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Chapter 11: Q. 14 (page 871)

Completethefollowingdefinition:Ifr(t)=x(t),y(t),z(t)isatwice-differentiablepositionfunction,thentheaccelerationvectora(t)is....

Short Answer

Expert verified

Ifr(t)=x(t),y(t),z(t)isatwicedifferentiablepositionfunction.Then,a(t)=x''(t),y''(t),z''(t)

Step by step solution

01

Step 1. Given information is:

r(t)=x(t),y(t),z(t)beatwicedifferentiablepositionfunction.

02

Step 2. Acceleration Vector

Letcbethecurvedefinedbyr(t)=x(t),y(t),z(t),whichisatwicedifferentiablepositionfunctionontheintervalI⊂R.Thentheaccelerationvectorisgivenby:a(t)=r''(t)a(t)=v'(t)=x''(t),y''(t),z''(t)Thus,a(t)=x''(t),y''(t),z''(t)

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Most popular questions from this chapter

Let r1(f)and r2(t) be differentiable vector functions with three components each. Prove that

ddtr1(t)-r2(t)=r1'(t)-r2(t)+r1(t)-r2'(t)∣(This is Theorem 11.11 (c).)

Let r(t)=x(t),y(t),t∈[a,∞),be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as t→∞?Provide an example illustrating your answer.

Given a twice-differentiable vector-valued function r(t)and a point t0in its domain, what is the osculating plane at rt0?

Letr(t)=x(t),y(t),z(t).Provideadefinitionforthecontinuityofthevectorfunctionratapointcinthedomainofr.

For each of the vector-valued functions, find the unit tangent vector.

r(t)=(t,t2,t3)
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