Lecture 4 - Computer Science Department - University of Kentucky
Lecture 4 - Computer Science Department - University of Kentucky
Lecture 4 - Computer Science Department - University of Kentucky
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Error Analysis (3)<br />
Using the Mean‐Value Theorem for Integrals<br />
So we have<br />
∫<br />
1<br />
0<br />
∫<br />
1<br />
f "[ ξ ( x)]<br />
x(<br />
x −1)<br />
dx =<br />
f "[ ξ ( s)]<br />
0 0<br />
f ( x)<br />
dx −<br />
1<br />
2<br />
[<br />
f<br />
(0)<br />
+<br />
= −<br />
We then do a change <strong>of</strong> variable, and let<br />
f<br />
1<br />
6<br />
(1)]<br />
∫<br />
1<br />
f "( ζ )<br />
=<br />
x(<br />
x −1)<br />
dx<br />
−<br />
1<br />
12<br />
f "( ζ )<br />
g( t)<br />
= f [ a + t(<br />
b − a)],<br />
x = a + ( b − a)<br />
t<br />
dx<br />
= ( b<br />
− a)<br />
dt,<br />
g'(<br />
t)<br />
'[ a<br />
g"(<br />
t<br />
) =<br />
f<br />
"[ a<br />
+<br />
t<br />
(<br />
b − a<br />
)](<br />
b − a<br />
)<br />
=<br />
f<br />
+ t(<br />
b<br />
2<br />
− a)](<br />
b<br />
−<br />
a)<br />
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