Use zero- through fourth-order Taylor series expansions to predict
To calculate: The approximate value of
Answer to Problem 6P
Solution:
The zero first, second, third and fourth order Taylor series approximations for the function
The result shows that theTaylor series approximation could be generalised as,
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 2.5 for
The zero order approximation gives 0.
The exact value of the function at 2.5 would be:
Thus, the true relative percentage error would be:
The relative percentage error at this stage is 100%.
The first-order Taylor series approximation would be:
Now replace 2.5 for
The first order approximation gives 1.5.
Thus,
The relative percentage error at this stage is 38.91%.
The second-order Taylor series approximation would be:
Now replace 2.5 for
The second order approximation gives 0.375.
Thus,
The relative percentage error at this stage is 59.07%.
The third-order Taylor series approximation would be:
Now replace 2.5 for
The third order approximation gives 1.5.
Thus,
The relative percentage error at this stage is 63.7%
The fourth-order Taylor series approximation would be:
Now replace 2.5 for
The third order approximation gives 0.2344.
Thus,
The relative percentage error at this stage is 74.42%.
This implies that the Taylor series approximation could be generalised as,
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Chapter 4 Solutions
Numerical Methods For Engineers, 7 Ed
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