## Calculus (3rd Edition)

$$\frac{4}{17} .$$
Since $r(t) = \lt 4\cos t, t, \sin t\gt$, then $r'(t) = \lt -4\sin t,1,4\cos t\gt$ and hence $\|r'(t)\|=\sqrt{17 }$, $T(t)=\frac{r'(t)}{\|r'(t)\|}=\frac{ \lt -4\sin t,1,4\cos t\gt}{\sqrt{17}}$. Now, the curvature is given by $$\kappa(t)=\frac{1}{\|r'(t)\|}\|\frac{dT}{dt}\|=\frac{1}{\sqrt{17 }}\|\frac{ \lt -4\cos t,0,-4\sin t\gt}{ \sqrt{17 }}\|=\frac{4}{17} .$$