   Chapter 13.3, Problem 14E

Chapter
Section
Textbook Problem

# (a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P. and (b) find the point 4 units along the curve (in the direction of increasing t) from P.14. r ( t ) = e t sin t   i + e t cos t   j + 2 e t   k ,  P (0, 1,  2 )

(a)

To determine

To find: The arc length function for the curved measured equation r(t)=etsinti+etcostj+2etk from the point P(0,1,2).

Explanation

Given data:

r(t)=etsinti+etcostj+2etk,P(0,1,2).

Formula used:

Write the expression to find the arc length function of the curve r(t).

s(t)=at|r(u)|du (1)

Write the expression to find length of the curve L for the vector r(t).

L=ab|r(t)|dt (2)

Here,

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

s(t) is denoted as arc length function, and

[a,b] is parameter interval.

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

ddt[r(t)]=ddt(etsinti+etcostj+2etk)

r(t)=ddt(etsint),ddt(etcost),ddt(2et) (3)

Write the following formula to compute the expression for r(t).

ddt(etsint)i=et(cost+sint)iddt(etcost)j=et(sintcost)jddt(2et)k=2etkddt(constant)=0

Apply the corresponding formula in equation (3) to find r(t).

r(t)=ddt(etsint),ddt(etcost),ddt(2et)

r(t)=et(cost+sint),et(sintcost),2et (4)

Take magnitude on both sides of equation (4)

(b)

To determine

To find: The point 4 units along the curve from P.

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