Find r'(t), r"(t), r'(t) · r"(t), and r'(t) × r"(t). P(C) = 21²1 - 48 + ²P²H 2³k r(t) 3 (a) r(t) 3it + 2t²k - 4j (b) r"(t) 4kt + 3i (c) r'(t) r"(t) (d) r(t) x r"(t) X Your answer cannot be understood or graded. More Information
Find r'(t), r"(t), r'(t) · r"(t), and r'(t) × r"(t). P(C) = 21²1 - 48 + ²P²H 2³k r(t) 3 (a) r(t) 3it + 2t²k - 4j (b) r"(t) 4kt + 3i (c) r'(t) r"(t) (d) r(t) x r"(t) X Your answer cannot be understood or graded. More Information
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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### Vector Calculus Problem Set
#### Task: Compute derivatives and perform vector operations
Given the vector function:
\[ \mathbf{r}(t) = \frac{3}{2}t^2\mathbf{i} - 4t\mathbf{j} + \frac{2}{3}t^3\mathbf{k} \]
Find the following:
1. \(\mathbf{r}'(t)\)
2. \(\mathbf{r}''(t)\)
3. \(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)
4. \(\mathbf{r}'(t) \times \mathbf{r}''(t)\)
#### Solutions:
**(a) \(\mathbf{r}'(t)\)**
\[ \boxed{3t\mathbf{i} + 2t^2\mathbf{k} - 4\mathbf{j}} \]
This solution is correct.
**(b) \(\mathbf{r}''(t)\)**
\[ \boxed{4t\mathbf{k} + 3\mathbf{i}} \]
This solution is correct.
**(c) \(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)**
\[ \boxed{} \]
\[ \textcolor{red}{\text{Your answer cannot be understood or graded.}} \]
**(d) \(\mathbf{r}'(t) \times \mathbf{r}''(t)\)**
\[ \boxed{} \]
\[ \textcolor{red}{\text{Your answer cannot be understood or graded.}} \]
---
### Explanation of Graphs and Calculations
- **Part (a):**
- Compute the first derivative \(\mathbf{r}'(t)\) by differentiating each component of \(\mathbf{r}(t)\):
\[ \mathbf{r}'(t) = \frac{d}{dt} \left( \frac{3}{2}t^2 \mathbf{i} - 4t\mathbf{j} + \frac{2}{3}t^3\mathbf{k} \right) \]
\[ = 3t\mathbf{i} - 4\mathbf{j} + 2t^2\mathbf{k} \]
- **Part (b):**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe18dfc87-3c35-485c-a41e-ee4c781c7cac%2F3b5566ad-a0bc-4900-9535-cf3f94b2005d%2F7blp6h_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Vector Calculus Problem Set
#### Task: Compute derivatives and perform vector operations
Given the vector function:
\[ \mathbf{r}(t) = \frac{3}{2}t^2\mathbf{i} - 4t\mathbf{j} + \frac{2}{3}t^3\mathbf{k} \]
Find the following:
1. \(\mathbf{r}'(t)\)
2. \(\mathbf{r}''(t)\)
3. \(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)
4. \(\mathbf{r}'(t) \times \mathbf{r}''(t)\)
#### Solutions:
**(a) \(\mathbf{r}'(t)\)**
\[ \boxed{3t\mathbf{i} + 2t^2\mathbf{k} - 4\mathbf{j}} \]
This solution is correct.
**(b) \(\mathbf{r}''(t)\)**
\[ \boxed{4t\mathbf{k} + 3\mathbf{i}} \]
This solution is correct.
**(c) \(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)**
\[ \boxed{} \]
\[ \textcolor{red}{\text{Your answer cannot be understood or graded.}} \]
**(d) \(\mathbf{r}'(t) \times \mathbf{r}''(t)\)**
\[ \boxed{} \]
\[ \textcolor{red}{\text{Your answer cannot be understood or graded.}} \]
---
### Explanation of Graphs and Calculations
- **Part (a):**
- Compute the first derivative \(\mathbf{r}'(t)\) by differentiating each component of \(\mathbf{r}(t)\):
\[ \mathbf{r}'(t) = \frac{d}{dt} \left( \frac{3}{2}t^2 \mathbf{i} - 4t\mathbf{j} + \frac{2}{3}t^3\mathbf{k} \right) \]
\[ = 3t\mathbf{i} - 4\mathbf{j} + 2t^2\mathbf{k} \]
- **Part (b):**
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