20. Let z = (xo +x1)/2, h = (x, – xo)/2. To analyze the approximation (4.19), con- sider the error %3D = flxo, x1] – f' ( **): f(z +h) – f(z – h) – f'(2) 2 2h Expand f(z + h) and f(z – h) about z by using Taylor's theorem from Section 1.2, and include terms of degree <3 in the variable h. Use these expansions to show that h? G "(2) for small values of h. Thus, (4.19) is a good approximation when x1 - xo is relatively small.
20. Let z = (xo +x1)/2, h = (x, – xo)/2. To analyze the approximation (4.19), con- sider the error %3D = flxo, x1] – f' ( **): f(z +h) – f(z – h) – f'(2) 2 2h Expand f(z + h) and f(z – h) about z by using Taylor's theorem from Section 1.2, and include terms of degree <3 in the variable h. Use these expansions to show that h? G "(2) for small values of h. Thus, (4.19) is a good approximation when x1 - xo is relatively small.
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
![20. Let z = (xo+x1)/2, h = (x1 – xo)/2. To analyze the approximation (4.19), con-
sider the error
%3D
E = f(xo, x1] – f' (*) =
Xo + x1
f(z+h) – f(z – h)
– f'(2)
2
2h
Expand f(z + h) and f(z – h) about z by using Taylor's theorem from Section
1.2, and include terms of degree <3 in the variable h. Use these expansions to
show that
h?
E - "(2)
for small values of h. Thus, (4.19) is a good approximation when x1 - xo is
relatively small.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c87fee3-31eb-4201-90fd-3f41f32ea6e1%2F1b344c12-3cf1-461f-96fd-596a6c9830d4%2Fka5umuj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:20. Let z = (xo+x1)/2, h = (x1 – xo)/2. To analyze the approximation (4.19), con-
sider the error
%3D
E = f(xo, x1] – f' (*) =
Xo + x1
f(z+h) – f(z – h)
– f'(2)
2
2h
Expand f(z + h) and f(z – h) about z by using Taylor's theorem from Section
1.2, and include terms of degree <3 in the variable h. Use these expansions to
show that
h?
E - "(2)
for small values of h. Thus, (4.19) is a good approximation when x1 - xo is
relatively small.
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