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Find r'(t), r"(t), r'(t) · r"(t), and r'(t) × r"(t). r(t) = 5t²i - 5tj + 3t³k 2 2 (a) r(t) (b) r"(t) (c) r'(t) · r"(t) (d) r'(t) x r"(t)

Find r'(t), r"(t), r'(t) · r"(t), and r'(t) × r"(t). r(t) = 5t²i - 5tj + 3t³k 2 2 (a) r(t) (b) r"(t) (c) r'(t) · r"(t) (d) r'(t) x r"(t)

Calculus: Early Transcendentals
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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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): -
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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