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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Question
Answer only D please, I have a,b,c already
![To understand the mathematical concept of vector calculus, the task is to find the first and second derivatives of a given vector function, as well as perform specific vector operations. Here is the step-by-step guide:
### Given:
\[ \mathbf{r}(t) = \frac{5}{2}t^2\mathbf{i} - 5t\mathbf{j} + \frac{2}{3}t^3\mathbf{k} \]
### Tasks:
1. **Find** \(\mathbf{r}'(t)\)
2. **Find** \(\mathbf{r}''(t)\)
3. **Compute** \(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)
4. **Compute** \(\mathbf{r}'(t) \times \mathbf{r}''(t)\)
### Solutions:
(a) **\(\mathbf{r}'(t)\)** (First Derivative):
- Differentiate each component with respect to \(t\):
- \( \frac{d}{dt} \left(\frac{5}{2}t^2\right) = 5t \)
- \( \frac{d}{dt} (-5t) = -5 \)
- \( \frac{d}{dt} \left(\frac{2}{3}t^3\right) = 2t^2 \)
- Thus, \(\mathbf{r}'(t) = 5t\mathbf{i} - 5\mathbf{j} + 2t^2\mathbf{k}\).
(b) **\(\mathbf{r}''(t)\)** (Second Derivative):
- Differentiate \(\mathbf{r}'(t)\) with respect to \(t\):
- \( \frac{d}{dt} (5t) = 5 \)
- \( \frac{d}{dt} (-5) = 0 \)
- \( \frac{d}{dt} (2t^2) = 4t \)
- Thus, \(\mathbf{r}''(t) = 5\mathbf{i} + 0\mathbf{j} + 4t\mathbf{k}\).
(c) **\(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)** (Dot Product):
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3fcbcc6-ad88-4808-a051-b3d90496ccaa%2Fc91ff392-9f46-4cba-8cc7-eab9c77ef5af%2F5wdp9x_processed.png&w=3840&q=75)
Transcribed Image Text:To understand the mathematical concept of vector calculus, the task is to find the first and second derivatives of a given vector function, as well as perform specific vector operations. Here is the step-by-step guide:
### Given:
\[ \mathbf{r}(t) = \frac{5}{2}t^2\mathbf{i} - 5t\mathbf{j} + \frac{2}{3}t^3\mathbf{k} \]
### Tasks:
1. **Find** \(\mathbf{r}'(t)\)
2. **Find** \(\mathbf{r}''(t)\)
3. **Compute** \(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)
4. **Compute** \(\mathbf{r}'(t) \times \mathbf{r}''(t)\)
### Solutions:
(a) **\(\mathbf{r}'(t)\)** (First Derivative):
- Differentiate each component with respect to \(t\):
- \( \frac{d}{dt} \left(\frac{5}{2}t^2\right) = 5t \)
- \( \frac{d}{dt} (-5t) = -5 \)
- \( \frac{d}{dt} \left(\frac{2}{3}t^3\right) = 2t^2 \)
- Thus, \(\mathbf{r}'(t) = 5t\mathbf{i} - 5\mathbf{j} + 2t^2\mathbf{k}\).
(b) **\(\mathbf{r}''(t)\)** (Second Derivative):
- Differentiate \(\mathbf{r}'(t)\) with respect to \(t\):
- \( \frac{d}{dt} (5t) = 5 \)
- \( \frac{d}{dt} (-5) = 0 \)
- \( \frac{d}{dt} (2t^2) = 4t \)
- Thus, \(\mathbf{r}''(t) = 5\mathbf{i} + 0\mathbf{j} + 4t\mathbf{k}\).
(c) **\(\mathbf{r}'(t) \cdot \mathbf{r}''(t)\)** (Dot Product):
-
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