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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

Evaluate the limits in Exercises 42–45.

limt→π(sint,cost,sect)

Let r(t)=x(t),y(t),z(t),t∈[a,∞), be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in some sphere centered at the origin. (Hint: Consider the functions r1(t)=cost,sint,1/tand r2(t)=cost,sint,t,bothwithdomain[1,∞).)

Given a vector-valued function r(t) with domain ℝ, what is the relationship between the graph of r(t) and the graph of kr(t), where k > 1 is a scalar?

Evaluate and simplify the indicated quantities in Exercises 35–41.
5(cost,sint)

Find and graph the vector function determined r(t)=⟨x(t),y(t)⟩by the differential equation

x′(t)=−y,y′(t)=x,x(0)=1,y(0)=0. ( HINT: What familiar pair of functions have the given properties ?)
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