Worksheet: Derivatives of Vector Valued Functions

In this worksheet, we will practice determining the derivatives of vector-valued functions in one variable by taking the derivative of each component.

Q1:

Calculate 𝑓 ( 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 ( 𝑠 ) = ( 2 𝑠 , 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 1 , 0 , 0 ) + 𝑡 ( 0 , 1 , 1 ) :
  • B 𝑓 ( 𝑠 ) = ( 2 2 𝑠 , 2 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 0 , 2 , 1 ) + 𝑡 ( 1 , 0 , 0 ) :
  • C 𝑓 ( 𝑠 ) = ( 2 𝑠 , 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 0 , 1 , 1 ) + 𝑡 ( 1 , 0 , 0 ) :
  • D 𝑓 ( 𝑠 ) = ( 2 2 𝑠 , 2 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 1 , 0 , 0 ) + 𝑡 ( 0 , 2 , 1 ) :
  • E 𝑓 ( 𝑠 ) = ( 2 2 𝑠 , 2 2 𝑠 , 1 ) s i n c o s , 𝐿 ( 1 , 0 , 0 ) + 𝑡 ( 0 , 2 , 1 ) :

Q2:

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

  • A 𝑓 ( 𝑠 ) = 2 , 2 𝑠 + 1 , 3 𝑠 + 1 2 , 𝐿 ( 2 , 1 , 1 ) + 𝑡 ( 2 , 0 , 0 ) :
  • B 𝑓 ( 𝑠 ) = 1 , 2 𝑠 , 3 𝑠 2 , 𝐿 ( 1 , 0 , 0 ) + 𝑡 ( 1 , 1 , 1 ) :
  • C 𝑓 ( 𝑠 ) = 2 , 2 𝑠 + 1 , 3 𝑠 + 1 2 , 𝐿 ( 2 , 0 , 0 ) + 𝑡 ( 2 , 1 , 1 ) :
  • D 𝑓 ( 𝑠 ) = 1 , 2 𝑠 , 3 𝑠 2 , 𝐿 ( 1 , 1 , 1 ) + 𝑡 ( 1 , 0 , 0 ) :
  • E 𝑓 ( 𝑠 ) = ( 1 , 2 𝑠 , 3 𝑠 ) , 𝐿 ( 1 , 1 , 1 ) + 𝑡 ( 1 , 0 , 0 ) :

Q3:

Calculate f ( 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 ( 𝑠 ) = 𝑒 + 1 , 2 𝑒 + 1 , 2 𝑠 𝑒 + 1 𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 2 , 2 ) + 𝑡 ( 2 , 3 , 1 ) :
  • B f ( 𝑠 ) = 𝑒 , 2 𝑒 , 2 𝑠 𝑒 𝑠 2 𝑠 𝑠 2 , 𝐿 ( 1 , 2 , 0 ) + 𝑡 ( 2 , 2 , 2 ) :
  • C f ( 𝑠 ) = 𝑒 + 1 , 2 𝑒 + 1 , 2 𝑠 𝑒 + 1 𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 3 , 1 ) + 𝑡 ( 2 , 2 , 2 ) :
  • D f ( 𝑠 ) = 𝑒 , 2 𝑒 , 2 𝑠 𝑒 𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 2 , 2 ) + 𝑡 ( 1 , 2 , 0 ) :
  • E f ( 𝑠 ) = 𝑒 , 𝑒 , 𝑒 𝑠 2 𝑠 𝑠 2 , 𝐿 ( 2 , 2 , 2 ) + 𝑡 ( 1 , 1 , 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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