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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New calculation please
![**Vector Equation of a Tangent Line**
Consider the vector function:
\[ \vec{r}(t) = 2 \left( e^{1/t} \right) \ln t \, \hat{i} + \tan^{-1} t \, \hat{j} + \frac{t^3}{\cos(\pi t)} \, \hat{k} \]
We are asked to find the vector equation of the tangent line to \(\vec{r}(t)\) at the point where \( t = 1 \).
**Steps to find the tangent line:**
1. **Compute \(\vec{r}(1)\):**
\[ \vec{r}(1) = 2 \left( e^{1/1} \right) \ln 1 \, \hat{i} + \tan^{-1} 1 \, \hat{j} + \frac{1^3}{\cos(\pi \cdot 1)} \, \hat{k} \]
Simplify each component:
\[ \ln 1 = 0 \]
\[ \tan^{-1} 1 = \frac{\pi}{4} \]
\[ \cos(\pi) = -1 \]
Thus:
\[ \vec{r}(1) = 0 \, \hat{i} + \frac{\pi}{4} \, \hat{j} - 1 \, \hat{k} \]
\[ \vec{r}(1) = \frac{\pi}{4} \, \hat{j} - 1 \, \hat{k} \]
2. **Find the derivative \(\vec{r}'(t)\):**
\[ \vec{r}'(t) = \frac{d}{dt} \left[ 2 \left( e^{1/t} \right) \ln t \, \hat{i} + \tan^{-1} t \, \hat{j} + \frac{t^3}{\cos(\pi t)} \, \hat{k} \right] \]
Differentiate each component:
- For the \(\hat{i}\) component:
\[ \frac{d}{dt} \left( 2 e^{1/t} \ln t \right) = 2 \left( \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4bda451-1bcc-473c-9a91-77116df85d00%2Fa20f424b-3b7e-43ea-98da-53177da84805%2Fftdyvak_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector Equation of a Tangent Line**
Consider the vector function:
\[ \vec{r}(t) = 2 \left( e^{1/t} \right) \ln t \, \hat{i} + \tan^{-1} t \, \hat{j} + \frac{t^3}{\cos(\pi t)} \, \hat{k} \]
We are asked to find the vector equation of the tangent line to \(\vec{r}(t)\) at the point where \( t = 1 \).
**Steps to find the tangent line:**
1. **Compute \(\vec{r}(1)\):**
\[ \vec{r}(1) = 2 \left( e^{1/1} \right) \ln 1 \, \hat{i} + \tan^{-1} 1 \, \hat{j} + \frac{1^3}{\cos(\pi \cdot 1)} \, \hat{k} \]
Simplify each component:
\[ \ln 1 = 0 \]
\[ \tan^{-1} 1 = \frac{\pi}{4} \]
\[ \cos(\pi) = -1 \]
Thus:
\[ \vec{r}(1) = 0 \, \hat{i} + \frac{\pi}{4} \, \hat{j} - 1 \, \hat{k} \]
\[ \vec{r}(1) = \frac{\pi}{4} \, \hat{j} - 1 \, \hat{k} \]
2. **Find the derivative \(\vec{r}'(t)\):**
\[ \vec{r}'(t) = \frac{d}{dt} \left[ 2 \left( e^{1/t} \right) \ln t \, \hat{i} + \tan^{-1} t \, \hat{j} + \frac{t^3}{\cos(\pi t)} \, \hat{k} \right] \]
Differentiate each component:
- For the \(\hat{i}\) component:
\[ \frac{d}{dt} \left( 2 e^{1/t} \ln t \right) = 2 \left( \frac
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