Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Find the curvature \(\kappa(t)\) of the curve given by the parametric vector function:
\[
\mathbf{r}(t) = (4 \sin t) \mathbf{i} + (4 \sin t) \mathbf{j} + (5 \cos t) \mathbf{k}
\]
**Instructions:**
You need to calculate the curvature \(\kappa(t)\) of the provided curve using the standard curvature formula for a space curve.
**Concepts Covered:**
- Curvature of a space curve
- Parametric equations
- Vector calculus
**Relevant Formula:**
The curvature \(\kappa(t)\) of a curve defined by the vector function \(\mathbf{r}(t)\) is given by:
\[
\kappa(t) = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}
\]
Where:
- \(\mathbf{r}'(t)\) is the first derivative of \(\mathbf{r}(t)\)
- \(\mathbf{r}''(t)\) is the second derivative of \(\mathbf{r}(t)\)
- \(\times\) denotes the cross product
Through this problem, you will apply differentiation and vector operations to find the curvature.
**Solution Approach:**
1. Differentiate \(\mathbf{r}(t)\) to find \(\mathbf{r}'(t)\).
2. Differentiate \(\mathbf{r}'(t)\) to find \(\mathbf{r}''(t)\).
3. Calculate the cross product \(\mathbf{r}'(t) \times \mathbf{r}''(t)\).
4. Find the magnitudes \(|\mathbf{r}'(t)|\) and \(|\mathbf{r}'(t) \times \mathbf{r}''(t)|\).
5. Substitute these in the curvature formula to find \(\kappa(t)\).
By solving this, you will enhance your understanding of curvature and vector operations in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F569ff496-8bfa-46fa-9221-cf5ae21a3c2a%2F81d6d01e-2c0a-4caf-8d5d-6fc5b5e251ec%2F2yydwq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the curvature \(\kappa(t)\) of the curve given by the parametric vector function:
\[
\mathbf{r}(t) = (4 \sin t) \mathbf{i} + (4 \sin t) \mathbf{j} + (5 \cos t) \mathbf{k}
\]
**Instructions:**
You need to calculate the curvature \(\kappa(t)\) of the provided curve using the standard curvature formula for a space curve.
**Concepts Covered:**
- Curvature of a space curve
- Parametric equations
- Vector calculus
**Relevant Formula:**
The curvature \(\kappa(t)\) of a curve defined by the vector function \(\mathbf{r}(t)\) is given by:
\[
\kappa(t) = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}
\]
Where:
- \(\mathbf{r}'(t)\) is the first derivative of \(\mathbf{r}(t)\)
- \(\mathbf{r}''(t)\) is the second derivative of \(\mathbf{r}(t)\)
- \(\times\) denotes the cross product
Through this problem, you will apply differentiation and vector operations to find the curvature.
**Solution Approach:**
1. Differentiate \(\mathbf{r}(t)\) to find \(\mathbf{r}'(t)\).
2. Differentiate \(\mathbf{r}'(t)\) to find \(\mathbf{r}''(t)\).
3. Calculate the cross product \(\mathbf{r}'(t) \times \mathbf{r}''(t)\).
4. Find the magnitudes \(|\mathbf{r}'(t)|\) and \(|\mathbf{r}'(t) \times \mathbf{r}''(t)|\).
5. Substitute these in the curvature formula to find \(\kappa(t)\).
By solving this, you will enhance your understanding of curvature and vector operations in three-dimensional space.
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