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Chapter 11: Q. 36 (page 890)

Find the curvature of each of the functions defined by the parametric equations in Exercises 36鈥�38.

x=kcost,y=ksint

Short Answer

Expert verified

The curvature is1k.

Step by step solution

01

Step 1. Given information.

The given function isx=kcost,y=ksint.

02

Step 2. Required answer.

Curvature is given by,

k=T'(t)r'(t)r'(t)=鉄�-ksint,kcost鉄�r'(t)=(-ksint)2+(kcost)2=kT(t)=r'(t)r'(t)=鉄�-ksint,kcost鉄�k=鉄�-sint,cost鉄�T'(t)=鉄�-cost,-sint鉄�T'(t)=(-cost)2+(-sint)2=1k=T'(t)r'(t)=1k

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Most popular questions from this chapter

As we saw in Example 1, the graph of the vector-valued function r(t)=cost,sint,t,fort鈭�[0,2蟺] is a circular helix that spirals counterclockwise around the z-axis and climbs ast increases. Find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

As we saw in Example 1, the graph of the vector-valued function r(t)=cost,sint,t,fort鈭�[0,2蟺] is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

Findthederivativeofthevector-valuedfunctionr(t)=t2i鈭�6t2j,t>0,andshowthatthederivativeatanypointt0isascalarmultipleofthederivativeatanyotherpoint.Whatdoesthispropertysayaboutthegraphofr?Usethatinformationtosketchthegraphofr.

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=3sint,5cost,4sint,t=蟺

Given a twice-differentiable vector-valued function r(t), why does the principal unit normal vector N(t) point into the curve?
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