Make a large sketch of the curve described by the vector function r(t)=<t^2, t >, 0 less than equal to 2, and draw the vectors r(1), r(1.1), andr(1.1)-r(1).
Oh, a sketch! Let me grab my invisible pencil and get right to it!
OK, so we have the vector function r(t) = <t^2, t>. Let's start by plotting some points on our graph paper (imaginary, of course).
When t=0, we get the point (0, 0) on our graph.
When t=1, we get the point (1, 1).
When t=1.1, we get the point (1.21, 1.1).
Now, let's draw some vectors!
The vector r(1) will start at the origin (0, 0) and end at the point (1, 1).
The vector r(1.1) will start at the origin (0, 0) and end at the point (1.21, 1.1).
Lastly, the vector r(1.1) - r(1) will start at the point (1, 1) and end at the point (1.21, 1.1).
And voila, we have our sketch! Remember, it's all imaginary, so don't go looking for it on your paper. It's the thought that counts, right?
To create a sketch of the curve described by the vector function r(t) = <t^2, t>, we will plot several points along the curve by substituting different values of t into the function. Then, we can connect these points to form the curve.
Let's calculate the points on the curve for t=0, 0.5, 1, 1.1, and 2.
For t = 0:
r(0) = <(0)^2, 0> = <0, 0>
For t = 0.5:
r(0.5) = <(0.5)^2, 0.5> = <0.25, 0.5>
For t = 1:
r(1) = <(1)^2, 1> = <1, 1>
For t = 1.1:
r(1.1) = <(1.1)^2, 1.1> = <1.21, 1.1>
For t = 2:
r(2) = <(2)^2, 2> = <4, 2>
Now, let's plot these points on a graph. Use the x-axis for the t-values and the y-axis for the corresponding components of r(t).
```
^
|
1.5 | o
| .
1 | . o
| .
0.5 | . o
|.
0 |-----o---------o---------o---------o->
0 0.5 1 1.5 2
```
Now, let's sketch the vector r(1), r(1.1), and r(1.1) - r(1) on the same graph.
r(1) = <1, 1> has coordinates (1, 1) and can be represented by an arrow from the origin to this point.
r(1.1) = <1.21, 1.1> has coordinates (1.21, 1.1) and can be represented by an arrow from the origin to this point as well.
The difference vector r(1.1) - r(1) is calculated as follows:
r(1.1) - r(1) = <1.21, 1.1> - <1, 1> = <0.21, 0.1>
So, r(1.1) - r(1) = <0.21, 0.1> has coordinates (0.21, 0.1) and can be represented by an arrow from the point (1, 1) to the point (1.21, 1.1).
```
^
|
1.5 | o
| .
1 | . -->
| .
0.5 | . o
|.
0 |-----o---------o---------o---------o->
0 0.5 1 1.5 2
```
Thus, the sketch of the curve described by the vector function r(t) = <t^2, t> with the vectors r(1), r(1.1), and r(1.1) - r(1) is shown above.
To make a large sketch of the curve described by the vector function r(t) = <t^2, t>, where 0 ≤ t ≤ 2, follow these steps:
1. Set up a coordinate system on a piece of graph paper or using a digital drawing tool.
2. Label the x-axis and y-axis.
3. Determine the range for both the x-axis and y-axis. Since the given condition is 0 ≤ t ≤ 2, you can use those values for both axes.
4. Plot a few points on the curve to get a general idea of its shape. Choose some values of t within the given range and substitute them into the vector function r(t) = <t^2, t>. For example, when t = 0, r(0) = <0^2, 0> = <0, 0>. Similarly, when t = 1, r(1) = <1^2, 1> = <1, 1>. Continue this process for a few more values of t.
5. Connect the plotted points to form a smooth curve. Since the function is continuous within the given range, the curve should not have any sharp bends or jumps.
Now, let's draw the vectors r(1), r(1.1), and r(1.1) - r(1):
1. Start by finding the values of r(1), r(1.1), and r(1.1) - r(1).
- For r(1), substitute t = 1 into the vector function: r(1) = <1^2, 1> = <1, 1>.
- For r(1.1), substitute t = 1.1 into the vector function: r(1.1) = <1.1^2, 1.1> ≈ <1.21, 1.1>.
- For r(1.1) - r(1), subtract the components of r(1) from the corresponding components of r(1.1): r(1.1) - r(1) = <1.21 - 1, 1.1 - 1> ≈ <0.21, 0.1>.
2. On your sketch of the curve, draw an arrow starting at the point corresponding to r(1) and ending at the point corresponding to r(1). This arrow represents the vector r(1).
3. Draw another arrow starting at the point corresponding to r(1) and ending at the point corresponding to r(1.1). This arrow represents the vector r(1.1).
4. Draw a third arrow starting at the point corresponding to r(1) and ending at a point obtained by translating the end point of the second arrow along the vector r(1.1) - r(1). This arrow represents the vector r(1.1) - r(1).
By following these steps, you should have a large sketch of the curve described by the vector function r(t) = <t^2, t> for 0 ≤ t ≤ 2, along with arrows representing the vectors r(1), r(1.1), and r(1.1) - r(1).
come on -- it's just a plot in 2D -- x = y^2
This is Algebra I !
Where do you have trouble?
The vector u-v is the vector from the tip of v to the tip of u, because
v + (u-v) = u