Find the curvature K of the curve at the point P.

Question

Find the <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> curvature K of the curve at the point P.

Answer 1

Step 1 :

Formula for the curvature in the plane or in space :

Step 2 :

The function is and the point is .

Compare with .

and .

Substitute the point in .

The function is .

Apply derivative on each side with respect to t.

Substitute in above equation.

Find the magnitude  :

Substitute in above equation.

Step 3 :

Differentiate with respect t.

At , .

Find the cross product :

Find the magnitude :

Curvature is

Substitute and in above equation.

The curvature is

Solution :

The curvature is