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Q. 21

Expert-verified
Found in: Page 860

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

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

See the step by step solution

## 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 $t\in \mathrm{ℝ},r\left(2\right)=\left(1,1\right),r\left(0\right)=\left(-3,3\right),r\left(-2\right)=\left(-5,-5\right)\underset{t\to -\infty }{\mathrm{lim}}x\left(t\right)=\infty ,$

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

## Step 2. Sketch the graph.

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