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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14b.19 Please help me answer all parts to this math problem.
![**Objective:** Find the unit tangent vector **T** and the curvature κ for the following parameterized curve.
**Curve:**
\[
\mathbf{r}(t) = \langle \sqrt{21} \sin t, 10 \sin t, 11 \cos t \rangle
\]
**Explanation:**
The given parameterized curve is described using a vector function \(\mathbf{r}(t)\), which is composed of three components:
- The first component is \(\sqrt{21} \sin t\),
- The second component is \(10 \sin t\),
- The third component is \(11 \cos t\).
To find the unit tangent vector \(\mathbf{T}\), you will need to take the derivative of \(\mathbf{r}(t)\) with respect to \(t\) to get the velocity vector, and then normalize this vector.
To find the curvature \(\kappa\), you will need to use the formula that relates the derivatives of the vector function to the magnitude of the cross product of the derivatives.
This task involves understanding vector derivatives and their applications in determining geometric properties of curves.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3bd88a3-9d65-41d6-8e96-bcf2f13eb11b%2Fc326b646-f933-43e3-8a7b-d6131044783e%2Fitqjdjj_processed.png&w=3840&q=75)
Transcribed Image Text:**Objective:** Find the unit tangent vector **T** and the curvature κ for the following parameterized curve.
**Curve:**
\[
\mathbf{r}(t) = \langle \sqrt{21} \sin t, 10 \sin t, 11 \cos t \rangle
\]
**Explanation:**
The given parameterized curve is described using a vector function \(\mathbf{r}(t)\), which is composed of three components:
- The first component is \(\sqrt{21} \sin t\),
- The second component is \(10 \sin t\),
- The third component is \(11 \cos t\).
To find the unit tangent vector \(\mathbf{T}\), you will need to take the derivative of \(\mathbf{r}(t)\) with respect to \(t\) to get the velocity vector, and then normalize this vector.
To find the curvature \(\kappa\), you will need to use the formula that relates the derivatives of the vector function to the magnitude of the cross product of the derivatives.
This task involves understanding vector derivatives and their applications in determining geometric properties of curves.
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