Question

Find the curvature of each of the vector-valued functions defined in Exercises 39–44.

$\mathit{r}\left(t\right)=\left(t\sin t,t\cos t,2t\right)$

Short answer

The curvature of the given vector-valued function is$\kappa =\sqrt{\frac{t^4+8t^2+20}{{\left(t^2+5\right)}^3}}.$

Step-by-step solution

Step 1. Given Information. 

The given vector-valued function is$\mathit{r}\left(t\right)=\left(t\sin t,t\cos t,2t\right).$

Step 2. Find the curvature. 

To find the curvature of the given vector-valued function, we will use the formula for the Curvature of a Space Curve$\kappa =\frac{||r\text{'}\left(t\right)\times r\text{'}\text{'}\left(t\right)||}{{||r\text{'}\left(t\right)||}^3}.$

So,

Step 3. Calculate.  

Put the values of (a) and (b) in the formula of Curvature of a space curve,

$$\begin{gathered} \kappa =\frac{\sqrt{t^4+8t^2+20}}{{\left(\sqrt{t^2+5}\right)}^3} \\ \kappa =\sqrt{\frac{t^4+8t^2+20}{{\left(t^2+5\right)}^3}} \end{gathered}$$