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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![**Problem 4**
**a.** Find the arc length for the curve \( \mathbf{r}(t) = (\frac{t^2}{2}, \sqrt{2 \cdot t}, \ln(t)) \) for \( 1 \leq t \leq 5 \).
**b.** Find the curvature at the general point. The curvature \( K(t) \) is given by:
\[ K(t) = \frac{||\mathbf{r}'(t) \times \mathbf{r}''(t)||}{||\mathbf{r}'(t)||^3} \]
Note: The curve \( \mathbf{r}(t) \) is provided as a vector function with three components:
- The first component is \( \frac{t^2}{2} \)
- The second component is \( \sqrt{2 \cdot t} \)
- The third component is \( \ln(t) \)
To solve this problem, you need to:
1. Compute the derivatives \( \mathbf{r}'(t) \) and \( \mathbf{r}''(t) \)
2. Use the arc length formula:
\[ L = \int_{a}^{b} || \mathbf{r}'(t) || \, dt \]
3. Calculate the curvature \( K(t) \) using the provided formula.
This problem requires a solid understanding of vector calculus, specifically the concepts of arc length and curvature for vector-valued functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c88e3a7-d6d9-4180-9e78-6a7df1a2f887%2Fb95bd814-8797-443b-a3ff-6ff70638d7a2%2F23h3q9f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4**
**a.** Find the arc length for the curve \( \mathbf{r}(t) = (\frac{t^2}{2}, \sqrt{2 \cdot t}, \ln(t)) \) for \( 1 \leq t \leq 5 \).
**b.** Find the curvature at the general point. The curvature \( K(t) \) is given by:
\[ K(t) = \frac{||\mathbf{r}'(t) \times \mathbf{r}''(t)||}{||\mathbf{r}'(t)||^3} \]
Note: The curve \( \mathbf{r}(t) \) is provided as a vector function with three components:
- The first component is \( \frac{t^2}{2} \)
- The second component is \( \sqrt{2 \cdot t} \)
- The third component is \( \ln(t) \)
To solve this problem, you need to:
1. Compute the derivatives \( \mathbf{r}'(t) \) and \( \mathbf{r}''(t) \)
2. Use the arc length formula:
\[ L = \int_{a}^{b} || \mathbf{r}'(t) || \, dt \]
3. Calculate the curvature \( K(t) \) using the provided formula.
This problem requires a solid understanding of vector calculus, specifically the concepts of arc length and curvature for vector-valued functions.
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