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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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57095-12.5-22E-Question-Digital.docx

Finding Curvature In Exercises 19–22, find the curvature K of the curve, where s is thearc length parameter.

Curve in Exercise 18:

r ( t ) = 4 ( sin t t cos t ) , 4 ( cos t + t sin t ) , 3 2 t 2

To determine

To calculate: The curvature of the curve r(t)=4(sinttcost),4(cost+tsint),32t2.

Explanation

Given:

The equation of curve r(t)=4(sinttcost),4(cost+tsint),32t2.

Formula used:

The curvature of curve r(t) is,

K=T(t)r(t)

Calculation:

Consider the equation of the curve, r(t)=4(sinttcost),4(cost+tsint),32t2.

Differentiate r(t) with respect to t,

r(t)=4(cost(costtsint) i)+4(sint+(sint+tcost))j+ 3t k

Take the magnitude of r(t), That is,

r(t)=42(cost(costtsint) i)2+42(sint+(sint+tcost)j)2+ (3t k)2=16(tsint)2+(tcost)2+(3t)2=16t2(sin2t+cos2t)+9t2=16t2+9t2

Further simplify the above equation,

r(t)=25t2=5t

Find the value of tangent vector,

T

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