   Chapter 13.3, Problem 20E

Chapter
Section
Textbook Problem

# (a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature.20. r ( t ) = 〈 t ,   1 2 t 2 ,  t 2 〉

(a)

To determine

To find: The unit tangent vector T(t) and unit normal vector N(t) .

Explanation

Given data:

Consider the vector function r(t) ,

r(t)=t,12t2,t2 (1)

Differentiate equation (1) with respect to t,

r(t)=1,t,2t (2)

Take magnitude of equation (2),

|r(t)|=1+t2+4t2

|r(t)|=1+5t2 (3)

The expression for the unit tangent vector T(t) is,

T(t)=r(t)|r(t)| (4)

Substitute equation (2) and equation (3) in equation (1),

T(t)=11+5t21,t,2t (5)

Differentiate equation (5) with respect to t by using product rule T(t)=(ddt|r(t)|)r(t)+|r(t)|r(t) .

T(t)=(1+5t2)121,t,2t=12(1+5t2)32(0+10t)1,t,2t+11+5t20,1,2 {T(t)=[(ddt|r(t)|)r(t)+|r(t)|r(t)]}=5t(1+5t2)321,t,2t+11+5t20,1,2

Multiply and divide by the term 1+5t2 ,

T(t)=5t(1+5t2)321,t,2t+(1+5t2)(1+5t2)12(1+5t2)0,1,2

(b)

To determine

To find: The curvature k(t) .

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