91影视

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.

$\mathbf{r}\left(t\right)=鉄�t,伪\sin 尾t,伪\cos 尾t鉄�,t=0$

$r\left(t\right)=\left(t,t^2\right)$

$r\left(t\right)=鉄�\sin 3t,\cos 4t鉄�$

In Exercises 19鈥�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" $t鈮�0,r\left(0\right)=\left(0,3\right),r\left(1\right)=\left(-2,1\right),\underset{t鈫�鈭�}{\lim }x\left(t\right)=-5,\mathrm{and}\underset{\mathrm{t}鈫�鈭�}{\lim }\mathrm{y}\left(\mathrm{t}\right)=-鈭�.$

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.

$$\begin{gathered} \mathrm{What}\mathrm{is}\mathrm{the}\mathrm{relationship}\mathrm{between}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{a}\mathrm{differentiable}\mathrm{vector}\mathrm{function} \\ \mathrm{r}\left(\mathrm{t}\right)\mathrm{and}\mathrm{the}\mathrm{graph}\mathrm{of}鈭�\mathrm{r}\text{'}\left(\mathrm{t}\right)\mathrm{dt},\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{antiderivatives}\mathrm{of}\mathrm{r}\text{'}\left(\mathrm{t}\right)? \end{gathered}$$

Parametric equations for the unit circle: Find parametric equations for the unit circle centered at the origin of the xy-plane that satisfy the given conditions.

The graph is traced counterclockwise once on the interval[0, 2蟺] starting at the point (1, 0).

A decomposition of the acceleration vector: Find ${\mathrm{comp}}_{\mathbf{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}}\mathbf{a}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}$ wherev and a are the velocity and acceleration vectors, respectively, of the following functions.

Equation of a plane: Find the equation of the plane determined by the vectors (1, 鈭�3, 5)and (2, 4, 鈭�1) and containing the point(0, 3, 鈭�2).

$$\begin{gathered} \mathrm{Give}\mathrm{precise}\mathrm{mathematical}\mathrm{definitions}\mathrm{or}\mathrm{descriptions}\mathrm{of}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{concepts}\mathrm{that}\mathrm{follow}. \\ \mathrm{Then}\mathrm{illustrate}\mathrm{the}\mathrm{definition}\mathrm{with}\mathrm{a}\mathrm{graph}\mathrm{or}\mathrm{an}\mathrm{algebraic}\mathrm{example}. \\ \mathrm{a}\mathrm{vector}\mathrm{function}\mathrm{or}\mathrm{vector}-\mathrm{valued}\mathrm{function}\mathrm{in}{\mathrm{鈩�}}^2\mathrm{and}{\mathrm{鈩�}}^3 \end{gathered}$$