   Chapter 13.2, Problem 17E

Chapter
Section
Textbook Problem

# Find the unit tangent vector T(t) at the point with the given value of the parameter t.17. r ( t )  =  〈 t 2   −  2 t , 1 + 3 t ,  1 3 t 3  +  1 2 t 2 〉 ,   t  = 2

To determine

To find: The unit tangent vector T(t) for the vector r(t)=t22t,1+3t,13t3+12t2 at t=2.

Explanation

Formula used:

Write the expression to find unit tangent vector T(t) for the vector r(t) at a finite value of scalar parameter t.

T(t)=r(t)|r(t)| (1)

Here,

r(t) is the tangent vector, which is the derivative of vector r(t).

Find the tangent vector r(t) by differentiating each component of the vector r(t) as follows.

ddt[r(t)]=ddtt22t,1+3t,13t3+12t2r(t)=ddt(t22t),ddt(1+3t),ddt(13t3+12t2)

Use the following formula to compute the expression.

ddt[u(t)+v(t)]=ddtu(t)+ddtv(t)ddtt2=2tddtt3=3t2ddt(constant)=0

Compute the expression r(t)=ddt(t22t),ddt(1+3t),ddt(13t3+12t2) by using the formulae as follows.

r(t)=ddt(t22t),ddt(1+3t),ddt(13t3+12t2)=[ddt(t2)+ddt(2t)],[ddt(1)+ddt(3t)],[ddt(13t3)+ddt(

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