-ind the curvature of r` (t) = ti + t²j + t³ k at different valu oft. Which formula would you use to find curvature K: O k(t) = r(t) × ¹ (t) r' (t) Fox 3 all (+)
-ind the curvature of r` (t) = ti + t²j + t³ k at different valu oft. Which formula would you use to find curvature K: O k(t) = r(t) × ¹ (t) r' (t) Fox 3 all (+)
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...
Related questions
Question
What is |r′→(t)×r′′→(t)|
Find κ(0):
Find κ(1):
![**Curvature of a Parametric Curve**
To determine the curvature of the vector function \(\vec{r}(t) = t\hat{i} + t^2\hat{j} + t^3\hat{k}\) at various values of \(t\), we need to choose the correct formula for curvature \(\kappa\).
**Formula Options for Curvature \(\kappa\):**
1. \(\kappa(t) = \frac{|\vec{r}(t) \times \vec{r}'(t)|}{|\vec{r}'(t)|^3}\)
2. \(\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^2}\)
3. \(\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}(t)|}{|\vec{r}'(t)|^3}\)
4. \(\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}\) [**Selected Formula**]
The fourth formula is selected as the correct one for calculating the curvature \(\kappa(t)\) of the given vector function at different values of \(t\). This formula uses the cross product of the first and second derivatives of the vector function \(\vec{r}(t)\), divided by the cube of the magnitude of the first derivative.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc167378a-6772-4355-bdb0-a5fec1af8d0e%2F3fb170a3-e0ab-42dd-a83f-9dd51dc03d39%2Fgc28mk6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Curvature of a Parametric Curve**
To determine the curvature of the vector function \(\vec{r}(t) = t\hat{i} + t^2\hat{j} + t^3\hat{k}\) at various values of \(t\), we need to choose the correct formula for curvature \(\kappa\).
**Formula Options for Curvature \(\kappa\):**
1. \(\kappa(t) = \frac{|\vec{r}(t) \times \vec{r}'(t)|}{|\vec{r}'(t)|^3}\)
2. \(\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^2}\)
3. \(\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}(t)|}{|\vec{r}'(t)|^3}\)
4. \(\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}\) [**Selected Formula**]
The fourth formula is selected as the correct one for calculating the curvature \(\kappa(t)\) of the given vector function at different values of \(t\). This formula uses the cross product of the first and second derivatives of the vector function \(\vec{r}(t)\), divided by the cube of the magnitude of the first derivative.
![### Calculus Problem
**Question**: What is \(\vec{r}'(t)\)?
**Answer Format**: \(a, b, c\)
*[Answer Input Box]*
**Feedback**: If incorrect, a red cross (✗) will appear.
---
**Support Options**:
- [Check Answer/Save]
- [Step-By-Step Example]
- [Live Help]
---
**Question**: What is \(\vec{r}''(t)\)?
**Answer Format**: \(a, b, c\)
*[Answer Input Box]*
**Feedback**: If incorrect, a red cross (✗) will appear.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc167378a-6772-4355-bdb0-a5fec1af8d0e%2F3fb170a3-e0ab-42dd-a83f-9dd51dc03d39%2F2ixth9b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem
**Question**: What is \(\vec{r}'(t)\)?
**Answer Format**: \(a, b, c\)
*[Answer Input Box]*
**Feedback**: If incorrect, a red cross (✗) will appear.
---
**Support Options**:
- [Check Answer/Save]
- [Step-By-Step Example]
- [Live Help]
---
**Question**: What is \(\vec{r}''(t)\)?
**Answer Format**: \(a, b, c\)
*[Answer Input Box]*
**Feedback**: If incorrect, a red cross (✗) will appear.
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