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 oft.

Domainlocalid="1649578696830" $t\ge 0,r\left(0\right)=\left(0,3\right),r\left(1\right)=\left(-2,1\right),\underset{t\to \infty }{\lim }x\left(t\right)=-5,\mathrm{and}\underset{\mathrm{t}\to \infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-\infty .$

Short answer

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

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

Step 2. Sketch the graph.

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