Linear systems are so easy to work with that engineers often construct linear models of real (nonlinear) systems to assist in analysis and design. Such models are often surprisingly accurate over a limited range. For example, consider the simple exponential function ex. The Taylor series representation of this function is
(a) Construct a linear model for this function by truncating the Taylor series expansion alter the linear (first-order) term. (b) Evaluate your model function at x = 0.000005,.0.0005,0.05,0.5, and 5.0. (c) For which values of x does your model yield a “reasonable” approximation to ex? Explain your reasoning.
(a)
Construct a linear model for the given function by truncating the Taylor series expansion after the linear (first-order) term.
Answer to Problem 1E
Linear function for the given function is
Explanation of Solution
Given Data:
The Taylor series expansion of the given function is
Calculation:
A linear function is a simple function composed of constant terms and simple variable without exponent.
To make given function linear, remove all terms for which power of
So,
Conclusion:
Thus, linear function for the given function is
(b)
Evaluate linear function for given values of
Answer to Problem 1E
The percentage change in given function for
Explanation of Solution
Given Data:
Values of
Calculation:
The expression for percentage change in given function is as follows.
Here,
For
Substitute
Substitute
So percentage change in given function for
For
Substitute
Substitute
So percentage change in given function for
For
Substitute
Substitute
So percentage change in given function for
For
Substitute
Substitute
So percentage change in given function for
For
Substitute
Substitute
So percentage change in given function for
Conclusion:
Thus, the percentage change in given function for
(c)
For which values of
Answer to Problem 1E
The values for which the approximation yields a reasonable result is
Explanation of Solution
Calculation:
For
is
As for
Conclusion:
Thus, the values for which the approximation yields a reasonable result is
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