Find an arc length parametrization r₁(s) of the curve r(t) = r₁(s) = (e' sin(t), e' cos(t), 12e'). (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)
Find an arc length parametrization r₁(s) of the curve r(t) = r₁(s) = (e' sin(t), e' cos(t), 12e'). (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Arc Length Parametrization Problem
**Problem Statement:**
Find an arc length parametrization \(\mathbf{r_1}(s)\) of the curve \(\mathbf{r}(t) = \langle e^t \sin(t), e^t \cos(t), 12e^t \rangle\).
(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)
\[ \mathbf{r_1}(s) = \]
**Explanation:**
To find the arc length parameter \(s\), we need to compute the length of the curve from the starting point up to a parameter value \(t\). This involves integrating the magnitude of the derivative of \(\mathbf{r}(t)\) with respect to \(t\).
1. **Compute \(\mathbf{r}'(t)\) (the derivative of \(\mathbf{r}(t)\))**:
\[ \mathbf{r}(t) = \langle e^t \sin(t), e^t \cos(t), 12e^t \rangle \]
\[ \mathbf{r}'(t) = \left\langle \frac{d}{dt}\left(e^t \sin(t)\right), \frac{d}{dt}\left(e^t \cos(t)\right), \frac{d}{dt}\left(12e^t\right) \right\rangle \]
2. **Compute each derivative**:
\[ \frac{d}{dt}\left(e^t \sin(t)\right) = e^t \sin(t) + e^t \cos(t) \]
\[ \frac{d}{dt}\left(e^t \cos(t)\right) = e^t \cos(t) - e^t \sin(t) \]
\[ \frac{d}{dt}\left(12e^t\right) = 12e^t \]
Thus,
\[ \mathbf{r}'(t) = \langle e^t(\sin(t) + \cos(t)), e^t(\cos(t) - \sin(t)), 12e^t \rangle \]
3. **Find the magnitude of \(\mathbf{r}'(t)\)**:
\[ \|\mathbf{r}'(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c410f98-c150-4333-801a-7de091a4db56%2Ff96715c5-5abc-4862-ba11-d2d3daa0b985%2F4u09y6o_processed.png&w=3840&q=75)
Transcribed Image Text:### Arc Length Parametrization Problem
**Problem Statement:**
Find an arc length parametrization \(\mathbf{r_1}(s)\) of the curve \(\mathbf{r}(t) = \langle e^t \sin(t), e^t \cos(t), 12e^t \rangle\).
(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)
\[ \mathbf{r_1}(s) = \]
**Explanation:**
To find the arc length parameter \(s\), we need to compute the length of the curve from the starting point up to a parameter value \(t\). This involves integrating the magnitude of the derivative of \(\mathbf{r}(t)\) with respect to \(t\).
1. **Compute \(\mathbf{r}'(t)\) (the derivative of \(\mathbf{r}(t)\))**:
\[ \mathbf{r}(t) = \langle e^t \sin(t), e^t \cos(t), 12e^t \rangle \]
\[ \mathbf{r}'(t) = \left\langle \frac{d}{dt}\left(e^t \sin(t)\right), \frac{d}{dt}\left(e^t \cos(t)\right), \frac{d}{dt}\left(12e^t\right) \right\rangle \]
2. **Compute each derivative**:
\[ \frac{d}{dt}\left(e^t \sin(t)\right) = e^t \sin(t) + e^t \cos(t) \]
\[ \frac{d}{dt}\left(e^t \cos(t)\right) = e^t \cos(t) - e^t \sin(t) \]
\[ \frac{d}{dt}\left(12e^t\right) = 12e^t \]
Thus,
\[ \mathbf{r}'(t) = \langle e^t(\sin(t) + \cos(t)), e^t(\cos(t) - \sin(t)), 12e^t \rangle \]
3. **Find the magnitude of \(\mathbf{r}'(t)\)**:
\[ \|\mathbf{r}'(
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