Worksheet: Derivatives of Vector-Valued Functions

In this worksheet, we will practice determining the derivatives of vector-valued functions and finding unit tangent vectors.

Q1:

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

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

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

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

Q4:

Consider the curve r(𝑠)=(2𝑠,2𝑠,2𝑠)sinsincos. Determine r(𝑠) and find the tangent 𝐿 to the curve when 𝑠=0.

  • A r ( 𝑠 ) = ( 2 2 𝑠 , 2 2 𝑠 , 2 𝑠 ) c o s s i n s i n , 𝐿 ( 2 , 0 , 0 ) + 𝑡 ( 0 , 0 , 2 ) :
  • B r ( 𝑠 ) = ( 2 𝑠 , 2 𝑠 , 2 𝑠 ) c o s c o s s i n , 𝐿 ( 0 , 0 , 2 ) + 𝑡 ( 2 , 1 , 0 ) :
  • C r ( 𝑠 ) = ( 2 2 𝑠 , 4 𝑠 , 2 𝑠 ) c o s c o s s i n , 𝐿 ( 2 , 2 , 0 ) + 𝑡 ( 0 , 0 , 2 ) :
  • D r ( 𝑠 ) = ( 2 2 𝑠 , 2 𝑠 𝑠 , 2 𝑠 ) c o s s i n c o s s i n , 𝐿 ( 0 , 0 , 2 ) + 𝑡 ( 2 , 2 , 0 ) :
  • E r ( 𝑠 ) = ( 2 2 𝑠 , 4 𝑠 𝑠 , 2 𝑠 ) c o s s i n c o s s i n , 𝐿 ( 0 , 0 , 2 ) + 𝑡 ( 2 , 0 , 0 ) :

Q5:

Given that rijk(𝑡)=𝑎𝑡+𝑡𝑒+𝑐𝑡sincos, where 𝑎 and 𝑏 are constants, find r(𝑡).

  • A 2 𝑎 𝑎 𝑡 𝑎 𝑡 + 𝑒 ( 1 + 𝑏 𝑡 ) + 2 𝑐 𝑐 𝑡 𝑐 𝑡 s i n c o s c o s s i n i j k
  • B 𝑎 𝑎 𝑡 𝑎 𝑡 + 𝑒 ( 1 + 𝑏 𝑡 ) 𝑐 𝑐 𝑡 𝑐 𝑡 s i n c o s c o s s i n i j k
  • C 2 𝑎 𝑡 𝑎 𝑡 + 𝑒 ( 1 + 𝑡 ) 2 𝑐 𝑡 𝑐 𝑡 s i n c o s c o s s i n i j k
  • D 𝑎 𝑎 𝑡 𝑎 𝑡 + 𝑒 ( 1 + 𝑡 ) + 𝑐 𝑐 𝑡 𝑐 𝑡 s i n c o s c o s s i n i j k
  • E 2 𝑎 𝑎 𝑡 𝑎 𝑡 + 𝑒 ( 1 + 𝑏 𝑡 ) 2 𝑐 𝑐 𝑡 𝑐 𝑡 s i n c o s c o s s i n i j k

Q6:

Find the derivative of a vector-valued function rijk(𝑡)=1+𝑡+5𝑡+1+𝑡+2.

  • A 3 𝑡 + ( 1 0 𝑡 ) + 3 𝑡 i j k
  • B 1 + 3 𝑡 + ( 1 0 𝑡 ) + 3 𝑡 i j k
  • C 6 𝑡 + 1 0 𝑡
  • D 3 𝑡 + 1 0 𝑡
  • E ( 3 𝑡 ) + ( 1 0 𝑡 ) + 3 𝑡 i j k

Q7:

Find the derivative of a vector-valued function rij(𝑡)=(32𝑡)+2𝑡+3𝑡2.

  • A 2 ( 4 𝑡 + 3 ) i j
  • B 4 𝑡 + 1
  • C 2 + ( 4 𝑡 + 3 ) i j
  • D 2 + ( 4 𝑡 + 3 ) i j
  • E 2 + ( 4 𝑡 ) i j

Q8:

Find the derivative of the vector-valued function rijk(𝑡)=𝑒+𝑒+3.

  • A 𝑒 𝑒 + 3 i j k
  • B 𝑒 + 𝑒 i j
  • C 𝑒 𝑒 i j
  • D 𝑒 𝑒 + i j k
  • E 𝑒 𝑒 i j

Q9:

Find the derivative of a vector-valued function 𝑟(𝑡)=(2𝑡)𝑖(𝑡)𝑗+𝑒𝑘sincos.

  • A 2 ( 𝑡 ) 𝑖 + ( 𝑡 ) 𝑗 + 𝑒 𝑘 c o s s i n
  • B c o s s i n ( 2 𝑡 ) 𝑖 ( 𝑡 ) 𝑗 + 𝑒 𝑘
  • C 2 ( 2 𝑡 ) 𝑖 ( 𝑡 ) 𝑗 + 𝑒 𝑘 c o s s i n
  • D c o s s i n ( 2 𝑡 ) 𝑖 + ( 𝑡 ) 𝑗 + 𝑒 𝑘
  • E 2 ( 2 𝑡 ) 𝑖 + ( 𝑡 ) 𝑗 + 𝑒 𝑘 c o s s i n

Q10:

Find the derivative of a vector-valued function rijk(𝑡)=𝑡++.

  • A 𝑡 + 2
  • B1
  • C i j k + +
  • D i
  • E3

Q11:

Find the derivative of the following vector-valued function: r(𝑡)=5𝑡+3𝑡2𝑒5(𝑡).cos

  • A 1 2 𝑒 1 0 𝑡 + 3 5 ( 𝑡 ) s i n
  • B 1 0 𝑡 + 3 2 𝑒 5 ( 𝑡 ) c o s
  • C 1 0 𝑡 + 3 1 2 𝑒 5 ( 𝑡 ) s i n
  • D 1 0 𝑡 + 3 1 2 𝑒 5 ( 𝑡 ) s i n
  • E 1 0 𝑡 + 3 2 𝑒 5 ( 𝑡 ) s i n

Q12:

Find the derivative of the vector-valued function rij(𝑡)=3𝑡𝑒.

  • A 1 3 3 𝑡 𝑒 9 i j
  • B 3 2 3 𝑡 9 𝑒 i j
  • C 1 3 𝑡 𝑒 i j
  • D 3 2 3 𝑡 + 9 𝑒 i j
  • E 1 3 𝑡 𝑒 9 i j

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