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Chapter 16: Problem 7

Sketch graphs of (a) \(y=\tanh x\) and (b) \(y=\operatorname{coth} x\) for values of \(x\) between \(-3\) and 3

Short Answer

Expert verified
\(y = \tanh x\) is S-shaped, \(y = \operatorname{coth} x\) has vertical asymptote at \(x=0\) with branches declining towards \(1\) for \(x>0\) and \(-1\) for \(x<0\).

Step by step solution

01

Understanding the Functions

The function \(y = \tanh x\) is the hyperbolic tangent function. It is defined as \(\frac{e^x - e^{-x}}{e^x + e^{-x}}\). The function \(y = \operatorname{coth} x\) is the hyperbolic cotangent function, defined as \(\frac{e^x + e^{-x}}{e^x - e^{-x}}\) with a vertical asymptote at \(x=0\).
02

Analyze the Behavior of \(y = \tanh x\)

The function \(y = \tanh x\) is an odd function that is continuous and differentiable everywhere. As \(x\to \infty\), \(\tanh x\) approaches 1, and as \(x\to -\infty\), \(\tanh x\) approaches -1. The function is symmetric around the origin.
03

Analyze the Behavior of \(y = \operatorname{coth} x\)

The function \(y = \operatorname{coth} x\) is defined for \(xeq 0\) with a vertical asymptote at \(x=0\). For \(x>0\), \(\operatorname{coth} x\) approaches 1 as \(x\to \infty\), and for \(x<0\), \(\operatorname{coth} x\) approaches -1 as \(x\to -\infty\).
04

Sketch the Graph of \(y = \tanh x\)

Plot a curve that starts from -1 as \(x\to -3\), passes through the origin, and approaches 1 as \(x\to 3\). The graph is S-shaped and increases steadily but slowly at larger \(|x|\) values.
05

Sketch the Graph of \(y = \operatorname{coth} x\)

Plot two curves indicating the hyperbolic cotangent. For \(x>0\), the curve starts from a value just above 1 at \(x = 0^+\) and declines towards 1 as \(x\to 3\). For \(x<0\), it starts at a large negative value and increases towards -1 as \(x\to -3\). The graph is undefined at \(x=0\) due to a vertical asymptote.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding \( \tanh x \)
The hyperbolic tangent function, denoted as \( \tanh x \), is an important hyperbolic function that models the relationship between exponential functions in a unique way. The formula is given by:
  • \( \tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)
This function is defined for all values of \( x \), and its behavior is particularly notable because:
  • It is an odd function, which means it is symmetric about the origin.
  • For any real number \( x \), \( \tanh x \) gives a value between -1 and 1.
As \( x \) approaches infinity, the output of \( \tanh x \) gets closer to 1, and as \( x \) approaches negative infinity, it gets closer to -1. The graph of \( \tanh x \) is smooth and S-shaped. Often, this characteristic curve is utilized in fields like neural networks and signal processing.
Exploring \( \coth x \)
The hyperbolic cotangent function, \( \coth x \), is another intriguing function defined as:
  • \( \coth x = \frac{e^x + e^{-x}}{e^x - e^{-x}} \)
This function exhibits some interesting properties:
  • It is defined everywhere except at \( x = 0 \), where it has a vertical asymptote.
  • For positive \( x \), \( \coth x \) approaches 1 as \( x \) increases, and for negative \( x \), it approaches -1 as \( x \) decreases.
  • Above or below 1 for most parts of its domain, depending on the direction of \( x \).
The hyperbolic cotangent is exceptional because it highlights how functions can behave starkly differently around asymptotes. Understanding \( \coth x \) is essential in solving differential equations and analyzing inverse hyperbolic functions.
Graphing Hyperbolic Functions
The graphical representation of hyperbolic functions like \( \tanh x \) and \( \coth x \) reveals their distinct behaviors and trends. Visualizing these functions on a graph provides insight into:
  • **Shapes and Symmetry**: The \( \tanh x \) graph is S-shaped and passes through the origin, displaying symmetry about the y-axis. Meanwhile, \( \coth x \) shows two separate branches due to its asymptote at \( x = 0 \).
  • **Range and Limit Behavior**: \( \tanh x \) remains bounded between -1 and 1 for all \( x \), indicating a limiting behavior as \( |x| \) becomes large. Conversely, \( \coth x \) diverges as it approaches the asymptote, with its values extending beyond the range of \( \tanh x \).
  • **Smoothness and Continuity**: \( \tanh x \) is continuous everywhere, while \( \coth x \) is continuous except where it speaks with infinity at \( x = 0 \).
Graphing these functions is a practical step to grasp the comparative continuity, asymptotic behavior, and amplitude limits across the real number line. Whether you're sketching by hand or using a computer, understanding these graphs deepens comprehension of hyperbolic properties.

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