To compute the Taylor series of f(x) = sin(x) about x =, we first need the values f(3) = 1 f"() = 0 f(¹) () = 1 f(6) () = -1 noticing that the odd-order derivatives f(2k+1) () = [ for k= 0, 1, 2. The Taylor expansion of f about x = to n = 7 terms is thus f(x) = If 0≤x≤ then since |f (8) (a) on [0,] is O increasing O decreasing the smallest upper estimate one can obtain from this analysis for f(8) (x) is We can use these results to obtain the estimate +R7. for sin(1). The least upper bound on the error in this approximation, based on the above bound, is given by |R7| ≤
To compute the Taylor series of f(x) = sin(x) about x =, we first need the values f(3) = 1 f"() = 0 f(¹) () = 1 f(6) () = -1 noticing that the odd-order derivatives f(2k+1) () = [ for k= 0, 1, 2. The Taylor expansion of f about x = to n = 7 terms is thus f(x) = If 0≤x≤ then since |f (8) (a) on [0,] is O increasing O decreasing the smallest upper estimate one can obtain from this analysis for f(8) (x) is We can use these results to obtain the estimate +R7. for sin(1). The least upper bound on the error in this approximation, based on the above bound, is given by |R7| ≤
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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![To compute the Taylor series of f(x) = sin(x) about = we first need the values
f() = 1
f" () = 0
f(¹) () = [1
f) () = -1
noticing that the odd-order derivatives
f(2k+1) () = [
for k = 0, 1, 2.
The Taylor expansion of f about x = to n = 7 terms is thus
f(x) =
If 0≤x≤ then since |f (8)(a) on [0,] is
O increasing
O decreasing
the smallest upper estimate one can obtain from this analysis for
f (8) (x) is
We can use these results to obtain the estimate
+R7.
for
sin(1). The least upper bound on the error in this approximation, based on the above
bound, is given by
|R7| ≤](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda0442f0-082c-40d3-95ab-8cd5dccad2c3%2F31a8e50f-eb6b-4992-9918-15beeeab3eec%2Fofn02c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To compute the Taylor series of f(x) = sin(x) about = we first need the values
f() = 1
f" () = 0
f(¹) () = [1
f) () = -1
noticing that the odd-order derivatives
f(2k+1) () = [
for k = 0, 1, 2.
The Taylor expansion of f about x = to n = 7 terms is thus
f(x) =
If 0≤x≤ then since |f (8)(a) on [0,] is
O increasing
O decreasing
the smallest upper estimate one can obtain from this analysis for
f (8) (x) is
We can use these results to obtain the estimate
+R7.
for
sin(1). The least upper bound on the error in this approximation, based on the above
bound, is given by
|R7| ≤
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