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The following series can be used to approximate
(a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion
(b) Use the Taylor series to estimate
(a)
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To prove: The Maclaurin series
Explanation of Solution
Given Information:
The Maclaurin series of exponential function
Formula Used:
The Taylor series expansion
Proof:
Let the function
For this function,
For all integer n.
Now apply this in the Taylor series expansion to obtain:
Now replace
Note that the remainder term
This gives,
Hence, proved that the provided Maclaurin series is an approximation of
(b)
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To calculate: The approximate value of
Answer to Problem 1P
Solution:
The zero first, second and third order Taylor series approximations for the function
Explanation of Solution
Given Information:
The function
Formula Used:
The Taylor series approximation of
Calculation:
Consider the zero-order approximation for the provided function,
Now replace 1 for
The zero order approximation gives 0.81873.
The exact value of the function at 1 would be:
Thus,
The relative percentage error at this stage is 122.55%.
The first-order Taylor series approximation would be:
Now replace 1 for
The first order approximation gives 0.16375.
Thus,
The relative percentage error at this stage is 55.49%.
The second-order Taylor series approximation would be:
Now replace 1 for
The second order approximation gives 0.42574.
Thus,
The relative percentage error at this stage is 15.73%.
The third-order Taylor series approximation would be:
Now replace 1 for
The third order approximation gives 0.35587.
Thus,
The relative percentage error at this stage is
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Chapter 4 Solutions
Numerical Methods for Engineers
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