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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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Finding the Principal Unit Normal Vector In Exercises 47-50, find the principal unit normal vector to the curse at the specified value of the parameter.

r ( t ) = 2 t i + 3 t 2 j , t = 1

To determine

To calculate: The principal unit normal vector to the curve r(t)=2ti+3t2j at the specified value of the parameter where, t=1.

Explanation

Given:

The vector-valued curve is: r(t)=2ti+3t2j and the parameter is t=1.

Formula used:

Principal unit normal vector:

N(t)=T(t)T(t)

Unit Tangent Vector:

T(t)= r(t)r(t)

The magnitude of a vector x=ai+bj is x=a2+b2.

Calculation:

Consider the provided vector-valued curve,

r(t)=2ti+3t2j

Differentiate the vector r(t).

r(t)=2i+6tjr(t)=4+36t2

At t=1

r(1)=4+36(1)2=4+36=210

Unit Tangent Vector:

T(t)= r(t)r(t)

T(t)=24+36t2i+6t4+36t2j

At t=1

T(1)= r(1)r(1) Now, calculate the unit tangent vector.

T(1)= r(1)r(1)= 2210i+6210j= 110i+310j

Now,

T(t)=22(4+36t2)32(36×2t)i+64+36t26t(36×2t24+36t2)(4+36t2)j=72t(4+36t2)32i+6(

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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