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Chapter 11: Q. 13 (page 871)

Completethefollowingdefinition:Ifr(t)=x(t),y(t),z(t)isadifferentiablepositionfunction,thenthevelocityvectorv(t)is....

Short Answer

Expert verified

Ifr(t)=x(t),y(t),z(t)isadifferentiablepositionfunction.Then,v(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)beadifferentiablepositionfunction.

02

Step 2. Velocity Vector

Letcbethespacecurvedefinedbyr(t)=x(t),y(t),z(t),whichisadifferentiablevectorvaluedfunction.Thenthevelocityoftheparticlealongcisgivenby:v(t)=r'(t)v(t)=dr(t)dt=dx(t),y(t),z(t)dt=x'(t),y'(t),z'(t)Thus,v(t)=x'(t),y'(t),z'(t)

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Most popular questions from this chapter

Let r1(t)and r2(t) both be differentiable three-component vector functions. Prove that

ddtr1(t)×r2(t)=r1'(t)×r2(t)+r1(t)×r2'(t) (This is Theorem 11.11 (d).)

In Exercises 19–21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domainlocalid="1649578783745" t≥0,r0=1,0,r2=0,-1,limt→∞x(t)=-2,andlimt→∞y(t)=-3.

Under what conditions does a differentiable vector-valued functionr(t) not have a unit tangent vector at a point in the domain of r(t)?

Let y = f(x). State the definition for the continuity of the function f at a point c in the domain of f .

Evaluate the limits in Exercises 42–45.

limt→0(sin3t,cos3t)
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