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Chapter 11: Q. 4 (page 859)

Letr(t)=x(t),y(t),z(t).Whatisthedefinitionoflimt鈫�cr(t)?

Short Answer

Expert verified

limt鈫�cr(t)=limt鈫�cx(t)i+limt鈫�cy(t)j+limt鈫�cz(t)k,whichisavector.

Step by step solution

01

Step 1. Given information is 

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

02

Step 2. Definition of limt→cr(t)

Thelimitofr(t)astapproachesc,denotedbylimt鈫�cr(t),isdefinedbylimt鈫�cr(t)=limt鈫�cx(t),y(t),z(t)=limt鈫�cx(t)i+limt鈫�cy(t)j+limt鈫�cz(t)k,wherelimt鈫�cx(t),limt鈫�cy(t)andlimt鈫�cz(t)exists.Thuslimt鈫�cr(t)=limt鈫�cx(t)i+limt鈫�cy(t)j+limt鈫�cz(t)k,whichisavector.

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Most popular questions from this chapter

Prove Theorem 11.7 for vectors in R2. That is, let x(t)and y(t)be two scalar functions, each of which is differentiable on an interval I 鈯� R, and let localid="1649578343519" r(t)=x(t),y(t)be a vector function. Prove that r'(f)=limh鈫�0r(t+h)-r(t)h.

Using the definitions of the normal plane and rectifying plane in Exercises 20 and 21, respectively, find the equations of these planes at the specified points for the vector functions in Exercises 40鈥�42. Note: These are the same functions as in Exercises 35, 37, and 39.

r(t)=et,e鈭�t,2tatt=0

Let f(t)be a differentiable scalar function and r(t)be a differentiable vector function. Prove that ddt(f(t)r(t))=f'(t)r(t)+f(t)r'(t). (This is Theorem 11.11 (b).)

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

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=3sint,5cost,4sint,t=蟺
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