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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the unit tangent vector T(t) at the point with the given value of the parameter t.

18. r(t) = ⟨tan-1 t, 2e 2t, 8tet⟩, t = 0

To determine

To find: The unit tangent vector T(t) for the vector r(t)=tan1t,2e2t,8tet at t=0.

Explanation

Formula used:

Write the expression to find unit tangent vector T(t) for the vector r(t) at a finite value of scalar parameter t.

T(t)=r(t)|r(t)| (1)

Here,

r(t) is the tangent vector, which is the derivative of vector r(t).

Find the tangent vector r(t) by differentiating each component of the vector r(t) as follows.

ddt[r(t)]=ddttan1t,2e2t,8tetr(t)=ddt(tan1t),ddt(2e2t),ddt(8tet)

Use the following formula to compute the expression.

ddttan1t=11+t2ddt[u(t)v(t)]=u(t)ddt[v(t)]+v(t)ddt[u(t)]ddtet=etddte2t=2et

Compute the expression r(t)=ddt(tan1t),ddt(2e2t),ddt(8tet) by using the formulae as follows.

r(t)=ddt(tan1t),2ddt(e2t),ddt(8tet)=11+t2,2(2)e2t,8[tddt(e

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Chapter 13 Solutions

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