1. a) Use commonly known Taylor series to find the Taylor series of f(x) = cos(x²) %3D centered at a = 0. Show how you obtained the Taylor series and express your answer in summation notation

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
icon
Related questions
Question
1. a) Use commonly known Taylor series to find the Taylor series of f(x) = cos(x²)
centered at a =
0. Show how you obtained the Taylor series and express your answer
in summation notation
b) Use the Taylor series found in part a) to represent the integral cos(x²)dx as an
infinite series.
express the series in summation notation.
c) Approximate the value of the integral with an error less than 10-5. Justify that
your approximation satisfies the error bound.
Transcribed Image Text:1. a) Use commonly known Taylor series to find the Taylor series of f(x) = cos(x²) centered at a = 0. Show how you obtained the Taylor series and express your answer in summation notation b) Use the Taylor series found in part a) to represent the integral cos(x²)dx as an infinite series. express the series in summation notation. c) Approximate the value of the integral with an error less than 10-5. Justify that your approximation satisfies the error bound.
Expert Solution
Step 1

part (a)

given

fx=cosx2

to find 

taylor series of fx=cosx2 centered at a=0

solution

as we know

taylor series of function fx centered at a is defined as 

fx=fa+f'a1!x-a+f''a2!x-a2+f'''a3!x-a3+f''''a4!x-a4+.....

according to question

a=0

series becomes

fx=f0+f'01!x+f''02!x2+f'''03!x3+f''''04!x4+.....      ...i

Step 2

now as fx=cosx2

put x=0

f0=cos0=1

differentiate fx with respect to x

f'x=-sinx2×2xf'0=-sin0×0          =0

differentiate f'x again with respect to x

we will use product rule

f''x=-2sinx2+xcosx2×2xf''0=-2sin0+0cos0×0           =0

steps

Step by step

Solved in 4 steps

Blurred answer
Knowledge Booster
Power Series
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage