In the following, assume f: R → R has 3 continuous derivatives. A common usage of Taylor's theorem is to construct finite difference schemes for approximating the derivatives of f. Let's take a look at some basic problems in this vein: (a) Let x ER and h> 0. Show that there exists constants M₁, M₂, possibly depend- ing on x, h, and f and its derivatives, such that M₂h ≤ (1₁ (x) — 1(x + h) - S(x)) : h This shows the "first-order accuracy" of the forward-difference approximation. (Hint: Use Taylor's theorem for x,x + h) 0. Show that there exists constants M3, M4, possibly depend- ing on x, h, and f and its derivatives, such that Mgh² ≤ (f'(x) — f(x + h) — ƒ (x − h) 2h - h |ƒ′(z) – f(x + h) − f (x) | |f² (2) _ f(x + h) = f(x − h) - | 2h This shows the "second-order accuracy" of the centered-difference approximation. (Hint: Use Taylor's theorem for x,x+h and x, x-h) (c) Letk € R, and let f(x) = sin(kx). Show that for all x ER and h> 0, 3/2. This might happen if your function f has "high-frequency noise".

Given : f : ℝ→ℝ has 3 continuous derivatives(a) Let x∈ℝ and h>0. Using Taylor series expansion for fx+h, we have fx+h=fx+h f'x+h22!f''ξ, x<ξ<x+h⇒fx+h-fx=h f'x+h22!f''ξ⇒fx+h-fxh=f'x+h2!f''ξ⇒fx+h-fxh-f'x=h2!f''ξ Thus, the error in the finite difference approximation can be bounded by a multiple of the step size. This means fx+h-fxh-f'x≤Mh or f'x-fx+h-fxh≤Mh where M=max 12f''ξ Since h>0, we get f'x-fx+h-fxh≤M h ------ (A) ⇒M1h≤f'x-fx+h-fxh≤M2h where M1=-max 12f''ξ, M2=max 12f''ξ(b) Let x∈ℝ and h>0. Using Taylor series expansions for fx+h and fx-h, we have fx+h=fx+h f'x+h22!f''x+h33!f'''ξ1, x<ξ1<x+h 1fx-h=fx-h f'x+h22!f''x-h33!f'''ξ2, x-h<ξ2<x 2 Subtracting (2) from (1), we get fx+h-fx-h=2h f'x+h33!f'''ξ1+f'''ξ2⇒fx+h-fx-h2h=f'x+h22·3!f'''ξ1+f'''ξ2⇒fx+h-fx-h2h-f'x=h212f'''ξ1+f'''ξ2 Thus, the error in the centered finite difference approximation to second derivative can be bounded by a multiple of the step size. This means fx+h-fx-h2h-f'x≤Mh2 or f'x-fx+h-fx-h2h≤M h2 where M=max 112f'''ξ1+f'''ξ2. Thus, we have f'x-fx+h-fx-h2h≤M h2 ------------ (B) ⇒M3h2≤f'x-fx+h-fx-h2h≤M4h2 where M3=-max 12f'''ξ1+f'''ξ2, M4=max 12f'''ξ1+f'''ξ2(c) Given : k∈ℝ and fx :=sinkx To show : for all x∈ℝ, h>0 f'x-fx+h-fxh≤k2h2 and f'x-fx+h-fx-h2h≤k3h23 Proof : From part (a), we have equation (A) as f'x-fx+h-fxh≤M h where M=max 12f''ξ. We know that fx=sinkxf'x=k coskxf''x=-k2sinkx and also sinkx≤1 ⇒-k2sinkx≤k2 Therefore, we get M=max 12f''ξ=max 12-k2sinkξ=k22 Thus, we have f'x-fx+h-fxh≤k2h2From part (b), we have equation (B) as f'x-fx+h-fx-h2h≤M h2 where M=max 112f'''ξ1+f'''ξ2. We have f'''x=-k3coskx and coskx≤1 ⇒ -k3coskx≤k3 So, we have M=max 112f'''ξ1+f'''ξ2=max 112k3+k3=max 1122k3=max 16k3≤k33 Thus, we have f'x-fx+h-fx-h2h≤k3h23

Answered: In the following, assume f: R → R has 3 continuous derivatives. A common usage of Taylor's theorem is to construct finite difference schemes for approximating…

Advanced Engineering Mathematics

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ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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3. In the following, assume \( f : \mathbb{R} \rightarrow \mathbb{R} \) has 3 continuous derivatives. A common usage of Taylor’s theorem is to construct finite difference schemes for approximating the derivatives of \( f \). Let’s take a look at some basic problems in this vein:

(a) Let \( x \in \mathbb{R} \) and \( h > 0 \). Show that there exist constants \( M_1, M_2 \), possibly depending on \( x, h, \) and \( f \) and its derivatives, such that

\[
M_1 h \leq \left( f'(x) - \frac{f(x+h) - f(x)}{h} \right) \leq M_2 h
\]

This shows the “first-order accuracy” of the forward-difference approximation.

*(Hint: Use Taylor’s theorem for \( x, x+h \))*

(b) Let \( x \in \mathbb{R} \) and \( h > 0 \). Show that there exist constants \( M_3, M_4 \), possibly depending on \( x, h, \) and \( f \) and its derivatives, such that

\[
M_3 h^2 \leq \left( f'(x) - \frac{f(x+h) - f(x-h)}{2h} \right) \leq M_4 h^2
\]

This shows the “second-order accuracy” of the centered-difference approximation.

*(Hint: Use Taylor’s theorem for \( x, x+h \) and \( x, x-h \))*

(c) Let \( k \in \mathbb{R} \), and let \( f(x) := \sin(kx) \). Show that for all \( x \in \mathbb{R} \) and \( h > 0 \),

\[
\left| f'(x) - \frac{f(x+h) - f(x)}{h} \right| \leq \frac{k^2 h}{2}
\]

\[
\left| f'(x) - \frac{f(x+h) - f(x-h)}{2h} \right| \leq \frac{k^3 h^2}{3}
\]

**Remark:** Problem (c

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