91影视

Given a differentiable vector-valued function $r\left(t\right)$, what is the relationship between $r\text{'}\left(t_0\right)$and $T\left(t_0\right)$at a point$t_0$in the domain of $r\left(t\right)$?

Given that an object is moving along the graph of a vector function r(t) defined on an interval [a,b], what is meant by the displacement of the object from t = a to t = b?

Finding limits: Find the given limits if they exist. If a limit does not exist, explain why.

What makes Definition 11.21 for curvature easy to understand? What makes it difficult to use?

For constants $伪,尾$, and $纬$, the graph of a vector-valued function of the form

$\mathbf{r}\left(t\right)=伪e^{纬t}\left(\cos t\right)\mathbf{i}+伪e^{纬t}\left(\sin t\right)\mathbf{j}+尾e^{纬t}\mathbf{k},t鈮�0$

Prove that if a particle moves along a curve at a constant speed, then the velocity and acceleration vectors are orthogonal.

If an object is moving along the graph of a vector function r(t) defined on an interval [a,b], how can the distance the object travels from t = a to t = b be calculated?

Most of the parametric equations and vector-valued functions we have studied have component functions that are continuous. What happens when one of the component functions is discontinuous at a point? For example, the 鈥渇loor鈥� function $z\left(t\right)=鈱�t鈱�$has a jump discontinuity for every integer $t$. What is the graph of the equations $x=\cos 2蟺t,y=\sin 2蟺t,z=t,t鈭�R$?

$$\begin{gathered} \mathrm{Give}\mathrm{precise}\mathrm{mathematical}\mathrm{definitions}\mathrm{or}\mathrm{descriptions}\mathrm{of}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{concepts}\mathrm{that}\mathrm{follow}. \\ \mathrm{Then}\mathrm{illustrate}\mathrm{the}\mathrm{definition}\mathrm{with}\mathrm{a}\mathrm{graph}\mathrm{or}\mathrm{an}\mathrm{algebraic}\mathrm{example}. \\ \mathrm{the}\mathrm{unit}\mathrm{tangent}\mathrm{function}\mathrm{for}\mathrm{a}\mathrm{differentiable}\mathrm{vector}\mathrm{function}\mathrm{r}\left(\mathrm{t}\right) \end{gathered}$$