4. Using the alternative formula for curvature k(t) = = for the helix F(t) = (4 sin t, 4 cos t, 0). and the circle ↓ F(t) = (3 cos t, 3 sin t, 4t) 1 |* (t) × (t) 3 (t)\ " compute the curvatures
4. Using the alternative formula for curvature k(t) = = for the helix F(t) = (4 sin t, 4 cos t, 0). and the circle ↓ F(t) = (3 cos t, 3 sin t, 4t) 1 |* (t) × (t) 3 (t)\ " compute the curvatures
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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![**Curvature Computation: A Mathematical Exercise**
**Section 4: Calculating Curvature Using an Alternative Formula**
To compute the curvature \(\kappa(t)\) using the alternative formula, use the expression:
\[
\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}
\]
Apply this formula to the following cases:
**For the Helix:**
The position vector is given by:
\[
\vec{r}(t) = \langle 4 \sin t, 4 \cos t, 0 \rangle
\]
**For the Circle:**
The position vector is given by:
\[
\vec{r}(t) = \langle 3 \cos t, 3 \sin t, 4t \rangle
\]
Compute the curvature for each case using the given expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec2755b6-7089-4f85-a844-ede57d3c63f1%2Fef4c28cd-056d-4d6e-876b-e7ff5acf3ee3%2Fahydjja.jpeg&w=3840&q=75)
Transcribed Image Text:**Curvature Computation: A Mathematical Exercise**
**Section 4: Calculating Curvature Using an Alternative Formula**
To compute the curvature \(\kappa(t)\) using the alternative formula, use the expression:
\[
\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}
\]
Apply this formula to the following cases:
**For the Helix:**
The position vector is given by:
\[
\vec{r}(t) = \langle 4 \sin t, 4 \cos t, 0 \rangle
\]
**For the Circle:**
The position vector is given by:
\[
\vec{r}(t) = \langle 3 \cos t, 3 \sin t, 4t \rangle
\]
Compute the curvature for each case using the given expression.
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