Question

The hyperbolic cosine function, denoted cosh \(x\), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as \(\cosh x=\frac{e^{x}+e^{-x}}{2}\). a. Determine its end behavior by evaluating \(\lim \cosh x\) and \(\lim _{x \rightarrow-\infty} \cosh x\). b. Evaluate cosh 0. Use symmetry and part (a) to sketch a plausible graph for \(y=\cosh x\).

Short answer

Question: Determine the end behavior of the function \(y=\cosh x\) and sketch its graph. Solution: The end behavior of \(y=\cosh x\) is as follows: 1. As \(x\rightarrow \infty\), \(\cosh x \rightarrow \infty\). 2. As \(x\rightarrow -\infty\), \(\cosh x \rightarrow \infty\). The graph of \(y=\cosh x\) is symmetric about the y-axis, passes through the point \((0,1)\), and has branches extending upward to the right and left along the positive and negative x-axes as \(x\) goes to \(\pm \infty\). The graph resembles an upward-facing parabola opening upwards but is slightly flatter in the middle.

Step-by-step solution

Determine the end behavior of \(\cosh x\)

To determine the end behavior of \(\cosh x\), we need to evaluate the limit as \(x\rightarrow \infty\) and \(x\rightarrow -\infty\). By the definition of \(\cosh x\), we have: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\) As \(x\rightarrow \infty\), \(e^{x}\rightarrow \infty\) and \(e^{-x}\rightarrow 0\). Thus, \(\lim_{x\rightarrow \infty}\cosh x = \lim_{x\rightarrow \infty}\left( \frac{e^{x} + e^{-x}}{2}\right) = \left( \frac{\infty + 0}{2}\right) = \infty\) As \(x\rightarrow -\infty\), \(e^{x}\rightarrow 0\) and \(e^{-x}\rightarrow \infty\). Thus, \(\lim_{x\rightarrow -\infty}\cosh x = \lim_{x\rightarrow -\infty}\left( \frac{e^{x} + e^{-x}}{2}\right) = \left( \frac{0 + \infty}{2}\right) = \infty\)

Evaluate \(\cosh 0\)

We can use the definition of \(\cosh x\) to evaluate \(\cosh 0\): \(\cosh 0 = \frac{e^{0} + e^{-0}}{2} = \frac{1 + 1}{2} = 1\)

Sketch a plausible graph for \(y=\cosh x\)

Based on our findings in Steps 1 and 2, we know that: 1. As \(x\rightarrow \infty\) and \(x\rightarrow -\infty\), \(\cosh x \rightarrow \infty\). 2. \(\cosh 0 = 1\) Additionally, \(\cosh x\) has an even symmetry, since \(\cosh (-x) = \cosh x\). Taking into account the end behavior, symmetry, and the value of \(\cosh 0\), we can sketch a plausible graph for \(y=\cosh x\): Start at the point \((0,1)\) and draw two symmetric branches extending upwards to the right and left along the positive and negative x-axes, approaching \(\infty\) as \(x\) goes to \(\pm \infty\). The graph should look like an upward-facing parabola opening upwards, but slightly flatter in the middle.

Key concepts

End Behavior of Hyperbolic Functions

To understand the end behavior of the hyperbolic cosine function, \(\cosh x\), we analyze how the function behaves as \(x\) approaches positive and negative infinity. The formula for \(\cosh x\) is given by:

• \(\cosh x = \frac{e^{x} + e^{-x}}{2}\)

As \(x \) tends towards infinity:

• \(e^{x} \to \infty\), which dominates \(e^{-x} \to 0\).
• Thus, \(\lim_{x \to \infty} \cosh x = \frac{\infty + 0}{2} = \infty\).

Conversely, as \(x \) approaches negative infinity:

• \(e^{x} \to 0\), while \(e^{-x} \to \infty\).
• Therefore, \(\lim_{x \to -\infty} \cosh x = \frac{0 + \infty}{2} = \infty\).

In summary, for both extreme ends of the x-axis, the end behavior of \(\cosh x\) heads towards infinity.

Symmetry of Hyperbolic Cosine

Symmetry in mathematics helps us determine how a graph reflects over the x or y-axis. For the function \(\cosh x\), it is important to identify any symmetry which is crucial in graph sketching.

• \(\cosh x = \cosh (-x)\)
• This property indicates even symmetry.

Even functions, like \(\cosh x\), are symmetric about the y-axis. This means whatever the graph does on one side of the y-axis, it mirrors on the opposite side. Observing this can help simplify graphing because you only need to analyze the function on one side and reflect it to the other.

Limits of Hyperbolic Functions

In calculus, limits allow us to understand the behavior of a function as it approaches a specific input value. Evaluating limits for hyperbolic functions like \(\cosh x\) provides significant insight.

• At \(x = 0\):\[ \cosh 0 = \frac{e^{0} + e^{-0}}{2} = 1 \]

Additionally, given the behavior as \(x\) goes to infinity (both directions):

• The limit as \(x\to \infty\) or \(x\to -\infty\) is \(\infty\).

These limits are foundational when predicting the graph of \(\cosh x\) and understanding its long-range behavior.

Graphing Hyperbolic Functions

Graphs illustrate how a function behaves visually. For \(\cosh x\), starting with the value at \(x = 0\) helps:

• At this point, the function value is 1.
• Combine this with even symmetry and end behavior.

The graph starts at \( (0, 1) \) and reflects symmetrically across the y-axis. As \(x\) moves towards positive or negative infinity:

• \(\cosh x\) grows infinitely large, with branches curving upwards.

Imagining the shape: It's somewhat like a parabola opening upwards, flatter around the center near the y-axis. The clarity on symmetry and end behavior ensures accurate graph depiction.