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Lesson: Derivatives of Vector Valued Functions

In this lesson, we will learn how to determine the derivatives of vector-valued functions in one variable by taking the derivative of each component.

Worksheet • 5 Questions

Q1:

Calculate 𝑓 β€² ( 𝑠 ) , and find the vector form of the equation of the tangent line 𝐿 at 𝑓 ( 0 ) for 𝑓 ( 𝑠 ) = ( 2 𝑠 , 2 𝑠 , 𝑠 ) c o s s i n .

  • A 𝑓 β€² ( 𝑠 ) = ( βˆ’ 2 2 𝑠 , 2 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 1 , 0 , 0 ) + 𝑑 ( 0 , 2 , 1 ) :
  • B 𝑓 β€² ( 𝑠 ) = ( 2 2 𝑠 , βˆ’ 2 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 1 , 0 , 0 ) + 𝑑 ( 0 , βˆ’ 2 , 1 ) :
  • C 𝑓 β€² ( 𝑠 ) = ( βˆ’ 2 2 𝑠 , 2 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 0 , 2 , 1 ) + 𝑑 ( 1 , 0 , 0 ) :
  • D 𝑓 β€² ( 𝑠 ) = ( βˆ’ 2 𝑠 , 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 0 , 1 , 1 ) + 𝑑 ( 1 , 0 , 0 ) :
  • E 𝑓 β€² ( 𝑠 ) = ( βˆ’ 2 𝑠 , 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 1 , 0 , 0 ) + 𝑑 ( 0 , 1 , 1 ) :

Q2:

Calculate 𝑓 β€² ( 𝑠 ) , and find the vector form of the equation of the tangent line at 𝑓 ( 0 ) for 𝑓 ( 𝑠 ) = ο€Ή 𝑠 + 1 , 𝑠 + 1 , 𝑠 + 1  2 3 .

  • A 𝑓 β€² ( 𝑠 ) = ο€Ή 1 , 2 𝑠 , 3 𝑠  2 , 𝐿 ( 1 , 1 , 1 ) + 𝑑 ( 1 , 0 , 0 ) :
  • B 𝑓 β€² ( 𝑠 ) = ( 1 , 2 𝑠 , 3 𝑠 ) , 𝐿 ( 1 , 1 , 1 ) + 𝑑 ( 1 , 0 , 0 ) :
  • C 𝑓 β€² ( 𝑠 ) = ο€Ή 1 , 2 𝑠 , 3 𝑠  2 , 𝐿 ( 1 , 0 , 0 ) + 𝑑 ( 1 , 1 , 1 ) :
  • D 𝑓 β€² ( 𝑠 ) = ο€Ή 2 , 2 𝑠 + 1 , 3 𝑠 + 1  2 , 𝐿 ( 2 , 0 , 0 ) + 𝑑 ( 2 , 1 , 1 ) :
  • E 𝑓 β€² ( 𝑠 ) = ο€Ή 2 , 2 𝑠 + 1 , 3 𝑠 + 1  2 , 𝐿 ( 2 , 1 , 1 ) + 𝑑 ( 2 , 0 , 0 ) :

Q3:

Calculate f β€² ( 𝑠 ) , and find the vector form of the equation of the tangent line at f ( 0 ) for f ( 𝑠 ) = ( 𝑒 + 1 , 𝑒 + 1 , 𝑒 + 1 ) 𝑠 2 𝑠 𝑠 2 .

  • A f β€² ( 𝑠 ) = ο€Ί 𝑒 , 2 𝑒 , 2 𝑠 β‹… 𝑒  𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 2 , 2 ) + 𝑑 ( 1 , 2 , 0 ) :
  • B f β€² ( 𝑠 ) = ο€Ί 𝑒 , 𝑒 , 𝑒  𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 2 , 2 ) + 𝑑 ( 1 , 1 , 1 ) :
  • C f β€² ( 𝑠 ) = ο€Ί 𝑒 , 2 𝑒 , 2 𝑠 β‹… 𝑒  𝑠 2 𝑠 𝑠 2 , 𝐿 ( 1 , 2 , 0 ) + 𝑑 ( 2 , 2 , 2 ) :
  • D f β€² ( 𝑠 ) = ο€Ί 𝑒 + 1 , 2 𝑒 + 1 , 2 𝑠 β‹… 𝑒 + 1  𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 3 , 1 ) + 𝑑 ( 2 , 2 , 2 ) :
  • E f β€² ( 𝑠 ) = ο€Ί 𝑒 + 1 , 2 𝑒 + 1 , 2 𝑠 β‹… 𝑒 + 1  𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 2 , 2 ) + 𝑑 ( 2 , 3 , 1 ) :

Q4:

Consider the curve . Determine and find the tangent to the curve when .

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q5:

Given that , where and are constants, find .

  • A
  • B
  • C
  • D
  • E
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