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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![**Finding the Domain of the Vector Function**
Given the vector function:
\[
\vec{r}(t) = \left< \sqrt{t + 4}, \frac{1}{\sqrt{t + 4}}, \log(11 - t) \right>
\]
We need to determine the domain of \(\vec{r}(t)\).
**Considerations for the Domain:**
1. **Square Root:**
- The expression \(\sqrt{t + 4}\) requires that \(t + 4 \geq 0\).
- Therefore, \(t \geq -4\).
2. **Fraction with Square Root:**
- The expression \(\frac{1}{\sqrt{t + 4}}\) requires that \(\sqrt{t + 4} \neq 0\).
- Therefore, \(t + 4 > 0\) which means \(t > -4\).
3. **Logarithm:**
- The expression \(\log(11 - t)\) requires that \(11 - t > 0\).
- Therefore, \(t < 11\).
**Combining All Conditions:**
Taking into account all these conditions, the domain of \(\vec{r}(t)\) is:
\[
\text{Domain: } \{ t \mid -4 < t < 11 \}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a1263ae-327d-4428-a0aa-137e6771516c%2Fd4c62156-cb68-4310-a394-eaaf109215e2%2F3lhtmk5_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Domain of the Vector Function**
Given the vector function:
\[
\vec{r}(t) = \left< \sqrt{t + 4}, \frac{1}{\sqrt{t + 4}}, \log(11 - t) \right>
\]
We need to determine the domain of \(\vec{r}(t)\).
**Considerations for the Domain:**
1. **Square Root:**
- The expression \(\sqrt{t + 4}\) requires that \(t + 4 \geq 0\).
- Therefore, \(t \geq -4\).
2. **Fraction with Square Root:**
- The expression \(\frac{1}{\sqrt{t + 4}}\) requires that \(\sqrt{t + 4} \neq 0\).
- Therefore, \(t + 4 > 0\) which means \(t > -4\).
3. **Logarithm:**
- The expression \(\log(11 - t)\) requires that \(11 - t > 0\).
- Therefore, \(t < 11\).
**Combining All Conditions:**
Taking into account all these conditions, the domain of \(\vec{r}(t)\) is:
\[
\text{Domain: } \{ t \mid -4 < t < 11 \}
\]
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