Given the vector function (1) = (e+²³, 2√²³ +1, 4arctan(1 − 1)) 1)), find the speed and the equation of the tangent line to this curve at to = 2, then graph the tangent line. Speed= l(t) =
Given the vector function (1) = (e+²³, 2√²³ +1, 4arctan(1 − 1)) 1)), find the speed and the equation of the tangent line to this curve at to = 2, then graph the tangent line. Speed= l(t) =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
Related questions
Question
Please do the graph in the question as well
![### Problem Statement
Given the vector function:
\[
\vec{r}(t) = \left\langle e^{4-t^2}, \ 2 \sqrt{t^3 + 1}, \ 4 \arctan(t-1) \right\rangle,
\]
find the **speed** and the **equation of the tangent line** to this curve at \( t_0 = 2 \), then **graph** the tangent line.
- **Speed:**
\[
\text{Speed} =
\]
- **Derivative:**
\[
\vec{r}'(t) =
\]
### Graph Description
- The provided graph is a 3D plot illustrating the vector function \(\vec{r}(t)\) within a coordinate system labeled with \(x\), \(y\), and \(z\) axes.
- A curve is plotted in blue, representing the trajectory of \(\vec{r}(t)\).
- The graph showcases the behavior of the function for different values of \(t\), giving a visual representation of how the vector's components \(x\), \(y\), and \(z\) vary with time.
### Instructions:
1. **Compute the speed** of the function which is the magnitude of the derivative \(\|\vec{r}'(t)\|\).
2. **Find the derivative** \(\vec{r}'(t)\) of the vector function \(\vec{r}(t)\).
3. **Determine the equation** of the tangent line at \( t_0 = 2 \).
4. **Graph the tangent line** alongside the vector function to visualize the point of tangency and the direction in which the curve is heading at that instance.
This information can be used to understand the dynamics of parametric curves in three-dimensional space and how to analyze their properties through calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59a94f0e-d22f-4993-9c92-37a006835338%2F32f830b6-3659-415b-8966-de19dbf6fbd5%2F0b8wqwm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given the vector function:
\[
\vec{r}(t) = \left\langle e^{4-t^2}, \ 2 \sqrt{t^3 + 1}, \ 4 \arctan(t-1) \right\rangle,
\]
find the **speed** and the **equation of the tangent line** to this curve at \( t_0 = 2 \), then **graph** the tangent line.
- **Speed:**
\[
\text{Speed} =
\]
- **Derivative:**
\[
\vec{r}'(t) =
\]
### Graph Description
- The provided graph is a 3D plot illustrating the vector function \(\vec{r}(t)\) within a coordinate system labeled with \(x\), \(y\), and \(z\) axes.
- A curve is plotted in blue, representing the trajectory of \(\vec{r}(t)\).
- The graph showcases the behavior of the function for different values of \(t\), giving a visual representation of how the vector's components \(x\), \(y\), and \(z\) vary with time.
### Instructions:
1. **Compute the speed** of the function which is the magnitude of the derivative \(\|\vec{r}'(t)\|\).
2. **Find the derivative** \(\vec{r}'(t)\) of the vector function \(\vec{r}(t)\).
3. **Determine the equation** of the tangent line at \( t_0 = 2 \).
4. **Graph the tangent line** alongside the vector function to visualize the point of tangency and the direction in which the curve is heading at that instance.
This information can be used to understand the dynamics of parametric curves in three-dimensional space and how to analyze their properties through calculus.
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