91Ó°ÊÓ

Eliminate the parameter \(t,\) write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. (Hint: Let \(x=2 t\) and \(y=t^{2} .\) Solve the first equation for \(x\) in terms of \(t\) and substitute this result into the second equation.) $$ \mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j} $$

Eliminate the parameter \(t\), write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. (Hint: Let \(x=2 t\) and \(y=t^{2} .\) Solve the first equation for \(x\) in terms of \(t\) and substitute this result into the second equation.) \(\mathbf{r}(t)=2(\sinh t) \mathbf{i}+2(\cosh t) \mathbf{j}, t>0\)

Use a graphing utility to sketch each of the following vector-valued functions: $$ \mathbf{r}(t)=2 \cos t^{2} \mathbf{i}+(2-\sqrt{t}) \mathbf{j} $$

Use a graphing utility to sketch each of the following vector-valued functions: $$ \mathbf{r}(t)=\left\langle e^{\cos (3 t)}, e^{-\sin (t)}\right\rangle $$

Use a graphing utility to sketch each of the following vector-valued functions: $$ \mathbf{r}(t)=\langle 2-\sin (2 t), 3+2 \cos t\rangle $$

Use a graphing utility to sketch each of the following vector-valued functions: \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle ;\) from left to right

The line through \(P\) and \(Q\) where \(P\) is \((1,4,-2)\) and \(Q\) is \((3,9,6)\)

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\) What is the initial point of the path corresponding to \(\mathbf{r}(0) ?\)

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\) What is \(\lim _{t \rightarrow \infty} \mathbf{r}(t) ?\)

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\) Use technology to sketch the curve.