   Chapter 13.3, Problem 16E

Chapter
Section
Textbook Problem

# Reparametrize the curve r ( t ) = ( 2 t 2 + 1 − 1 ) i + 2 t t 2 + 1 j with respect to arc length measured from the point (1, 0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the curve?

To determine

To express: The reparametrization in its simplest form and provide conclusion about the curve.

Explanation

Given data:

Consider the expression of curve.

r(t)=(2t2+11)i+2tt2+1j (1)

Differentiate equation (1) with respect to t,

r(t)=4t(t2+1)2i+2t2+2(t2+1)2j (2)

Take magnitude of equation (2),

|r(t)|=[4t(t2+1)2]2+[2t2+2(t2+1)2]2=4t4+8t2+4(t2+1)4=4(t4+2t2+1)(t2+1)4=4(t2+1)2(t2+1)4 {(a+b)2=a2+2ab+b2}

|r(t)|=4(t2+1)2

|r(t)|=2t2+1 (3)

Since the starting point (1,0) corresponds to t=0 . Then, the arc function is given by,

s(t)=0t|r(t)|du (4)

Substitute equation (3) in equation (4),

s(t)=0t2t2+1du=2tan1t {1t2+1=tan1t}

Re-arrange the equation,

12s(t)=tan1t

t=tan12s (5)

Substitute equation (5) in equation (1),

r(t)=(2(tan12s)2+11)i+2tan12s(tan12s)2+1j=(2tan212s+11)i+2tan12stan212s+1j=(2tan212s1tan212s+1)i+2tan12stan212s+1j

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