Q. 19 - 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 $t\ge 0, r\left(0\right)=\left(0,3\right), r\left(1\right)=\left(-2,1\right), \underset{t\to \infty }{\lim }x\left(t\right)=-5, \mathrm{and} \underset{\mathrm{t}\to \infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-\infty .$

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

Step 2. Sketch the graph.

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