[4] [Centered difference approximation of second derivative] Let fe C4 ([a, b]). (a) Use Taylor's theorem to write ƒ as a third order (cubic) Taylor poly- nomial plus a fourth order (quartic) remainder term. Expand about the point xo. (b) Using the result from (a), evaluate f(x) at the points x = xo+h and x = xo - h; then, add the two results together to derive the centered difference approxi- mation to the second derivative: f(xo+h) — 2ƒ (x0) + ƒ (x − h) h² h² = ƒ” (xo) + ^ (ƒ(¹) (E₁) + ƒ(¹) ({₂)) 4!
[4] [Centered difference approximation of second derivative] Let fe C4 ([a, b]). (a) Use Taylor's theorem to write ƒ as a third order (cubic) Taylor poly- nomial plus a fourth order (quartic) remainder term. Expand about the point xo. (b) Using the result from (a), evaluate f(x) at the points x = xo+h and x = xo - h; then, add the two results together to derive the centered difference approxi- mation to the second derivative: f(xo+h) — 2ƒ (x0) + ƒ (x − h) h² h² = ƒ” (xo) + ^ (ƒ(¹) (E₁) + ƒ(¹) ({₂)) 4!
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.CR: Chapter 4 Review
Problem 5CR: Determine whether each of the following statements is true or false, and explain why. The chain rule...
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Question
Taylor's theorem
![[4] [Centered difference approximation of second derivative]
Let f = C¹([a, b]).
(a) Use Taylor's theorem to write ƒ as a third order (cubic) Taylor poly-
nomial plus a fourth order (quartic) remainder term. Expand about the point
xo.
(b) Using the result from (a), evaluate f(x) at the points
x = xo + h
and
x = xo - h;
then, add the two results together to derive the centered difference approxi-
mation to the second derivative:
f(xo+h)-2f(xo) + f(xo - h)
h²
= ƒ” (To) + h (ƒ(4) ({1) + ƒ(4) ({2))
4!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F3ce4bf62-ec0d-4185-83c2-e200773260f2%2Fcxgmhy_processed.png&w=3840&q=75)
Transcribed Image Text:[4] [Centered difference approximation of second derivative]
Let f = C¹([a, b]).
(a) Use Taylor's theorem to write ƒ as a third order (cubic) Taylor poly-
nomial plus a fourth order (quartic) remainder term. Expand about the point
xo.
(b) Using the result from (a), evaluate f(x) at the points
x = xo + h
and
x = xo - h;
then, add the two results together to derive the centered difference approxi-
mation to the second derivative:
f(xo+h)-2f(xo) + f(xo - h)
h²
= ƒ” (To) + h (ƒ(4) ({1) + ƒ(4) ({2))
4!
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