Chapter 12.4, Problem 13E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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# Finding the Principal Unit Normal Vector In Exercises 15-20, find the principal unit normal vector to the curve at the specified value of the parameter. r ( t ) = t i + 1 2 t 2 j , t = 2

To determine

To calculate: The unit normal vector to curve r(t)=ti+12t2j at the specified value of the parameter t=2.

Explanation

Given:

The curve: r(t)=ti+12t2j

And the specified value of the parameter: t=2

Formula Used:

Let C be a smooth curve which makes the representation by r on an open interval I. The unit tangent vector at t is defined as:

T(t)=r(t)r(t),  r(t)0

Let C be a smooth curve which makes the representation r on an open interval I. Then, the unit normal vector at t is defined as:

N(t)=T(t)T(t),  T(t)0

Calculation:

Consider the specified curve:

r(t)=ti+12t2j

Differentiate r(t) as follows:

r(t)=(ddtt)i+(ddt12t2)j=i+tj

This gives the unit tangent vector as follows:

T(t)=r(t)r(t)=112+t2(i+tj)=11+t2(i+tj)

Now differentiate T(t) as follows:

T(t)=(ddt11+t2)i+(ddtt1+t2)j=(12(1+t2)3/22t)i+(1+t21t12(1+t2)1/22t(1+t2)2)j=t(1+t2)

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