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-(+2h)+16f (x+h)-30f(x)+165 (x-h)-(-2h)Suppose f"(x)3Show that the error is O(h4)

Question
-(+2h)+16f (x+h)-30f(x)+165 (x-h)-(-2h)
Suppose f"(x)
3
Show that the error is O(h4)
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-(+2h)+16f (x+h)-30f(x)+165 (x-h)-(-2h) Suppose f"(x) 3 Show that the error is O(h4)

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Step 1

It is given that,

1
x 2h)+16 f(x+h)-30f(x) +16S(x-h)-f(x-2h)]
(x)
12h2
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1 x 2h)+16 f(x+h)-30f(x) +16S(x-h)-f(x-2h)] (x) 12h2

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Step 2

Use Taylor’s series to compute f(x+2h) as follows.

(2h)(x)
(2h)
2
(2h)
-S" (х)+
2!
f(x+2h)- f+(2h)f (x)+
(2h)
f(x)+
3!
4!
(2h)
f(x)+
6!
(x)+
+
5!
16h
f(x)
24
8h3
4h2
f"(x
)
)+
f"(x)
=f+2hf(x
+
6
2
64hx)+
32h
f(x)+
720
+
120
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(2h)(x) (2h) 2 (2h) -S" (х)+ 2! f(x+2h)- f+(2h)f (x)+ (2h) f(x)+ 3! 4! (2h) f(x)+ 6! (x)+ + 5! 16h f(x) 24 8h3 4h2 f"(x ) )+ f"(x) =f+2hf(x + 6 2 64hx)+ 32h f(x)+ 720 + 120

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Step 3

Compute 16(f(x+h)) as foll...

h2
f+hf(x)
2!
h's
h2
4
f"(x)+
-f"(x)+
3!
4!
16f(x+h)16
5
6
x)+
5!
6!
h2
h's
ht
f"(x)+
24
f+hf (x)+x)+f"(x)
2
6
=-16
h6
f(x)+
720
5
120
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h2 f+hf(x) 2! h's h2 4 f"(x)+ -f"(x)+ 3! 4! 16f(x+h)16 5 6 x)+ 5! 6! h2 h's ht f"(x)+ 24 f+hf (x)+x)+f"(x) 2 6 =-16 h6 f(x)+ 720 5 120

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