Exercises: 1. Let f(x)=sin(x) and apply theorem(4.1) b.Show that if x| <1 then the approximation sin(x)× x 3! 5! 7! 9! 1 has the error bound! |E9(x)| < · < 2.755 74 × 10-7. 10! c.Use Xo and find P5(x), which involves powers of (x-÷). Theorem(4.1): (Taylor Polynomial Approximation) Assume that feCN+*[a, b] and Xo € [a, b]: is a fixed value.x e [a, b], then f(x)=PM(x)+Ex(x) (4.1) where PMx) is a polynomial that can be used to approximate f(x): f(R) (xo) (x – Xo)* f(x)~ Py(x) = Ex=o° (4.2) %3D k! The error term EMx) has the form: Exx)=fN*"(c) (x – xo)N+1 (N+1)! (4.3) for some value c=c(x) that lies between x and xo.
Exercises: 1. Let f(x)=sin(x) and apply theorem(4.1) b.Show that if x| <1 then the approximation sin(x)× x 3! 5! 7! 9! 1 has the error bound! |E9(x)| < · < 2.755 74 × 10-7. 10! c.Use Xo and find P5(x), which involves powers of (x-÷). Theorem(4.1): (Taylor Polynomial Approximation) Assume that feCN+*[a, b] and Xo € [a, b]: is a fixed value.x e [a, b], then f(x)=PM(x)+Ex(x) (4.1) where PMx) is a polynomial that can be used to approximate f(x): f(R) (xo) (x – Xo)* f(x)~ Py(x) = Ex=o° (4.2) %3D k! The error term EMx) has the form: Exx)=fN*"(c) (x – xo)N+1 (N+1)! (4.3) for some value c=c(x) that lies between x and xo.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Exercises:
1. Let f(x)=sin(x) and apply theorem(4.1)
b.Show that if |x| <1 then the approximation
sin(x)~ x
3!
5!
7!
9!
has the error bound'|E9(x)| < S 2.755 74 × 10-7.
c.Use xo
and find P5(x), which involves powers of (x--).
Theorem(4.1): (Taylor Polynomial Approximation)
Assume that
feCN+[a,b]
and Xo E [a, b]: is a fixed value.x E [a, b]. then
f(x)=PM(x)+Ex(x)
(4.1)
where P(x) is a polynomial that can be used to approximate f(x):
f(x)* Py(x) =D Σ¥=o
flR) (xo) (x – xo)*
(4.2)
k!
The error term Ex) has the form:
p(N+2)(C) (x – xo)N+1
Ex(x)=
(4.3)
(N+1)!
for some value c=c(x) that lies between x and xo.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5e7886c-d02b-47df-8394-801f13e65eab%2F14104131-47c2-4fd1-9b84-ea834e510a86%2Fqpmnir_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercises:
1. Let f(x)=sin(x) and apply theorem(4.1)
b.Show that if |x| <1 then the approximation
sin(x)~ x
3!
5!
7!
9!
has the error bound'|E9(x)| < S 2.755 74 × 10-7.
c.Use xo
and find P5(x), which involves powers of (x--).
Theorem(4.1): (Taylor Polynomial Approximation)
Assume that
feCN+[a,b]
and Xo E [a, b]: is a fixed value.x E [a, b]. then
f(x)=PM(x)+Ex(x)
(4.1)
where P(x) is a polynomial that can be used to approximate f(x):
f(x)* Py(x) =D Σ¥=o
flR) (xo) (x – xo)*
(4.2)
k!
The error term Ex) has the form:
p(N+2)(C) (x – xo)N+1
Ex(x)=
(4.3)
(N+1)!
for some value c=c(x) that lies between x and xo.
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