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
![**Problem Statement:**
Given the **vector function** \(\vec{r}(t) = \left\langle 3t^2 - 1, 1 - 8\sqrt{t^3}, 12t - 4 \right\rangle\), find the **speed** and **arc length** of the curve on the interval \(1 \leq t \leq 4\).
---
**Visual Explanation:**
The diagram on the bottom right of the image appears to be a 3D plot showcasing a curve, represented by vector function \(\vec{r}(t)\). The plot includes three axes:
- The x-axis (horizontal) is labeled with values trending from -54 to 24.
- The y-axis (vertical) is labeled on the upper right, with values ranging from -2 to 8.
- The z-axis appears to extend into the plane of the paper, with values going from -42 to 32.
It appears to be a curved line traced by the parametric vector function \(\vec{r}(t)\) within the specified interval \(1 \leq t \leq 4\).
**Computation Steps for Speed and Arc Length:**
1. **Speed (Magnitude of Velocity)**:
- Velocity \(\vec{v}(t)\) is the derivative of \(\vec{r}(t)\):
\[
\vec{v}(t) = \frac{d\vec{r}(t)}{dt} = \left\langle \frac{d}{dt}(3t^2 - 1), \frac{d}{dt}(1 - 8\sqrt{t^3}), \frac{d}{dt}(12t - 4) \right\rangle
\]
- Compute each component:
\[
\vec{v}(t) = \left\langle 6t, -12 t^{\frac{1}{2}}, 12 \right\rangle
\]
- Speed is the magnitude of \(\vec{v}(t)\):
\[
\text{Speed} = \|\vec{v}(t)\| = \sqrt{(6t)^2 + (-12 t^\frac{1}{2})^2 + 12^2}
\]
\[
\text{Speed} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1e27600-0eaa-4d61-9500-f3e4660bc89b%2F8b06a2ca-ac7f-4906-8361-655932d46673%2Fx6n8w8c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the **vector function** \(\vec{r}(t) = \left\langle 3t^2 - 1, 1 - 8\sqrt{t^3}, 12t - 4 \right\rangle\), find the **speed** and **arc length** of the curve on the interval \(1 \leq t \leq 4\).
---
**Visual Explanation:**
The diagram on the bottom right of the image appears to be a 3D plot showcasing a curve, represented by vector function \(\vec{r}(t)\). The plot includes three axes:
- The x-axis (horizontal) is labeled with values trending from -54 to 24.
- The y-axis (vertical) is labeled on the upper right, with values ranging from -2 to 8.
- The z-axis appears to extend into the plane of the paper, with values going from -42 to 32.
It appears to be a curved line traced by the parametric vector function \(\vec{r}(t)\) within the specified interval \(1 \leq t \leq 4\).
**Computation Steps for Speed and Arc Length:**
1. **Speed (Magnitude of Velocity)**:
- Velocity \(\vec{v}(t)\) is the derivative of \(\vec{r}(t)\):
\[
\vec{v}(t) = \frac{d\vec{r}(t)}{dt} = \left\langle \frac{d}{dt}(3t^2 - 1), \frac{d}{dt}(1 - 8\sqrt{t^3}), \frac{d}{dt}(12t - 4) \right\rangle
\]
- Compute each component:
\[
\vec{v}(t) = \left\langle 6t, -12 t^{\frac{1}{2}}, 12 \right\rangle
\]
- Speed is the magnitude of \(\vec{v}(t)\):
\[
\text{Speed} = \|\vec{v}(t)\| = \sqrt{(6t)^2 + (-12 t^\frac{1}{2})^2 + 12^2}
\]
\[
\text{Speed} =
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