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 analyzing \(\lim _{x \rightarrow \infty} \cosh x\) 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

In this exercise, we determined the end behavior of the hyperbolic cosine function by analyzing its limits as x approaches positive and negative infinity. We found that the function approaches infinity in both cases. We also evaluated the hyperbolic cosine function at x = 0, obtaining a value of 1. Based on this information and the function's symmetry, we sketched a graph of the hyperbolic cosine function, which resulted in a symmetric 'U' shape opening upwards with its lowest point at (0, 1) and asymptotically going to infinity as x goes to positive and negative infinity.

Step-by-step solution

(Understand the Hyperbolic Cosine Function)

The hyperbolic cosine function, denoted as \(\cosh x\), is defined as: \(\cosh x =\frac{e^{x}+e^{-x}}{2}\) We will be analyzing its end behavior, evaluating it at 0, and sketching its graph. #Step 2: Limit as x approaches positive infinity#

(Find limit as x approaches positive infinity)

To find the limit as x approaches positive infinity, we will analyze the function \(\cosh x= \frac{e^x + e^{-x}}{2}\) as \(x \rightarrow \infty\): \(\lim_{x \rightarrow \infty} \cosh x =\lim_{x \rightarrow \infty} \frac{e^x + e^{-x}}{2}\) As \(x\rightarrow\infty\), \(e^x\) goes to infinity and \(e^{-x}\) goes to 0. Thus, the limit becomes: \(\lim_{x \rightarrow \infty} \cosh x=\frac{\infty + 0}{2}=\infty\) #Step 3: Limit as x approaches negative infinity#

(Find limit as x approaches negative infinity)

To find the limit as x approaches negative infinity, we will analyze the function \(\cosh x= \frac{e^x + e^{-x}}{2}\) as \(x \rightarrow -\infty\) : \(\lim_{x \rightarrow -\infty} \cosh x =\lim_{x \rightarrow -\infty} \frac{e^x + e^{-x}}{2}\) As \(x\rightarrow-\infty\), \(e^x\) goes to 0 and \(e^{-x}\) goes to infinity. Thus, the limit becomes: \(\lim_{x \rightarrow -\infty} \cosh x=\frac{0 + \infty}{2}=\infty\) #Step 4: Evaluate cosh(0)#

(Evaluate the Hyperbolic Cosine at x = 0)

To evaluate \(\cosh(0)\), we will substitute 0 into the definition of the function: \(\cosh(0)=\frac{e^0 + e^{-0}}{2}=\frac{1+1}{2}=1\) #Step 5: Sketch the graph of y = cosh(x)#

(Sketch the Graph of the Hyperbolic Cosine Function)

Using the information from previous steps, we can sketch a graph of the hyperbolic cosine function: 1. The end behavior of the function tells us that as x goes to positive or negative infinity, the function goes to infinity. 2. The value of cosh(0) is 1, which is the lowest value of the function since it never goes to 0. 3. The hyperbolic cosine function is symmetric with respect to the y-axis because \(\cosh(-x)=\cosh(x)\). Based on this information, we can sketch a symmetric graph with an 'U' shape opening upwards, with the lowest point at (0, 1) and asymptotically going to infinity as x goes to positive and negative infinity.

Key concepts

End Behavior

The concept of end behavior in functions pertains to how the function behaves as the independent variable, often denoted as \(x\), moves towards positive or negative infinity. For the hyperbolic cosine function, \(\cosh x\), we determine its end behavior by examining the function's limits as \(x\) approaches these extremes.
To evaluate \(\lim_{x \to \infty} \cosh x\), observe that \(\cosh x = \frac{e^x + e^{-x}}{2}\). Here, \(e^x\) grows significantly larger than \(e^{-x}\), causing the fraction to gain a large positive value. Therefore, the limit as \(x\) approaches infinity is infinity itself.
Conversely, for \(\lim_{x \to -\infty} \cosh x\), \(e^{-x}\) becomes exceedingly large while \(e^x\) almost vanishes. Thus, the function still trends towards infinity as \(x\) becomes very negative.

This dual behavior implies that \(\cosh x\) continually increases to infinity in both directions, illustrating its asymptotic nature.

Symmetry in Functions

Symmetry in mathematical functions is an important property that can simplify understanding their graphs and characteristics. Specifically, a function is symmetric with respect to the y-axis if the function remains unchanged when \(x\) is replaced with \(-x\).
The hyperbolic cosine function \(\cosh x\) is a perfect example of even symmetry. This is because \(\cosh(-x) = \cosh x\). In other words, flipping to the negative side of the x-axis does not change the value of the function. This means the graph of \(\cosh x\) mirrors itself about the y-axis.

Such symmetry makes it easier to predict the behavior and characteristics of the function by looking at just one side of the graph. It's like having a mirror placed on the y-axis, and whatever is on one side is perfectly reflected over to the other side, creating a balanced and predictable shape.

Graphing Functions

When graphing functions, understanding their key characteristics like end behavior, symmetry, and specific points helps in accurately representing their behavior on a coordinate plane. The graph of the hyperbolic cosine function, \(\cosh x\), provides an elegant visual story of these properties.
Initially, we calculate \(\cosh(0)\) to find the function's specific value at \(x = 0\). This gives \(\cosh(0) = 1\), representing the graph's minimum point found at (0, 1). This serves as the base from which the curve expands upwards.
The hyperbolic cosine graph is symmetric around the y-axis, helping us double-check our graph by ensuring the left and right sides are mirror images. The end behavior analysis earlier, where \(\cosh x\) tends to infinity as \(x\) moves to either side, helps us sketch the curve properly, drawing an upward opening U-shape.

• Start at the lowest point (0, 1).
• Allow the curve to rise as the graph moves in both directions along the x-axis.
• Ensure symmetry, maintaining equal spacing on both sides of the y-axis.
• Eventually, the lines extend towards infinity, never leveling off but continuing to climb.

Referencing these properties during graphing can ease the process and ensure accuracy.