請高手幫我題解一高微題目(有關積分,20分)

Question

Let f ∈C'[a , b] . Let Pn be the usual partition with equally Spaced points Xj=a+jh , j=0,1,. . . n
where h=(b-a)/n ,and choose Sj∈ [X(j-1),Xj] .
Let S(f,Pn,S)=∑(j-1)^n▒f(Sj )h be a Riemann sum for f . Show that:

|∫_a^b▒f-S(f,Pn,S)|≤M(b-a)h
Where
M=max|f'(X) | , a≤x≤b



其中
(j-1)^n表示j-1到n
∫_a^b表示a積到b
S(f,Pn,S)內部的S其實是小s,我想讓後面的Sj看起來比較好分辨

請詳細幫我[解釋題目]的要求及[作法說明]
若有題意不懂可以討論
謝謝大家~

Answer 1

1. 可設f(x)在[a, b]內均為正數,否則討論函數 f(x)-min(f(x))即可


2. 設f(x)在[xj-1~xj]內最大值為f(xj*), xj* in [xj-1, xj]
∫(x=xj-1~xj) f(x) dx <= f(xj*)h


3. |∫ab f(x) dx - S(f,Pn,S)| <= ∑(j=1~n) |∫(x=xj-1~xj) f(x) dx - f(sj)h |

<= ∑(j=1~n) [ | f(xj*)-f(sj)|*h ]

= ∑(j=1~n) | f'(tj)(xj*-sj) *h | (MVT, tj 介於 xj* 與 sj之間)

<= ∑(j=1~n) M*h*h ( |xj*-sj|<=h, |f'(x)|<=M)

= M*h*h* n = M*h*(b-a)/n *n = M(b-a)h



得證