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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Sketching Vector-Valued Functions
**Topic: Graphical Representation of Vector-Valued Functions**
In this example, we will sketch the plane curve represented by the vector-valued function and identify the orientation of the curve.
The given function is:
\[ \mathbf{r}(t) = \frac{t}{7}\mathbf{i} + (t - 1)\mathbf{j} \]
### Detailed Explanation of Graphs
#### Graph Description:
Four identical graphs are provided, each displaying the curve represented by the vector-valued function \(\mathbf{r}(t) = \frac{t}{7}\mathbf{i} + (t - 1)\mathbf{j}\). Each graph is a linear plot on the \(xy\)-plane with axes labeled as \(x\) and \(y\), ranging from \(-10\) to \(10\).
#### Layout:
- The \(x\)-axis and \(y\)-axis intersect at the origin \((0,0)\).
- Each graph shows a straight line extending across the first and third quadrants, reflecting a positive slope.
- The orientation of the curve is indicated with arrows.
#### Orientation:
- The orientation arrows on each plot point from the lower left side \((-\infty, -\infty)\) to the upper right side \((\infty, \infty)\).
### Understanding the Function
1. **Vector Components**:
- \(\frac{t}{7}\mathbf{i}\): This represents the \(x\)-component of the vector as a function of \(t\), dividing \(t\) by 7.
- \((t - 1)\mathbf{j}\): This represents the \(y\)-component of the vector as a function of \(t\), subtracting 1 from \(t\).
2. **Curve Representation**:
- For different values of \(t\), the function traces out a line in the plane.
- As \(t\) increases, the \(x\)-coordinate increases more slowly (due to division by 7) in comparison to the \(y\)-coordinate.
3. **Graphical Traits**:
- The line on each graph starts below the origin and passes through the origin when \(t = 0\), then continues to the positive side of both axes as \(t\) increases.
- Because the coefficient of \(t\) in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ac392e2-4675-4f88-9d78-8c7af6559e48%2F4a87aabf-6f2d-4b70-852a-c16439b074d2%2Flfrjeoq_processed.png&w=3840&q=75)
Transcribed Image Text:### Sketching Vector-Valued Functions
**Topic: Graphical Representation of Vector-Valued Functions**
In this example, we will sketch the plane curve represented by the vector-valued function and identify the orientation of the curve.
The given function is:
\[ \mathbf{r}(t) = \frac{t}{7}\mathbf{i} + (t - 1)\mathbf{j} \]
### Detailed Explanation of Graphs
#### Graph Description:
Four identical graphs are provided, each displaying the curve represented by the vector-valued function \(\mathbf{r}(t) = \frac{t}{7}\mathbf{i} + (t - 1)\mathbf{j}\). Each graph is a linear plot on the \(xy\)-plane with axes labeled as \(x\) and \(y\), ranging from \(-10\) to \(10\).
#### Layout:
- The \(x\)-axis and \(y\)-axis intersect at the origin \((0,0)\).
- Each graph shows a straight line extending across the first and third quadrants, reflecting a positive slope.
- The orientation of the curve is indicated with arrows.
#### Orientation:
- The orientation arrows on each plot point from the lower left side \((-\infty, -\infty)\) to the upper right side \((\infty, \infty)\).
### Understanding the Function
1. **Vector Components**:
- \(\frac{t}{7}\mathbf{i}\): This represents the \(x\)-component of the vector as a function of \(t\), dividing \(t\) by 7.
- \((t - 1)\mathbf{j}\): This represents the \(y\)-component of the vector as a function of \(t\), subtracting 1 from \(t\).
2. **Curve Representation**:
- For different values of \(t\), the function traces out a line in the plane.
- As \(t\) increases, the \(x\)-coordinate increases more slowly (due to division by 7) in comparison to the \(y\)-coordinate.
3. **Graphical Traits**:
- The line on each graph starts below the origin and passes through the origin when \(t = 0\), then continues to the positive side of both axes as \(t\) increases.
- Because the coefficient of \(t\) in
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