Problem 15 - Precalculus Mathematics for Calculus

Question

The hyperbolic cosine function is defined by $$\cosh (x)=\frac{e^{x}+e^{-x}}{2}$$ (a) Sketch the graphs of the functions \(y=\frac{1}{2} e^{x}\) and \(y=\frac{1}{2} e^{-x}\) on the same axes, and use graphical addition (see Section 2.6 ) to sketch the graph of \(y=\cosh (x).\) (b) Use the definition to show that \(\cosh (-x)=\cosh (x).\)

Step-by-Step Solution

Answer

The graph of \(y = \cosh(x)\) is obtained by summing two exponentials, and \(\cosh(x)\) is even, meaning \(\cosh(-x) = \cosh(x)\).

Step 1: Graph each component function

To graph the functions \(y = \frac{1}{2} e^x\) and \(y = \frac{1}{2} e^{-x}\), start by plotting several values. The function \(y = \frac{1}{2} e^x\) is an exponential growth function that starts at \(y = \frac{1}{2}\) when \(x = 0\) and increases rapidly as \(x\) increases. The function \(y = \frac{1}{2} e^{-x}\) is an exponential decay function that also starts at \(y = \frac{1}{2}\) when \(x = 0\) and decreases towards zero as \(x\) increases. Plot these points and sketch the curves.

Step 2: Graphical addition

To sketch the graph of \(y = \cosh(x)\), use graphical addition by adding the corresponding y-values of \(y = \frac{1}{2} e^x\) and \(y = \frac{1}{2} e^{-x}\) for each x-value. This means that for a specific \(x\), if the first function gives a value \(y_1\) and the second function gives \(y_2\), then \(y_1 + y_2\) gives the value of \(y = \cosh(x)\) at that point. Draw the resulting curve from the summed points.

Step 3: Definition application for even property

To show \(\cosh(-x) = \cosh(x)\), substitute \(-x\) into the definition of \(\cosh(x)\):\[\cosh(-x) = \frac{e^{-x} + e^{x}}{2}\]. Due to the commutative property of addition, we can rearrange the terms as \(\frac{e^x + e^{-x}}{2}\), which is exactly \(\cosh(x)\). This demonstrates that \(\cosh(x)\) is an even function and thus \(\cosh(-x) = \cosh(x)\).

Key Concepts

Hyperbolic Cosine

The hyperbolic cosine function, often denoted as \(\cosh(x)\), is a fundamental building block in the study of hyperbolic functions. Its definition \(\cosh (x)=\frac{e^{x}+e^{-x}}{2}\) contains exponential components that, together, describe important mathematical behaviors resembling those found in hyperbolic geometry.
Understanding \(\cosh(x)\) begins with comprehending the nature of its components. The expression involves the sum of two exponential functions: \(\frac{1}{2} e^x\) and \(\frac{1}{2} e^{-x}\). These separate parts illustrate opposite but complementary actions; one depicts growth while the other showcases decay.
One fascinating aspect of \(\cosh(x)\) is its symmetry. It is an even function, meaning \(\cosh(-x) = \cosh(x)\). This property highlights a central feature of \(\cosh(x)\): it does not change when \(x\) is flipped across the y-axis. This is instrumental when modeling physical phenomena or solving differential equations, where symmetry simplifies the processes involved.

Exponential Growth and Decay

Exponential growth and decay are critical concepts in a myriad of fields, from population dynamics to radioactive decay.
For \(\frac{1}{2} e^x\), we observe exponential growth, characterized by a rapid increase as \(x\) becomes larger. Starting from \(x = 0\) with a base value of \(\frac{1}{2}\), this function swiftly rises, illustrating how quantities can expand at increasing rates under certain conditions.
On the other hand, \(\frac{1}{2} e^{-x}\) illustrates exponential decay, where values diminish as \(x\) increases. Similarly beginning from \(1/2\), it heads toward zero, showing the gradual reduction typical of decay processes, such as cooling or discharging over time.
Through graphing these two distinct behaviors, one gains insight into the balancing act that forms the basis of the hyperbolic cosine, providing a nuanced understanding of compound exponential behaviors.

Graphical Addition

Graphical addition is a key technique utilized to comprehend the combination of functions visually.
For this method, we overlay the graphs of \(y = \frac{1}{2} e^x\) and \(y = \frac{1}{2} e^{-x}\) on the same axes. These graphs reveal complementary exponential movements that contrast one another along the x-axis.
By summing the corresponding y-values of the two functions at each specific \(x\), we can construct the graph of \(y = \cosh(x)\). For instance, at a given \(x\), take the y-value from \(\frac{1}{2} e^x\) and add it to the y-value from \(\frac{1}{2} e^{-x}\). Plot this summed value for \(y\).
This step-by-step addition forms a new curve that encapsulates the essence of both the growth and decay properties inherent in \(\cosh(x)\). The resulting graph demonstrates the smooth, symmetrical bell-shaped curve characteristic of hyperbolic cosine functions, important in both theoretical and applied mathematics.