Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Hyperbolic Functions and Parametric Equations**
**Problem Statement:**
Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\).
(Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.)
**Equations:**
\[x(t) = \quad \rule{20em}{0.5pt}\]
\[y(t) = \quad \rule{20em}{0.5pt}\]
\[z(t) = \quad \rule{20em}{0.5pt}\]
**Instructions:**
1. **Identify the Parameterization Technique:**
- For the variable \(x\), use the hyperbolic cosine function (\(\cosh(t)\)).
- For the variable \(y\), use the hyperbolic sine function (\(\sinh(t)\)).
2. **Determine Parametric Equations:**
- Substitute these hyperbolic functions into the equation \(x^2 - y^2 = 4\).
- Ensure the equations satisfy both the given surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\).
3. **Input the Parametric Equations:**
- Write the correct parametric equations for \(x(t)\), \(y(t)\), and \(z(t)\) based on the parameter \(t\).
**Note:** The hyperbolic functions \(\cosh(t)\) and \(\sinh(t)\) are defined as:
\[
\cosh(t) = \frac{e^t + e^{-t}}{2}
\]
\[
\sinh(t) = \frac{e^t - e^{-t}}{2}
\]](https://content.bartleby.com/qna-images/question/96584006-fab9-4c6d-81e5-caf655fa5a7f/ab28fbc8-9be3-41aa-a963-86cbf598cb23/pqdb50a_processed.png)
Transcribed Image Text:**Hyperbolic Functions and Parametric Equations**
**Problem Statement:**
Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\).
(Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.)
**Equations:**
\[x(t) = \quad \rule{20em}{0.5pt}\]
\[y(t) = \quad \rule{20em}{0.5pt}\]
\[z(t) = \quad \rule{20em}{0.5pt}\]
**Instructions:**
1. **Identify the Parameterization Technique:**
- For the variable \(x\), use the hyperbolic cosine function (\(\cosh(t)\)).
- For the variable \(y\), use the hyperbolic sine function (\(\sinh(t)\)).
2. **Determine Parametric Equations:**
- Substitute these hyperbolic functions into the equation \(x^2 - y^2 = 4\).
- Ensure the equations satisfy both the given surfaces \(x^2 - y^2 = 4\) and \(z = 5xy\).
3. **Input the Parametric Equations:**
- Write the correct parametric equations for \(x(t)\), \(y(t)\), and \(z(t)\) based on the parameter \(t\).
**Note:** The hyperbolic functions \(\cosh(t)\) and \(\sinh(t)\) are defined as:
\[
\cosh(t) = \frac{e^t + e^{-t}}{2}
\]
\[
\sinh(t) = \frac{e^t - e^{-t}}{2}
\]
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