Question

End behavior for transcendental functions. 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 analyzing 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

Short Answer: The hyperbolic cosine function, \(\cosh x = \frac{e^{x} + e^{-x}}{2}\), has an end behavior where it approaches ∞ as x approaches both ∞ and -∞. The function has a minimum value of 1 at x = 0 and exhibits even symmetry about the y-axis. The graph can be sketched with these properties in mind.

Step-by-step solution

a. Analyzing the end behavior of the hyperbolic cosine function

To analyze the end behavior of the function, we need to find its limits as x approaches ∞ and -∞. We have: • \(\lim_{x \to \infty} \cosh x = \lim_{x \to \infty} \frac{e^{x} + e^{-x}}{2}\) • \(\lim_{x \to -\infty} \cosh x = \lim_{x \to -\infty} \frac{e^{x} + e^{-x}}{2}\) Let's find these limits separately.

Finding \(\lim_{x \to \infty} \cosh x\)

To find the limit as x approaches ∞, we can analyze the terms \(e^x\) and \(e^{-x}\) separately: • As x → ∞, \(e^x\) is growing exponentially larger. • As x → ∞, \(e^{-x}\) is approaching 0. So, the function becomes: \(\lim_{x \to \infty} \cosh x = \lim_{x \to \infty} \frac{e^{x} + e^{-x}}{2} ≈ \lim_{x \to \infty} \frac{e^{x}}{2}\) Since \(e^x\) is growing exponentially larger as x → ∞: \(\lim_{x \to \infty} \cosh x = \infty\)

Finding \(\lim_{x \to -\infty} \cosh x\)

Similarly, for the limit as x approaches -∞, we analyze \(e^x\) and \(e^{-x}\): • As x → -∞, \(e^x\) is approaching 0. • As x → -∞, \(e^{-x}\) is growing exponentially larger. So, the function becomes: \(\lim_{x \to -\infty} \cosh x = \lim_{x \to -\infty} \frac{e^{x} + e^{-x}}{2} ≈ \lim_{x \to -\infty} \frac{e^{-x}}{2}\) Since \(e^{-x}\) is growing exponentially larger as x → -∞: \(\lim_{x \to -\infty} \cosh x = \infty\)

b. Evaluating cosh(0) and sketching the graph

Now, let's evaluate \(\cosh 0\): \(\cosh 0 = \frac{e^{0} + e^{-0}}{2} = \frac{1 + 1}{2} = 1\) We know that the function has even symmetry, as it satisfies the condition \(\cosh(-x) = \cosh(x)\): \(\cosh(-x) = \frac{e^{-x} + e^{x}}{2} = \cosh x\) From this information and the end behavior, we can sketch a graph of \(y = \cosh x\): 1. The function has a minimum value of 1 at x = 0. 2. The function is symmetric with respect to the y-axis (even symmetry). 3. As x approaches ∞ or -∞, the function also approaches ∞. Using these characteristics, we can sketch a plausible graph of \(y = \cosh x\).

Key concepts

Hyperbolic Cosine Function

The hyperbolic cosine function, denoted as \(\cosh x\), is a type of transcendental function that has important applications in various fields such as physics, engineering, and mathematics. This function is defined by the equation \(\cosh x = \frac{e^{x} + e^{-x}}{2}\), where \(e\) represents the base of the natural logarithm.

The hyperbolic cosine function models the shape of a hanging cable, like a telephone wire, which is a visual example of the function's practical application. Another notable feature of this function is its symmetry; specifically, \(\cosh x\) exhibits even symmetry, meaning that \(\cosh(-x) = \cosh(x)\), which results in the graph being mirrored across the y-axis.

A key property of the hyperbolic cosine function is its behavior as \(x\) approaches infinity or negative infinity. Unlike trigonometric functions, which oscillate, \(\cosh x\) increases exponentially as \(x\) grows large in either the positive or negative direction. Understanding this end behavior is crucial for graphing the hyperbolic cosine function accurately.

Limits of Transcendental Functions

Transcendental functions, such as the hyperbolic cosine function, often require the understanding of limits for proper analysis and graph sketching.

The limit of a function as \(x\) approaches a certain value is a foundational concept in calculus. It describes the value that the function approaches as the input \(x\) gets infinitely close to that value. When dealing with transcendental functions, the limits as \(x\) approaches infinity (\(\infty\)) or negative infinity (\( -\infty\)) describe the function's end behavior.

Specifically, in the case of \(\cosh x\), the limits are both infinite, meaning the function grows without bound as \(x\) approaches either positive or negative infinity. This growth is a direct result of the exponential components within the hyperbolic cosine function. When approaching positive infinity, the \(e^x\) term dominates, and when approaching negative infinity, the \(e^{-x}\) term does. The understanding of these limits is key to understanding the nature of transcendent functions and their graphical representations.

Graph Sketching

Graph sketching is an important skill in mathematics that aids in visualizing the properties of functions, including transcendental ones such as \(\cosh x\).

In order to sketch the graph of \(y = \cosh x\), one should start by identifying critical points, including intercepts, maxima, minima, and points of inflection. In this function's case, the graph passes through the point (0, 1), which is the minimum value due to the function's even symmetry.

Furthermore, graph sketching involves understanding the end behavior, which, for \(\cosh x\), means acknowledging that the graph ascends towards infinity in both directions along the x-axis. A symmetry check is also a must – for \(\cosh x\), any value and its negative counterpart will yield the same result, leading to a mirrored effect across the y-axis.

By considering these factors, along with evaluating a few key values, one can draw a reasonably accurate graph of the hyperbolic cosine function, which resembles the shape of a hanging cable, gradually rising away from the x-axis as \(x\) moves away from zero in both the positive and negative directions.