Problem 68
Find a curve given by a polynomial \(P_{5}(x)\) that provides a smooth transition between \(y=0\) for \(x \leq 0\) and \(y=x\) for \(x \geq 1\)
Problem 68
Find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P .\) $$ \mathbf{n}=\langle 0,0,1\rangle ; P(1,2,-3) $$
Problem 70
Find the equation of the plane through \((-1,2,-3)\) and parallel to the plane \(2 x+4 y-z=6\).
Problem 71
Find the equation of the plane passing through \((-4,-1,2)\) and parallel to (a) the \(x y\) -plane (b) the plane \(2 x-3 y-4 z=0\)
Problem 73
Find the distance from \((1,-1,2)\) to the plane \(x+3 y+z=7\).
Problem 74
Find the distance from \((2,6,3)\) to the plane \(-3 x+2 y+z=9\).
Problem 75
Find the distance between the parallel planes \(-3 x+2 y+z=9\) and \(6 x-4 y-2 z=19\).
Problem 76
. Show that the curvature of the polar curve \(r=e^{6 \theta}\) is proportional to \(1 / r\).
Problem 76
Find the distance between the parallel planes \(5 x-3 y-2 z=5\) and \(-5 x+3 y+2 z=7\).
Problem 77
Show that the curvature of the polar curve \(r^{2}=\cos 2 \theta\) is directly proportional to \(r\) for \(r>0\).
