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Chapter 11: Q. 38 (page 890)

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

x=cost+tsint,y=sint-tcost

Short Answer

Expert verified

The curvature is1t.

Step by step solution

01

Step 1. Given information.

The given function is,

x=cost+tsint,y=sint-tcost

02

Step 2. Curvature.

Now,

r(t)=鉄�cost+tsint,sint-tcost鉄�r'(t)=鉄�tcost,tsint鉄�r'(t)=(tcost)2+(tsint)2=t2cos2t+sin2t=tT(t)=r'(t)r'(t)=鉄�tcost,tsint鉄�t=鉄�cost,sint鉄�T'(t)=(-sint)2+cos2t=1Curvature=k=T'(t)r'(t)=1t

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Most popular questions from this chapter

Evaluate the limits in Exercises 42鈥�45.

limt鈫�蟺(sint,cost,sect)

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.
t(sint,cost)

Find and graph the vector function determined r(t)=鉄�x(t),y(t)鉄�by the differential equation

x鈥�(t)=鈭�y,y鈥�(t)=x,x(0)=1,y(0)=0. ( HINT: What familiar pair of functions have the given properties ?)

Find and graph the vector function determined r(t)=鉄�x(t),y(t)鉄�by the differential equation

role="math" localid="1649566464308" x鈥�(t)=1+x2,y鈥�(t)=x2,x(0)=0,y(0)=1. ( HINT: Start by solving the initial-value problemrole="math" localid="1649566360577" x鈥�(t)=1+x2,x(0)=0.)

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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