Theoretical assignment Consider hyperbolic tangent function exp(x) – exp(-æ) exp(x)+ exp(-æ) tanh(x) = where a e [-2, 2]. Use 5 nodes to uniformly cut the interval [–2, 2] into 4 sub-intervals.
Theoretical assignment Consider hyperbolic tangent function exp(x) – exp(-æ) exp(x)+ exp(-æ) tanh(x) = where a e [-2, 2]. Use 5 nodes to uniformly cut the interval [–2, 2] into 4 sub-intervals.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Based on these 5 nodes, use linear spline function to approximate f(x) =
tanh(x), x ∈ [−2, 2] . The specific liner splines for each subintervals are required.
![**Theoretical Assignment**
Consider the hyperbolic tangent function:
\[
\tanh(x) = \frac{\exp(x) - \exp(-x)}{\exp(x) + \exp(-x)}
\]
where \( x \in [-2, 2] \). Use 5 nodes to uniformly cut the interval \([-2, 2]\) into 4 sub-intervals.
### Explanation:
The hyperbolic tangent function, \(\tanh(x)\), is defined as the difference between the exponential of \(x\) and the exponential of \(-x\), divided by their sum. This function is often used in mathematical modeling and neural networks due to its properties.
The task involves dividing the interval from \(-2\) to \(2\) into 4 equal parts using 5 nodes, which means identifying points at equal distances along this interval to delineate the sub-intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F015fa499-4229-4065-9077-9d1ac4317e71%2F802bebab-e7b5-456d-bac7-c1f183c518d2%2Fjvo69wq_processed.png&w=3840&q=75)
Transcribed Image Text:**Theoretical Assignment**
Consider the hyperbolic tangent function:
\[
\tanh(x) = \frac{\exp(x) - \exp(-x)}{\exp(x) + \exp(-x)}
\]
where \( x \in [-2, 2] \). Use 5 nodes to uniformly cut the interval \([-2, 2]\) into 4 sub-intervals.
### Explanation:
The hyperbolic tangent function, \(\tanh(x)\), is defined as the difference between the exponential of \(x\) and the exponential of \(-x\), divided by their sum. This function is often used in mathematical modeling and neural networks due to its properties.
The task involves dividing the interval from \(-2\) to \(2\) into 4 equal parts using 5 nodes, which means identifying points at equal distances along this interval to delineate the sub-intervals.
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