91影视

Given a vector-valued function r(t) with domain $\mathrm{鈩�},$what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.

$\mathbf{r}\left(t\right)=鉄�t-\sin t,1-\cos t鉄�,[0,2蟺]$

For each of the vector-valued functions, find the unit tangent vector.

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

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

Explain why the graph of every vector-valued function $r\left(t\right)=\left(\cos t,\sin t,f\left(t\right)\right)$lies on the surface of the cylinder $x^2+y^2=1$for every continuous functionf.

In Exercises 24鈥�29 a vector function and a point on the graph of the function are given. Find an equation for the line tangent to the curve at the specified point, and then find an equation for the plane orthogonal to the tangent line containing the given point.

Binormal vectors and osculating planes: Find the binormal vector and equation of the osculating plane for the given function at the specified value of t.

In Exercises 24鈥�29 a vector function and a point on the graph of the function are given. Find an equation for the line tangent to the curve at the specified point, and then find an equation for the plane orthogonal to the tangent line containing the given point.

$\text{25.}\mathbf{r}\left(t\right)=t{e}^t\mathbf{i}+t\ln t\mathbf{j},(e,0,0)$

For each of the vector-valued functions in Exercises 22鈥�28, find the unit tangent vector.

$r\left(t\right)=({\cos }^{3}t,{\sin }^{3}t)$

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.

$\mathbf{r}\left(t\right)=\left(4\sin t,t^{3/2},-4\cos t\right),[0,4]$