Find r'(t). cos 4 cos³(t)i + 2 sin³ (t)j + k r(t) = 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Find r’(t) for the function provided in the photo
![Below is the transcription of the provided image, formatted for an educational website.
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**Problem Statement:**
Find \(\mathbf{r}'(t)\).
**Given Vector Function:**
\[
\mathbf{r}(t) = 4 \cos^3(t) \mathbf{i} + 2 \sin^3(t) \mathbf{j} + \mathbf{k}
\]
**Details:**
In the given vector function, \(\mathbf{r}(t)\) is defined as a combination of trigonometric functions of \(t\), and the vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) denote the unit vectors in the x, y, and z directions, respectively. The function comprises terms with \( \cos^3(t) \) and \( \sin^3(t) \), each multiplied by different constants, and a constant unit vector \(\mathbf{k}\) in the z direction.
1. \(4 \cos^3(t) \mathbf{i}\) suggests the x-component of the vector.
2. \(2 \sin^3(t) \mathbf{j}\) suggests the y-component of the vector.
3. \(\mathbf{k}\) suggests the z-component of the vector is constant.
**Objective:**
The task is to find the derivative of the vector function \(\mathbf{r}(t)\).
---
This transcription retains the essential details needed for a student to understand the problem and the given information.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fc33cc1-e822-44c8-a303-0b5fd2ff9f49%2F5a1fa374-f36a-4dc2-b00d-4b8a95085671%2F8srifz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription of the provided image, formatted for an educational website.
---
**Problem Statement:**
Find \(\mathbf{r}'(t)\).
**Given Vector Function:**
\[
\mathbf{r}(t) = 4 \cos^3(t) \mathbf{i} + 2 \sin^3(t) \mathbf{j} + \mathbf{k}
\]
**Details:**
In the given vector function, \(\mathbf{r}(t)\) is defined as a combination of trigonometric functions of \(t\), and the vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) denote the unit vectors in the x, y, and z directions, respectively. The function comprises terms with \( \cos^3(t) \) and \( \sin^3(t) \), each multiplied by different constants, and a constant unit vector \(\mathbf{k}\) in the z direction.
1. \(4 \cos^3(t) \mathbf{i}\) suggests the x-component of the vector.
2. \(2 \sin^3(t) \mathbf{j}\) suggests the y-component of the vector.
3. \(\mathbf{k}\) suggests the z-component of the vector is constant.
**Objective:**
The task is to find the derivative of the vector function \(\mathbf{r}(t)\).
---
This transcription retains the essential details needed for a student to understand the problem and the given information.
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