Approximate e^{-0.75} using 5th degree Taylor Polynomials and determine the maximum error.

arenceabigns

arenceabigns

Answered question

2021-08-31

Approximate e0.75 using 5th degree Taylor Polynomials and determine the maximum error.

Answer & Explanation

Clara Reese

Clara Reese

Skilled2021-09-01Added 120 answers

The provided expression is e0.75
Power series formula is given below:
ex=n=0xnn!=1+x1!+x22!+x33!++xnn!+
The Taylor polynomial of degree m for a function f(x) with power series is given below:
Sm(x)=n=0mxnn!=1+x+x22!++xmm!
Consider the function as f(x)=ex
Evaluate the power series of the above function by replacing x with x-as below:
ex=n=0(1)nxnn!
Substitute m=5 in the Sm(x) and obtain the 5th degree Taylor Polynomials.
S5(x)=n=05(1)nxnn!
=1x+x22!x3!+x44!x55!
=1x+x22x36+x424x5120
Find the approximate value of the 5th degree Taylor polynomial of the exponential function at x=0.75 as follows:
S5(0.75)=10.75+(0.75)22(0.75)36+(0.754)24(0.75)5120
0.47379
Now, obtain the maximum error as follows.
Apply remainder theorem and evaluate R5 at x=0.75
R5=f(5+1)(z)(5+1)!(0.750)5+1
=f(6)(z)(6)!(0.75)6
=(1)

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