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Chapter 11: Q. 1 (page 900)

Fillintheblankstocompleteeachofthefollowingtheoremstatements:Thederivativeofavectorfunctionr(t)isgivenbyr'(t)=limh鈫�0______

Short Answer

Expert verified

r'(t)=limh鈫�0r(t+h)-r(t)h

Step by step solution

01

Step 1. Given information is:

A vector function r(t)

02

Step 2. Derivative of vector function

Letr(t)beadifferentiablevectorfunctionwitheithertwoorthreecomponents.Thederivativeofr(t)isgivenby:r'(t)=limh鈫�0r(t+h)-r(t)h

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Most popular questions from this chapter

Prove that the tangent vector is always orthogonal to the position vector for the vector-valued function.

r(t)=sint,sintcost,cos2t

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

Prove that the cross product of two orthogonal unit vectors is a unit vector.

Find parametric equations for each of the vector-valued functions in Exercises 26鈥�34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(sint,cos2t)fort鈭�[0,2蟺]

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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