =Rb

af(x) dx, 並說明為什麼証明了n = 2的情況即可。

47. Hint: 將Tn, Mn, T2n的定義寫清楚。

48. Hint: 將Tn, Mn, S2n的定義寫清楚

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