   Chapter 13.3, Problem 13E

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.13. r(t) = (5 − t)i + (4t − 3)j + 3t k, P(4, 1, 3)

(a)

To determine

To find: The arc length function for the curved measured equation r(t)=(5t)i+(4t3)j+3tk from the point P(4,1,3) .

Explanation

Given data:

r(t)=(5t)i+(4t3)j+3tk,P(4,1,3) (1)

Formula used:

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

s(t)=at|r(u)|du (2)

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

L=ab|r(t)|dt (3)

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(5t)i+(4t3)j+3tk

r(t)=ddt(5t)i,ddt(4t3)j,ddt(3t)k (4)

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

ddt(5t)i=iddt(4t3)j=4jddt(3t)k=3kddt(constant)=0

Apply the corresponding formula in equation (4) to find r(t) .

r(t)=i,+4j,3k (5)

Take magnitude on both sides of equation (5).

|r(t)|=|i,+4j,3k|=(1)2+(4)2+(3)2=1+16+9=26

The point P(4,1,3) is corresponding to t=1

(b)

To determine

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

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