Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.3: Maxima And Minima
Problem 19E
Related questions
Question
![**Problem Statement:**
Determine \( \mathbf{r}(t) \) given the following conditions:
\[ \mathbf{r}'(t) = t^7 \mathbf{i} + e^t \mathbf{j} + 3te^{3t} \mathbf{k} \]
and
\[ \mathbf{r}(0) = \mathbf{i} + \mathbf{j} + \mathbf{k}. \]
**Solution:**
\[ \mathbf{r}(t) = \boxed{} \]
**Instructions:**
1. **Identify the components:**
- The problem involves finding the vector function \( \mathbf{r}(t) \) given its derivative and an initial condition at \( t=0 \).
2. **Integration per component:**
- \( \mathbf{r}(t) \) is broken down into three functions: \( \mathbf{r}_x(t), \mathbf{r}_y(t), \mathbf{r}_z(t) \).
- Integrate each component of \( \mathbf{r}'(t) \) separately.
3. **Add constants:**
- Use the initial condition to determine the constants of integration.
**Detailed Explanation of Graphs/Diagrams (if any):**
(No graphs or diagrams are present in the image; only mathematical text.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2f5b0ad-df1d-48c6-a9a3-5338b8167143%2Fb2985b86-a271-4e40-b8e8-e765b9da40ad%2Fl9kl35d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine \( \mathbf{r}(t) \) given the following conditions:
\[ \mathbf{r}'(t) = t^7 \mathbf{i} + e^t \mathbf{j} + 3te^{3t} \mathbf{k} \]
and
\[ \mathbf{r}(0) = \mathbf{i} + \mathbf{j} + \mathbf{k}. \]
**Solution:**
\[ \mathbf{r}(t) = \boxed{} \]
**Instructions:**
1. **Identify the components:**
- The problem involves finding the vector function \( \mathbf{r}(t) \) given its derivative and an initial condition at \( t=0 \).
2. **Integration per component:**
- \( \mathbf{r}(t) \) is broken down into three functions: \( \mathbf{r}_x(t), \mathbf{r}_y(t), \mathbf{r}_z(t) \).
- Integrate each component of \( \mathbf{r}'(t) \) separately.
3. **Add constants:**
- Use the initial condition to determine the constants of integration.
**Detailed Explanation of Graphs/Diagrams (if any):**
(No graphs or diagrams are present in the image; only mathematical text.)
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