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 Statement
Find the speed over the path \( \mathbf{r}(t) = \langle \cosh(t), \cosh(t), 9t \rangle \) at \( t = 2 \).
(Use decimal notation. Give your answer to four decimal places.)
### Given Answer
\[ v(2) \approx 10.3630 \]
**Status:** Incorrect
---
### Explanation of the Problem
You are given a parametric path described by the vector function \( \mathbf{r}(t) \) with components involving the hyperbolic cosine function \( \cosh(t) \) and a linear term \( 9t \). The goal is to find the speed at a specific time, \( t = 2 \).
### Solution Outline
1. **Compute the derivative \( \mathbf{r}'(t) \)**:
\[
\mathbf{r}'(t) = \left\langle \frac{d}{dt} \cosh(t), \frac{d}{dt} \cosh(t), \frac{d}{dt} 9t \right\rangle
\]
2. **Evaluate the derivative**:
\[
\mathbf{r}'(t) = \langle \sinh(t), \sinh(t), 9 \rangle
\]
3. **Find the magnitude of the derivative (speed)**:
\[
v(t) = \|\mathbf{r}'(t)\| = \sqrt{(\sinh(t))^2 + (\sinh(t))^2 + 9^2}
\]
Simplify using \(\sinh^2(t) + \sinh^2(t)\):
\[
v(t) = \sqrt{2\sinh^2(t) + 81}
\]
4. **Substitute \( t = 2 \)**:
\[
v(2) = \sqrt{2\sinh^2(2) + 81}
\]
Compute \( \sinh(2) \):
\[
\sinh(2) \approx 3.6269
\]
Square the value:
\[
\sinh^2(2) \approx 13.1577
\]
5. **Final Calculation**:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d5d9e19-d852-4310-9ba2-2b2974e3f2aa%2F3251b688-5437-44b9-a882-2d752d6a01cd%2Fa8jlo1_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the speed over the path \( \mathbf{r}(t) = \langle \cosh(t), \cosh(t), 9t \rangle \) at \( t = 2 \).
(Use decimal notation. Give your answer to four decimal places.)
### Given Answer
\[ v(2) \approx 10.3630 \]
**Status:** Incorrect
---
### Explanation of the Problem
You are given a parametric path described by the vector function \( \mathbf{r}(t) \) with components involving the hyperbolic cosine function \( \cosh(t) \) and a linear term \( 9t \). The goal is to find the speed at a specific time, \( t = 2 \).
### Solution Outline
1. **Compute the derivative \( \mathbf{r}'(t) \)**:
\[
\mathbf{r}'(t) = \left\langle \frac{d}{dt} \cosh(t), \frac{d}{dt} \cosh(t), \frac{d}{dt} 9t \right\rangle
\]
2. **Evaluate the derivative**:
\[
\mathbf{r}'(t) = \langle \sinh(t), \sinh(t), 9 \rangle
\]
3. **Find the magnitude of the derivative (speed)**:
\[
v(t) = \|\mathbf{r}'(t)\| = \sqrt{(\sinh(t))^2 + (\sinh(t))^2 + 9^2}
\]
Simplify using \(\sinh^2(t) + \sinh^2(t)\):
\[
v(t) = \sqrt{2\sinh^2(t) + 81}
\]
4. **Substitute \( t = 2 \)**:
\[
v(2) = \sqrt{2\sinh^2(2) + 81}
\]
Compute \( \sinh(2) \):
\[
\sinh(2) \approx 3.6269
\]
Square the value:
\[
\sinh^2(2) \approx 13.1577
\]
5. **Final Calculation**:
\[
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