Question

The hyperbolic sine function is defined as \(\sinh x=\frac{e^{x}-e^{-x}}{2}\) a. Determine its end behavior by analyzing \(\lim _{x \rightarrow \infty} \sinh x\) and \(\lim _{x \rightarrow-\infty} \sinh x\) b. Evaluate sinh 0. Use symmetry and part (a) to sketch a plausible graph for \(y=\sinh x\)

Short answer

Question: Determine the end behavior of the hyperbolic sine function \(\sinh x\) and sketch its graph. Answer: The end behavior of the hyperbolic sine function is \(\lim_{x \rightarrow \infty} \sinh x = \infty\) and \(\lim_{x \rightarrow-\infty} \sinh x = -\infty\). The graph of \(y = \sinh x\) has origin symmetry and passes through the point (0,0). It curves upward to the right and downward to the left, increasing without bound as \(x\) approaches infinity and decreasing without bound as \(x\) approaches negative infinity.

Step-by-step solution

a. Finding the end behavior of the hyperbolic sine function

To find the end behavior of the function, we will find the limits as x approaches infinity and negative infinity. We have the definition \(\sinh x = \frac{e^x - e^{-x}}{2}\). Now, let's find the limits: 1. \(\lim_{x \rightarrow \infty} \sinh x = \lim_{x \rightarrow \infty} \frac{e^x - e^{-x}}{2}\) As \(x\) approaches infinity, \(e^x\) increases without bound while \(e^{-x}\) approaches 0. Therefore, the limit becomes: 2. \(\lim_{x \rightarrow \infty} \sinh x = \frac{\infty - 0}{2} = \infty\) Now, let's find the limit as \(x\) approaches negative infinity: 3. \(\lim_{x \rightarrow -\infty} \sinh x = \lim_{x \rightarrow -\infty} \frac{e^x - e^{-x}}{2}\) As \(x\) approaches negative infinity, \(e^x\) approaches 0 while \(e^{-x}\) increases without bound. Therefore, the limit becomes: 4. \(\lim_{x \rightarrow -\infty} \sinh x = \frac{0 - \infty}{2} = -\infty\) So, the end behavior of the hyperbolic sine function is \(\lim_{x \rightarrow \infty} \sinh x = \infty\) and \(\lim_{x \rightarrow-\infty} \sinh x = -\infty\).

b. Evaluating \(\sinh 0\) and sketching the graph

To evaluate \(\sinh 0\), we simply plug in \(0\) for \(x\) in the definition of the function: 5. \(\sinh 0 = \frac{e^0 - e^{-0}}{2} = \frac{1 - 1}{2} = 0\) Now, let's use the end behavior and symmetry to sketch the graph of \(y = \sinh x\): 6. From the end behavior, we know that as \(x\) approaches infinity, the function increases without bound, and as \(x\) approaches negative infinity, the function decreases without bound. 7. Since \(\sinh x = \frac{e^x - e^{-x}}{2}\), we know that \(\sinh (-x) = \frac{e^{-x} - e^x}{2} = -\frac{e^x - e^{-x}}{2} = -\sinh x\). This means that the function has origin symmetry: it is symmetric with respect to the origin. 8. With the end behavior, symmetry, and the value of \(\sinh 0\), we can now sketch a plausible graph for \(y = \sinh x\). The graph will look like a curve that passes through the origin (0,0), curves upward to the right, and curves downward to the left. There will be symmetry with respect to the origin, and the function will increase without bound as \(x\) approaches infinity and decrease without bound as \(x\) approaches negative infinity.

Key concepts

End Behavior of Functions

Understanding the end behavior of functions is crucial when analyzing how a function behaves as its input, or x-values, go towards infinity or negative infinity. In the case of the hyperbolic sine function, \(\sinh x = \frac{e^x - e^{-x}}{2}\), we can determine its end behavior by examining the limits at infinity.

As the input grows larger and larger, the term \(e^x\) in the numerator dominates since it increases without bounds much faster than \(e^{-x}\) decreases to zero. This is why, when we take the limit as x approaches infinity, \(\sinh x\) goes to infinity as well. Conversely, as the input becomes more and more negative, the term \(e^{-x}\) becomes significant, and \(e^x\) approaches zero. This causes \(\sinh x\) to approach negative infinity as x goes towards negative infinity.

Understanding this concept helps in predicting the long-term behavior of functions, which is essential in calculus and higher-level math.

Limits at Infinity

The concept of limits at infinity is a foundational idea in calculus that explores the value a function approaches as the input either increases or decreases without bound. For the hyperbolic sine function, we see diverse behaviors for \(\lim_{x \rightarrow \infty} \sinh x\), which equates to infinity, and for \(\lim_{x \rightarrow -\infty} \sinh x\), which is negative infinity.

These limits indicate not just the end behavior, but also provide insights into the growth rate of the function and its asymptotic tendencies. This knowledge proves pivotal in various aspects, such as finding horizontal asymptotes, and is often used to study the stability and control of systems in engineering.

Graphing Functions

Graphing functions serves as a visual representation of the relationship between variables in an equation. To graph the hyperbolic sine function, it is essential to consider points we know for certain, like \(\sinh 0 = 0\), and the end behaviors derived from the limits at infinity. The curve of \(y = \sinh x\) starts at the origin, and, guided by the end behaviors, it rises indefinitely in the first quadrant and falls indefinitely in the third quadrant, creating a smooth, continuous curve.

Graphing not only helps in visualizing the behavior of functions but also assists in identifying properties like intercepts, turning points, and overall shape, making it a powerful tool for predictions and calculations in both pure and applied mathematics.

Origin Symmetry

Origin symmetry is a concept where a function's graph is a mirror image across the origin, meaning that for every point \( (x, y) \) on the graph there is a corresponding point \( (-x, -y) \) also on the graph. This symmetry is a characteristic of odd functions, which satisfy the relation \( f(-x) = -f(x) \). The hyperbolic sine function \(\sinh x\) exhibits origin symmetry because it meets this criterion: \(\sinh (-x) = -\sinh x\).

Recognizing origin symmetry simplifies graphing and understanding functions as it provides structure and constraints to the possible shape of the graph. Moreover, this symmetry plays a role in solving equations and in the analysis of physical systems where such symmetrical behavior is mirrored in real-world phenomena.