91影视

$$\begin{gathered} \mathrm{Find}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{the}\mathrm{vector}-\mathrm{valued}\mathrm{function}\mathrm{r}\left(\mathrm{t}\right)={\mathrm{t}}^2\mathrm{i}鈭�6{\mathrm{t}}^2\mathrm{j},\mathrm{t}>0, \\ \mathrm{and}\mathrm{show}\mathrm{that}\mathrm{the}\mathrm{derivative}\mathrm{at}\mathrm{any}\mathrm{point}{\mathrm{t}}_0\mathrm{is}\mathrm{a}\mathrm{scalar}\mathrm{multiple}\mathrm{of}\mathrm{the}\mathrm{derivative} \\ \mathrm{at}\mathrm{any}\mathrm{other}\mathrm{point}.\mathrm{What}\mathrm{does}\mathrm{this}\mathrm{property}\mathrm{say}\mathrm{about}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{r}?\mathrm{Use}\mathrm{that} \\ \mathrm{information}\mathrm{to}\mathrm{sketch}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{r}. \end{gathered}$$

Given a twice-differentiable vector-valued function $\mathrm{r}\left(\mathrm{t}\right)$and a point $t_0$in its domain, what is the osculating plane at $\mathit{r}\left(t_0\right)$?

Show that the curvature of the function of $y=\sqrt{1鈭�x^2},x鈭�(鈭�1,1)$, is constant, but its second derivative varies with x.

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

$\mathbf{r}\left(t\right)=鉄�t,5\sin 3t,5\cos 3t鉄�,t=\frac{蟺}{6}$

Carefully outline the steps you would use to find the equation of the osculating plane at a point $r\left(t_0\right)$on the graph of a vector function.

What is the dot product of the vector functions $r_1\left(t\right)=\left(x_1\left(t\right),y_1\left(t\right)\right)\mathrm{and}{r}_2\left(t\right)=\left(x_2\left(t\right),y_2\left(t\right)\right)?$

Find

(a) the displacement vectors from r(a) tor(b),

(b) the magnitude of the displacement vector, and

(c) the distance travelled by a particle on the curve from a to b.

r(t) = 伪 sin尾t,伪 cos尾t,纬t, a = 0, b = 1

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

$\mathbf{r}\left(t\right)=鉄�t,2t\sin t,2t\cos t鉄�,t=蟺$

Compute the cross product of the vector functions $r_1\left(t\right)=\left(x_1\left(t\right),y_1\left(t\right)\right)\mathrm{and}\left(r_2\left(t\right)=x_2\left(t\right),y_2\left(t\right)\right)$by thinking of ${\mathrm{鈩�}}^2$ as the xy-plane in ${\mathrm{鈩�}}^3.$That is, let $r_1\left(t\right)=\left(x_1\left(t\right),y_1\left(t\right),0\right)\mathrm{and}{r}_2\left(t\right)=\left(x_2\left(t\right),y_2\left(t\right),0\right),$and take the cross product of these vector functions.