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Chapter 12: Problem 31
Give a geometric description of the following sets of points. $$x^{2}+y^{2}+z^{2}-2 y-4 z-4=0$$
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Evaluate the following limits. $$\lim _{t \rightarrow \pi / 2}\left(\cos 2 t \mathbf{i}-4 \sin t \mathbf{j}+\frac{2 t}{\pi} \mathbf{k}\right)$$
For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle-1,2,3\rangle, \mathbf{v}=\langle 2,1,1\rangle\)
Find the points (if they exist) at which the following planes and curves intersect. $$y+x=0 ; \mathbf{r}(t)=\langle\cos t, \sin t, t\rangle, \text { for } 0 \leq t \leq 4 \pi$$
Evaluate the following limits. $$\lim _{t \rightarrow \ln 2}\left(2 e^{t} \mathbf{i}+6 e^{-t} \mathbf{j}-4 e^{-2 t} \mathbf{k}\right)$$
An object moves clockwise around a circle centered at the origin with radius \(5 \mathrm{m}\) beginning at the point (0,5) a. Find a position function \(\mathbf{r}\) that describes the motion if the object moves with a constant speed, completing 1 lap every 12 s. b. Find a position function \(\mathbf{r}\) that describes the motion if it occurs with speed \(e^{-t}\)
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