91Ó°ÊÓ

Osculating circles: Find the equation of the osculating circle to the given function at the specified value of t.

Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34

$\text{31.}\mathbf{r}\left(t\right)=t{e}^t\mathbf{i}+t\ln t\mathbf{j}$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=({\cos }^{2}t,4int,t)fort∈[0,2\mathrm{π}]$

Osculating circles: Find the equation of the osculating circle to the given function at the specified value of t.

$\mathbf{r}\left(t\right)=⟨3\sin 2t,3\cos 2tâŸ\copyright ,t=\frac{\mathrm{Ï€}}{3}$

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.

$r\left(t\right)=\left({\cos }^3\left(t\right),{\sin }^3\left(t\right)\right),t=\frac{\mathrm{Ï€}}{4}$

Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34

$32.\mathrm{r}\left(t\right)=⟨\sec t,\frac{1}{t},e^t\ln tâŸ\copyright$

In Exercises 31–35 find the curvature of the given function at the indicated value of x. Then sketch the curve and the osculating circle at the indicated point.

$y=\csc x,x=\frac{\mathrm{Ï€}}{2}$

Osculating circles: Find the equation of the osculating circle to the given function at the specified value of t.

$\mathbf{r}\left(t\right)=⟨\mathrm{Î\pm }\sin \mathrm{β}t,\mathrm{Î\pm }\cos \mathrm{β}tâŸ\copyright ,t=0$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=({\cos }^{2}t,\sin 2t)fort∈[0,2\mathrm{π}]$

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.

$r\left(t\right)=\left(3\sin \left(t\right),5\cos \left(t\right),4\sin \left(t\right)\right),t=\mathrm{Ï€}$