91Ó°ÊÓ

Find the length of the parametrized curve. $$ x=5 t, \quad y=4 t^{2}, \quad z=3 t^{2} ; \quad 0 \leq t \leq 2 $$

Exer. \(19-22:\) For the given curve and point \(P,\) /a \(\mid\) find the radius of curvature, (b) find the center of curvature, and (c) sketch the graph and the circle of curvature for \(P\). $$ y=\sin x ; \quad P(\pi / 2,1) $$

Find the \(x\) -coordinates of the points on the graph of \(y=x^{3}-3 x\) at which the curvature is a maximum.

Find the length of the parametrized curve. $$ x=t^{2}, \quad y=t \sin t, \quad z=t \cos t ; \quad 0 \leq t \leq 1 $$

A curve \(C\) is given parametrically. Find parametric equations for the tangent line to \(C\) at \(P\). $$ x=t \sin t, \quad y=t \cos t, \quad z=t ; \quad P(\pi / 2,0, \pi / 2) $$

Exer. \(19-22:\) For the given curve and point \(P,\) /a \(\mid\) find the radius of curvature, (b) find the center of curvature, and (c) sketch the graph and the circle of curvature for \(P\). $$ y=\sec x ; \quad P(0,1) $$

If \(C\) is the graph of \(y=\cosh x,\) find an equation of the circle of curvature for the point \(P(0,1)\)

Exer. \(19-22:\) For the given curve and point \(P,\) /a \(\mid\) find the radius of curvature, (b) find the center of curvature, and (c) sketch the graph and the circle of curvature for \(P\). $$ y=e^{x} ; \quad P(0,1) $$

Find the length of the parametrized curve. $$ x=e^{t} \cos t, \quad y=e^{t}, \quad z=e^{t} \sin t ; \quad 0 \leq t \leq 2 \pi $$

Exer. 21-22: A curve \(C\) is given parametrically. Find two unit tangent vectors to \(C\) at \(P\). $$ x=e^{2 t}, \quad y=e^{-t}, \quad z=t^{2}+4 ; \quad P(1,1,4) $$