91Ó°ÊÓ

The region between the \(y z\) -plane and the vertical plane \(x=5\)

Use a graphing calculator or computer to graph both the curve and its curvature function \(\kappa(x)\) on the same screen. Is the graph of \(\kappa\) what you would expect? $$y=x^{-2}$$

Plot the space curve and its curvature function \(\kappa(t)\). Comment on how the curvature reflects the shape of the curve. $$\mathbf{r}(t)=\langle t-\sin t, 1-\cos t, 4 \cos (t / 2)\rangle, \quad 0 \leqslant t \leqslant 8 \pi$$

Try to sketch by hand the curve of intersection of the parabolic cylinder \(y=x^{2}\) and the top half of the ellipsoid \(x^{2}+4 y^{2}+4 z^{2}=16 .\) Then find parametric equations for this curve and use these equations and a computer to graph the curve.

Find the tangential and normal components of the acceleration vector. $$\mathbf{r}(t)=t \mathbf{i}+\cos ^{2} t \mathbf{j}+\sin ^{2} t \mathbf{k}$$

(a) Find a nonzero vector orthogonal to the plane through the points \(P, Q,\) and \(R,\) and (b) find the area of triangle \(P Q R .\) $$P(-1,3,1), \quad Q(0,5,2), \quad R(4,3,-1)$$

Three forces act on an object. Two of the forces are at an angle of \(100^{\circ}\) to each other and have magnitudes 25 \(\mathrm{N}\) and 12 \(\mathrm{N} .\) The third is perpendicular to the plane of these two forces and has magnitude 4 \(\mathrm{N.}\) Calculate the magnitude of the force that would exactly counterbalance these three forces.

\(21-32=\) Find an equation of the plane. The plane that passes through the line of intersection of the planes \(x-z=1\) and \(y+2 z=3\) and is perpendicular to the plane \(x+y-2 z=1\)

\(29-32\) Find the scalar and vector projections of b onto a. $$\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{b}=\mathbf{i}-\mathbf{j}+\mathbf{k}$$

Find an equation for the surface consisting of all points \(P\) for which the distance from \(P\) to the \(x\) -axis is twice the distance from \(P\) to the \(y z\) -plane. Identify the surface.