[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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