2. f(x) = sin(2x) + x², you are only given the following data x f(x) -0.1 0 0.1 0.2 -0.1887 0 0.2087 0.4294 (2-point) Forward-Difference 3-point midpoint 3-point forward endpoint 3-point backward endpoint f'(x) = f'(x) = f'(x) = f'(x) = f(x +h)-f(x) h f(x+h)-f(xh) 2h -3f(x) + 4f(x+h)-f(x + 2h) 2h 3f(x)-4f(xh) + f(x-2h) 2h (a) Use the two-point forward difference formula to approximate f'(0.1). Compare your actual error with the error bound. (b) Use the most accurate three-point formula to find f'(-0.1), f'(o), and f'(0.2). Also compare your actual errors with the error bounds. (c) For parts (a) and (b), do you observe the actual error is ≤ error bound for each case? Explain why "actual error>error bound" may happen, and how to solve the issue?
2. f(x) = sin(2x) + x², you are only given the following data x f(x) -0.1 0 0.1 0.2 -0.1887 0 0.2087 0.4294 (2-point) Forward-Difference 3-point midpoint 3-point forward endpoint 3-point backward endpoint f'(x) = f'(x) = f'(x) = f'(x) = f(x +h)-f(x) h f(x+h)-f(xh) 2h -3f(x) + 4f(x+h)-f(x + 2h) 2h 3f(x)-4f(xh) + f(x-2h) 2h (a) Use the two-point forward difference formula to approximate f'(0.1). Compare your actual error with the error bound. (b) Use the most accurate three-point formula to find f'(-0.1), f'(o), and f'(0.2). Also compare your actual errors with the error bounds. (c) For parts (a) and (b), do you observe the actual error is ≤ error bound for each case? Explain why "actual error>error bound" may happen, and how to solve the issue?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:2. f(x) = sin(2x) +x², you are only given the following data
x f(x)
-0.1
0
0.1
0.2
-0.1887
0
0.2087
0.4294
(2-point) Forward-Difference
3-point midpoint
3-point forward
endpoint
3-point backward
endpoint
f'(x) =
f'(x) =
f'(x) =
f'(x) =
f(x +h)-f(x)
h
f(x+h)-f(xh)
2h
-3f(x) + 4f(x+h)-f(x + 2h)
2h
3f (x) - 4f(xh) + f(x-2h)
2h
(a) Use the two-point forward difference formula to approximate ƒ'(0.1). Compare your actual
error with the error bound.
(b) Use the most accurate three-point formula to find ƒ'(-0.1), ƒ'( 0), and ƒ'(0.2). Also
compare your actual errors with the error bounds.
(c) For parts (a) and (b), do you observe the actual error is ≤ error bound for each case?
Explain why "actual error>error bound” may happen, and how to solve the issue?
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