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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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57095-12.4-38E-Question-Digital.docx

Finding Vectors In Exercises 37–42, find T(t), N(t), a T , and a N at the given time t for the space curve r(t). [Hint: Find a(t), T(t), a T , and a N . Solve for N in the equation a(t) = a N T + a N N.]

Vector-valued function    Time

r ( t ) = cos t i + sin t j + 2 t k t = π 3

To determine

To calculate: The vectors T(t),N(t),aT and aN, of the curve, r(t)=costi+sintj+2tk at t=π3.

Explanation

Given:

r(t)=costi+sintj+2tk and t=π3.

Formula used:

If r(t) is a curve then,

T(t)=r(t)r(t),N(t)=T(t)T(t),aT=ddt[v(t)] and aN=v(t)T(t).

And a(t)=aTT+aNN

Calculation:

Consider the curve, r(t)=costi+sintj+2tk.

Now, r(t)=sinti+costj+2k

And,

r(t)=(sint)2+(cost)2+(2)2=1+4=5

Now, find T(t)=r(t)r(t).

So,

T(t)=sinti+costj+2k5=5sint5i+5cost5j+255k

Now, at t=π3

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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 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