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(1 point) Find the curvature (t) of the curve r(t) = (4 sin t)i + (4 sin t)j + (5 cost) k

(1 point) Find the curvature (t) of the curve r(t) = (4 sin t)i + (4 sin t)j + (5 cost) k

Calculus: Early Transcendentals
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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**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.
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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