Question

[T] Find expressions for \(\cosh x+\sinh x\) and \(\cosh x-\sinh x .\) Use a calculator to graph these functions and ensure your expression is correct.

Short answer

\(\cosh x + \sinh x = e^x\); \(\cosh x - \sinh x = e^{-x}\).

Step-by-step solution

Understand Hyperbolic Functions

The hyperbolic functions are defined as follows:\[ \cosh(x) = \frac{e^x + e^{-x}}{2} \]\[ \sinh(x) = \frac{e^x - e^{-x}}{2} \]These are the basic definitions we will use to find the expressions for the given questions.

Simplify \(\cosh x + \sinh x\)

Using the definitions from Step 1, we have:\[ \cosh(x) + \sinh(x) = \frac{e^x + e^{-x}}{2} + \frac{e^x - e^{-x}}{2} \]Combine the fractions:\[ = \frac{(e^x + e^{-x}) + (e^x - e^{-x})}{2} = \frac{2e^x}{2} = e^x \]So, \(\cosh x + \sinh x = e^x\).

Simplify \(\cosh x - \sinh x\)

Using the definitions from Step 1, we have:\[ \cosh(x) - \sinh(x) = \frac{e^x + e^{-x}}{2} - \frac{e^x - e^{-x}}{2} \]Combine the fractions:\[ = \frac{(e^x + e^{-x}) - (e^x - e^{-x})}{2} = \frac{2e^{-x}}{2} = e^{-x} \]So, \(\cosh x - \sinh x = e^{-x}\).

Graph the Functions

Using a graphing calculator or software, graph the functions \(e^x\) and \(e^{-x}\). Ensure that the behavior of these graphs matches your expectations for \(\cosh x + \sinh x\) and \(\cosh x - \sinh x\) respectively, with \(e^x\) increasing and \(e^{-x}\) decreasing as x increases.

Confirm the Results

Both expressions simplify directly to exponential functions \(e^x\) and \(e^{-x}\). If graphed correctly, the comparisons should confirm these results as the graphs should reflect the growth and decay typical of exponentials.

Key concepts

cosh and sinh identities

Hyperbolic functions often confused with trigonometric functions, play a crucial role in mathematics. These functions connect the dots between exponential functions and curves, similar to how regular sin and cos connect to circular functions. Let's demystify two key hyperbolic functions:

• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)
• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)

These formulas are central in understanding the problem. When combined, such as in \(\cosh x + \sinh x\), they simplify elegantly due to their definition:- For \(\cosh x + \sinh x\), adding the definitions gives \(e^x\).- For \(\cosh x - \sinh x\), subtracting them results in \(e^{-x}\).The identities show the profound link between hyperbolic and exponential expressions, simplifying computing efforts significantly, whether dealing with growth models or physical phenomena.

exponential functions

Exponential functions are powerful tools often seen in natural growth or decay contexts. They are characterized by the base constant \(e\), approximately equal to 2.71828. The fundamental behavior of these functions includes:- \(e^x\) indicates exponential growth since the function grows faster as \(x\) increases.- Conversely, \(e^{-x}\) shows exponential decay, decreasing with increasing \(x\).Relating it back to hyperbolic functions, we learned that \(\cosh x + \sinh x = e^x\), representing exponential growth, and \(\cosh x - \sinh x = e^{-x}\), representing decay. This mapping of hyperbolic identities to exponential functions allows for easier analysis in both mathematics and applied sciences. This correspondence simplifies calculations and gives deeper insights into the role of exponential functions beyond routine operations, bridging various mathematical fields effectively.

graphing functions

Graphing helps visualize mathematical concepts, bringing abstract numbers to life. In the context of our hyperbolic functions:

• When you graph \(e^x\): it rises steeply, capturing exponential growth.
• Graphing \(e^{-x}\): results in a steep decline, reflecting exponential decay.

Using graphing calculators or software, such as Desmos or a graphing calculator app, you can visually match these functions to their hyperbolic counterparts.- \(\cosh x + \sinh x\) will mirror the incline of \(e^x\).- \(\cosh x - \sinh x\) mirrors the decline of \(e^{-x}\).Graphing provides not only visual confirmation but also helps in understanding the behavior of these functions under varying conditions. It's a useful skill for students to develop as they engage with more complex functions. You can experiment with graphs by adjusting parameters to see how sensitive they are to changes, gaining a better understanding through visual feedback.