   Chapter 13.2, Problem 22E

Chapter
Section
Textbook Problem

# If r(t) = 〈e2t, e−2t, te2t〉, find T(0), r"(0), and r'(t) · r"(t).

To determine

To find: The unit tangent vector T(0) for the vector r(t)=e2t,e2t,te2t.

Explanation

Formula used:

Write the expression to find unit tangent vector T(0) for the vector r(t).

T(0)=r(0)|r(0)| (1)

Here,

r(0) is the tangent vector, which is the derivative of the vector r(t).

Calculation of tangent vector r(t):

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=e2t,e2t,te2t as follows.

ddt[r(t)]=ddt(e2t),ddt(e2t),ddt(te2t)

Use the following formula to compute the expression.

ddtet=etddte2t=2e2tddte2t=2e2tddt[u(t)v(t)]=u(t)ddt[v(t)]+v(t)ddt[u(t)]

Compute the expression ddt[r(t)]=ddt(e2t),ddt(e2t),ddt(te2t) by using the formulae as follows.

r(t)=2e2t,2e2t,[tddt(e2t)+e2tddt(t)]=2e2t,2e2t,(2te2t+e2t)

Calculation of tangent vector r(0):

Substitute 0 for t in the expression r(t)=2e2t,2e2t,(2te2t+e2t)

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