Q. 21 - Calculus

Question

In Exercises 19–21 sketch the graph of a vector-valued function $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right)$ with the specified properties. Be sure to indicate the direction of increasing values of t.

Domain

 $$\begin{gathered} t\in \mathrm{ℝ}, r\left(2\right)=\left(1,1\right), r\left(0\right)=\left(-3,3\right), r\left(-2\right)=\left(-5,-5\right) \underset{t\to -\infty }{\lim }x\left(t\right)=\infty , \\ \underset{\mathrm{t}\to -\infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-\infty , \mathrm{and} \underset{t\to \infty }{\lim }\mathrm{r}\left(\mathrm{t}\right)=\left(0,0\right). \end{gathered}$$

Step-by-Step Solution

Answer

The graph of the given vector-valued function with the specified properties is

Step 1. Given Information.

The given vector-valued function is $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right).$ The given specified properties are $t\in \mathrm{ℝ}, r\left(2\right)=\left(1,1\right), r\left(0\right)=\left(-3,3\right), r\left(-2\right)=\left(-5,-5\right) \underset{t\to -\infty }{\lim }x\left(t\right)=\infty ,$

$\underset{\mathrm{t}\to -\infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-\infty , \mathrm{and} \underset{t\to \infty }{\lim }\mathrm{r}\left(\mathrm{t}\right)=\left(0,0\right).$

Step 2. Sketch the graph.

The graph of the given vector-valued function with the specified properties is