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Chapter 11: Q. 8 (page 900)

Giveprecisemathematicaldefinitionsordescriptionsofeachoftheconceptsthatfollow.Thenillustratethedefinitionwithagraphoranalgebraicexample.theunittangentfunctionforadifferentiablevectorfunctionr(t)

Short Answer

Expert verified

T(t)=r'(t)r'(t)

Step by step solution

01

Step 1. Given information is:

A vector function r(t)

02

Step 2. Unit Tangent Function

Letr(t)beadifferentiablevectorfunctiononsomeintervalI⊆Rsuchthatr'(t)≠0onI.Theunittangentfunctionisdefinedtobe:T(t)=r'(t)r'(t)

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Most popular questions from this chapter

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).)

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=cosαt,sinαt,whereα>0,t=π

Carefully outline the steps you would use to find the equation of the osculating plane at a point rt0on the graph of a vector function.

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.

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=π
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