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Chapter 11: Q. 5 (page 900)

Giveprecisemathematicaldefinitionsordescriptionsofeachoftheconceptsthatfollow.Thenillustratethedefinitionwithagraphoranalgebraicexample.thederivativeofavectorfunctionr(t)

Short Answer

Expert verified

r'(t)=x'(t),y'(t),z'(t)=x'(t)i+y'(t)j+z't)k

Step by step solution

01

Step 1. Given information is:

Avectorfunctionr(t)

02

Step 2. Derivative of r(t)

Letx=x(t),y=y(t),andz=z(t)bethreereal-valuedfunctions,eachofwhichisdifferentiableateverypointinsomeintervalI⊆R.Thederivativeofthevectorfunctionr(t)=x(t),y(t),z(t)isr'(t)=x'(t),y'(t),z'(t)=x'(t)i+y'(t)j+z't)k.

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Most popular questions from this chapter

For constants α,β, and γ, the graph of a vector-valued function of the form

r(t)=αeγt(cost)i+αeγt(sint)j+βeγtk,t≥0

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

r(t)=(3+t)i+(3-1t)jfort>0

What is the dot product of the vector functions r1(t)=x1(t),y1(t)andr2(t)=x2(t),y2(t)?

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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 40–42. Note: These are the same functions as in Exercises 35, 37, and 39.

r(t)=et,e−t,2tatt=0
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