Problem 10 - Calculus Early Transcendentals

Question

How does the graph of the catenary \(y=a \cosh (x / a)\) change as \(a>0\) increases?

Step-by-Step Solution

Answer

Answer: As the parameter \(a\) increases in the catenary function, the minimum value of the function increases, moving the lowest point on the graph upward; the shape of the graph becomes less curved and more 'gentle'; and the function's behavior at infinity becomes less steep, as the horizontal scaling makes the graph wider.

Step 1: Understand the Catenary Function

The catenary function is given by \(y = a \cosh(x/a)\), where \(a\) is a positive constant and \(\cosh\) is the hyperbolic cosine function, defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).

Step 2: Plot the Catenary Function for Different Values of \(a\)

We can begin our investigation by plotting the catenary function for different values of \(a\). Use graphing software to plot the function for values of \(a\) ranging from small values (e.g., \(a=1\)) to larger values (e.g., \(a=5\)), and notice how the shape and position of the graph changes.

Step 3: Analyze the Minimum Value

The catenary function is symmetric about the x-axis. The function has a minimum value at \(x=0\), which is \(y = a\cosh(0) = a\). As \(a\) increases, the minimum value of the function also increases. This means that the lowest point on the graph will move upward as \(a\) increases.

Step 4: Analyze the Shape of the Catenary Function

The catenary function has a very distinct shape, resembling a hanging chain or a parabola. As \(a\) increases, the graph spreads out more horizontally. This can be mathematically seen by evaluating the second derivative of \(y\) with respect to \(x\) which is \(y''(x) = \cosh(x/a)\). The second derivative tells us about the concavity of the function. Notice that \(a\) essentially scales the \(x\)-axis. As \(a\) becomes larger, the function becomes more 'gentle,' and less curved.

Step 5: Analyze the Asymptotic Behavior

Analyze the graph's asymptotic behavior as \(x\) goes to positive and negative infinity. As \(x\) goes to infinity, we know that \(\cosh(x/a)\) goes to infinity as well. Therefore, \(y\) also goes to infinity. The function's behavior for large positive and negative values of x becomes less steep as the value of \(a\) increases due to the horizontal scaling of the graph.

Step 6: Summary

As \(a\) increases in the catenary function \(y = a \cosh(x/a)\): 1. The minimum value of the function increases, moving the lowest point on the graph upward. 2. The shape of the graph becomes less curved and more 'gentle'. 3. The function's behavior at infinity becomes less steep, as the horizontal scaling makes the graph wider. By following these steps, you can visualize and understand the changes in the graph of the catenary function as the parameter \(a\) increases.