Worksheet: Arc Length of Vector-Valued Functions

In this worksheet, we will practice finding the length of the curve of a vector-valued function and using that to find the arc-length parameterization of a curve.

Q1:

Calculate the arc length of 𝑓 ( 𝑡 ) = 2 3 𝑡 , 2 3 𝑡 , 2 𝑡 c o s s i n over the given interval [ 0 , 1 ] .

  • A 9 2
  • B 2 5 8
  • C 2 9 2
  • D 5 8
  • E 2 3

Q2:

Calculate the arc length of 𝑓 ( 𝑡 ) = ( 3 2 𝑡 , 3 2 𝑡 , 3 𝑡 ) c o s s i n over the given interval 0 , 𝜋 2 .

  • A 3 𝜋 5 2
  • B 𝜋 5 2
  • C 𝜋 6 2
  • D 6 𝜋
  • E 3 𝜋 6 2

Q3:

Calculate the arc length of 𝑓 ( 𝑡 ) = 𝑡 + 1 𝑡 , 𝑡 + 1 𝑡 , 2 2 𝑡 c o s s i n over the given interval [ 0 , 1 ] .

  • A 7 2
  • B 1 0 3
  • C4
  • D2
  • E 5 2

Q4:

Parametrize the curve 𝑓 ( 𝑡 ) = ( 3 2 𝑡 , 3 2 𝑡 , 3 𝑡 ) c o s s i n for 𝑡 in 0 , 𝜋 2 using arc length parameterization.

  • A 𝑓 ( 𝑠 ) = 3 2 5 𝑠 1 5 , 3 2 5 𝑠 1 5 , 5 𝑠 5 c o s s i n for all 𝑠 in 0 , 𝜋 2
  • B 𝑓 ( 𝑠 ) = 3 2 𝑠 3 , 3 2 𝑠 3 , 2 𝑠 3 c o s s i n for all 𝑠 in 0 , 3 𝜋 2
  • C 𝑓 ( 𝑠 ) = 3 2 5 𝑠 1 5 , 3 2 5 𝑠 1 5 , 5 𝑠 5 c o s s i n for all 𝑠 in 0 , 3 𝜋 5 2
  • D 𝑓 ( 𝑠 ) = 3 𝑠 3 , 3 𝑠 3 , 2 𝑠 3 c o s s i n for all 𝑠 in 0 , 3 𝜋 2
  • E 𝑓 ( 𝑠 ) = 3 5 𝑠 1 5 , 3 5 𝑠 1 5 , 5 𝑠 5 c o s s i n for all 𝑠 in 0 , 3 𝜋 5 2

Q5:

Determine the arc length parameterization of the curve 𝑓 ( 𝑡 ) = 2 3 𝑡 , 2 3 𝑡 , 2 𝑡 c o s s i n for 𝑡 in [ 0 , 1 ] .

  • A 𝑓 ( 𝑠 ) = 2 𝑠 + 1 6 2 4 , 2 𝑠 + 1 6 2 4 , 𝑠 c o s s i n for all 𝑠 in 0 , 2 5 9
  • B 𝑓 ( 𝑠 ) = 2 𝑠 2 4 , 2 𝑠 2 1 2 , 𝑠 4 c o s s i n for all 𝑠 in [ 0 , 1 ]
  • C 𝑓 ( 𝑠 ) = 2 𝑠 2 4 , 2 𝑠 2 4 , 𝑠 1 6 c o s s i n for all 𝑠 in 1 6 , 2 5
  • D 𝑓 ( 𝑠 ) = 2 3 𝑠 2 1 2 , 2 3 𝑠 2 1 2 , 𝑠 1 6 c o s s i n for all 𝑠 in [ 0 , 1 ]
  • E 𝑓 ( 𝑠 ) = 2 𝑠 + 1 6 2 4 , 2 𝑠 + 1 6 2 4 , 2 𝑠 + 1 6 2 4 c o s s i n for all 𝑠 in 0 , 2 5 9

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