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Question 11

This document provides the steps to: 1) Use Taylor's formula to find the Taylor series of the function f(x)=cos(x)e^x at the center π/2 up to degree 8. 2) Use the Taylor polynomial of order 4 to approximate f(π/2 + π/90) and calculate the maximum error of this approximation. 3) Derive that the maximum error occurs when considering the 5th derivative and involves finding the maximum value of the 5th derivative between π/2 and π/2 + π/90.

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Waqar Mirza
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98 views2 pages

Question 11

This document provides the steps to: 1) Use Taylor's formula to find the Taylor series of the function f(x)=cos(x)e^x at the center π/2 up to degree 8. 2) Use the Taylor polynomial of order 4 to approximate f(π/2 + π/90) and calculate the maximum error of this approximation. 3) Derive that the maximum error occurs when considering the 5th derivative and involves finding the maximum value of the 5th derivative between π/2 and π/2 + π/90.

Uploaded by

Waqar Mirza
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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EXERCISE 11:

Let the function f (x)=cos ⁡x e x .


π
a/ Use Taylor formula to find its Taylor series at center (develop until degree 8 , no
2
need to find the general term).
b/ Use the Taylor Polynomial of order 4 to find an approximation of f ( π2 + 90π ). Use
π /2
e ≈ 4.8105
c/ What is the maximum error done in the approximation of question c/.

a/ To find the Taylor series of the function f(x) = cos⁡x * e^x at the center π/2,
we will expand it up to degree 8.

Step 1: Find the derivatives of f (x) up to the 8 th derivative.

¿ ∧
f (x)=cos ⁡x e x
¿
Step 2: Evaluate the derivatives at x=π /2.
∧ ∧
f (π /2)=cos ⁡(π /2)∗e (π /2)=e (π /2)
' ∧ ∧ ∧
f (π /2)=(cos ⁡(π /2)−sin ⁡(π /2))∗e (π /2)=e (π /2)−e (π /2)=0
¿
Step 3: Write out the Taylor series up to degree 8.
The Taylor series of f (x)=cos ⁡x ¿ e∧ x at the center π /2 is given by:

Simplifying, we have:

b/ To find an approximation of f (π /2+π /90) using the Taylor Polynomial of


order 4 , we substitute x=π /2+ π /90 into the Taylor series obtained in part
(a).

Substituting x=π /2+ π /90 :

¿
c/ To find the maximum error in the approximation obtained in part (b), we
need to consider the error term of the Taylor Polynomial. The error term is
given by:
∧
E(x )=(1/(n+ 1))f (n+1)(c)¿

where f ∧ (n+1)(c ) is the (n+1) th derivative of f (x) evaluated at some point c between
π /2 and π /2+ π /90

In our case, since we used a Taylor Polynomial of order 4, the maximum error occurs
when considering the 5 th derivative.

To find the maximum error, we need to find the maximum value of (cosc + sinc ¿ e ∧ c for
c between π /2 and π /2+ π /90. This can be done by considering the critical points of the
function within that interval.

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