Worksheet: Taylor Series

In this worksheet, we will practice finding Taylor series of a function and finding the radius of convergence of the series.

Q1:

The following table shows the value of function 𝑓 and some of its derivatives at 𝑥 = 2 .

𝑓 ( 2 ) 𝑓 ( 2 ) 𝑓 ( 2 ) ( ) 𝑓 ( 2 ) ( ) 𝑓 ( 2 ) ( )
3 1 5 8 6

Write the first 5 terms of the Taylor series of 𝑓 .

  • A 3 ( 𝑥 + 2 ) + 5 2 ( 𝑥 + 2 ) + 8 3 ( 𝑥 + 2 ) 3 2 ( 𝑥 + 2 )
  • B 3 ( 𝑥 2 ) + 5 2 ( 𝑥 2 ) + 4 3 ( 𝑥 2 ) 1 4 ( 𝑥 2 )
  • C 3 ( 𝑥 + 2 ) + 5 ( 𝑥 + 2 ) + 8 ( 𝑥 + 2 ) 6 ( 𝑥 + 2 )
  • D 3 ( 𝑥 2 ) + 5 ( 𝑥 2 ) + 8 ( 𝑥 2 ) 6 ( 𝑥 + 2 )
  • E 3 ( 𝑥 + 2 ) + 5 2 ( 𝑥 + 2 ) + 4 3 ( 𝑥 + 2 ) 1 4 ( 𝑥 + 2 )

Q2:

For the function 𝑓 : 𝑓 ( 3 ) = 2 , 𝑓 ( 3 ) = 7 and 𝑓 ( 3 ) = 1 2 𝑛 𝑓 ( 3 ) ( ) ( ) for 𝑛 2 .

Find the first five terms of the Taylor series representation of 𝑓 at 𝑥 = 3 .

  • A 2 + 7 ( 𝑥 3 ) 7 2 ( 𝑥 3 ) + 7 8 ( 𝑥 3 ) 7 9 6 ( 𝑥 3 )
  • B 2 + 7 ( 𝑥 3 ) 7 ( 𝑥 3 ) + 2 1 2 ( 𝑥 3 ) 2 1 ( 𝑥 3 )
  • C 2 + 7 ( 𝑥 3 ) 7 4 ( 𝑥 3 ) + 7 2 4 ( 𝑥 3 ) 7 1 9 2 ( 𝑥 3 )
  • D 2 + 7 ( 𝑥 3 ) 7 2 ( 𝑥 3 ) + 7 4 ( 𝑥 3 ) 7 8 ( 𝑥 3 )
  • E 2 + 7 ( 𝑥 3 ) 1 2 ( 𝑥 3 ) 1 4 ( 𝑥 3 ) 1 1 2 ( 𝑥 3 )

Q3:

Consider the function 𝑓 ( 𝑥 ) = ( 2 + 𝑥 ) .

Find the Taylor series representation of the function 𝑓 ( 𝑥 ) = ( 2 + 𝑥 ) at 𝑥 = 1 .

  • A 1 + 1 2 ( 𝑥 + 1 ) ( 1 2 ) ( 1 1 ) ( 𝑥 + 1 ) 2 + ( 1 2 ) ( 1 1 ) ( 1 0 ) ( 𝑥 + 1 ) 6 ( 1 2 ) ( 1 1 ) ( 1 0 ) ( 9 ) ( 𝑥 + 1 ) 2 4 +
  • B 1 + 1 2 ( 2 ) ( 𝑥 + 1 ) + ( 1 2 ) ( 1 1 ) ( 2 ) ( 𝑥 + 1 ) 2 + ( 1 2 ) ( 1 1 ) ( 1 0 ) ( 2 ) ( 𝑥 + 1 ) 6 + ( 1 2 ) ( 1 1 ) ( 1 0 ) ( 9 ) ( 2 ) ( 𝑥 + 1 ) 2 4 +
  • C 1 + ( 𝑥 + 1 ) + ( 𝑥 + 1 ) 2 + ( 𝑥 + 1 ) 6 + ( 𝑥 + 1 ) 2 4 +
  • D 1 + ( 𝑥 + 1 ) ( 𝑥 + 1 ) 2 + ( 𝑥 + 1 ) 6 ( 𝑥 + 1 ) 2 4 +
  • E 1 + 1 2 ( 𝑥 + 1 ) + ( 1 2 ) ( 1 1 ) ( 𝑥 + 1 ) 2 + ( 1 2 ) ( 1 1 ) ( 1 0 ) ( 𝑥 + 1 ) 6 + ( 1 2 ) ( 1 1 ) ( 1 0 ) ( 9 ) ( 𝑥 + 1 ) 2 4 +

Are the terms of the Taylor series representation for the function 𝑓 finite or infinite?

  • AInfinite
  • BFinite

Q4:

Consider the function 𝑓 ( 𝑥 ) = ( 𝑥 ) c o s .

Find the Taylor series expansion of 𝑓 ( 𝑥 ) = ( 𝑥 ) c o s at 𝑥 = 𝜋 .

  • A 1 ( 𝑥 𝜋 ) 2 ( 𝑥 𝜋 ) 2 4 ( 𝑥 𝜋 ) 7 2 0
  • B 1 + ( 𝑥 𝜋 ) 2 ( 𝑥 𝜋 ) 2 4 + ( 𝑥 𝜋 ) 7 2 0
  • C ( 𝑥 𝜋 ) ( 𝑥 𝜋 ) 6 ( 𝑥 𝜋 ) 1 2 0
  • D ( 𝑥 𝜋 ) + ( 𝑥 𝜋 ) 6 ( 𝑥 𝜋 ) 1 2 0 +
  • E 1 + ( 𝑥 𝜋 ) ( 𝑥 𝜋 ) 2 + ( 𝑥 𝜋 ) 6 ( 𝑥 𝜋 ) 2 4 +

Write the Taylor series expansion of 𝑓 ( 𝑥 ) in sigma notation.

  • A ( 𝑥 𝜋 ) ( 2 𝑚 + 1 ) !
  • B ( 𝑥 𝜋 ) ( 2 𝑚 ) !
  • C ( 1 ) ( 𝑥 𝜋 ) 𝑚 !
  • D ( 1 ) ( 𝑥 𝜋 ) ( 2 𝑚 + 1 ) !
  • E ( 1 ) ( 𝑥 𝜋 ) ( 2 𝑚 ) !

Q5:

For a function 𝑓 : 𝑓 ( 4 ) = 6 , 𝑓 ( 4 ) = 6 and 𝑓 ( 4 ) = 1 𝑛 𝑓 ( 4 ) ( ) ( ) for 𝑛 2 . Find the Taylor series expansion of 𝑓 at 𝑥 = 4 .

  • A ( 1 ) 6 ( 𝑛 ! ) ( 𝑥 + 4 )
  • B ( 1 ) 6 ( 𝑛 ! ) ( 𝑥 + 4 )
  • C ( 1 ) 6 ( 𝑛 ) ( 𝑛 ! ) ( 𝑥 + 4 )
  • D ( 1 ) 6 ( 𝑛 ! ) ( 𝑥 + 4 )
  • E ( 1 ) 6 ( 𝑛 ! ) ( 𝑥 + 4 )

Q6:

The Taylor series representation of the function 𝑓 is given by 𝑓 ( 𝑥 ) = 2 4 ( 𝑥 7 ) + 3 ( 𝑥 7 ) ( 𝑥 7 ) 6 + .

Find the value of the fourth derivative of 𝑓 at 𝑥 = 7 .

  • A 1 8
  • B 0
  • C 3
  • D 6
  • E 72

Find the value of the third derivative of 𝑓 at 𝑥 = 7 .

Q7:

What is the necessary and sufficient condition for the function 𝑓 to have a Taylor series representation about 𝑥 = 𝑎 ?

  • A 𝑓 ( 𝑎 ) is not equal to 0.
  • B 𝑓 ( 𝑎 ) ( ) is not equal to 𝑓 ( 𝑎 ) for any positive integer 𝑛 . ( 𝑓 ( ) is the 𝑛 t h derivative of 𝑓 .)
  • C 𝑓 ( 𝑎 ) ( ) exists for any positive integer 𝑛 . ( 𝑓 ( ) is the 𝑛 t h derivative of 𝑓 .)
  • D 𝑓 ( 𝑎 ) exists.
  • E 𝑓 ( 𝑎 ) ( ) is not equal to 0 for any positive integer 𝑛 . ( 𝑓 ( ) is the 𝑛 t h derivative of 𝑓 .)

Q8:

Consider the function 𝑓 ( 𝑥 ) = ( 𝑥 ) l n .

Find the Taylor series representation of 𝑓 about 𝑥 = 1 .

  • A ( 1 ) 𝑛 ( 𝑥 1 ) 𝑛 !
  • B ( 1 ) ( 𝑛 1 ) ( 𝑥 1 ) 𝑛 !
  • C ( 1 ) ( 𝑥 1 ) 𝑛
  • D ( 1 ) ( 𝑥 1 ) 𝑛
  • E ( 1 ) ( 𝑛 1 ) ( 𝑥 1 ) 𝑛 !

Find the interval of convergence of the Taylor series representation of 𝑓 about 𝑥 = 1 .

  • A [ 0 , 2 ]
  • B ( , )
  • C ( 1 , 1 ]
  • D ( 0 , 2 ]
  • E [ 1 , 1 ]

What is the radius of convergence of the Taylor series representation of 𝑓 about 𝑥 = 1 ?

  • A
  • B1
  • C 1 2
  • D2
  • E0

Q9:

What are the first four terms of the Taylor series of the function 𝑓 ( 𝑥 ) = 𝑥 about 𝑥 = 4 ?

  • A 𝑓 ( 𝑥 ) = 2 + 1 4 ( 𝑥 4 ) 1 6 4 ( 𝑥 4 ) + 1 5 1 2 ( 𝑥 4 )
  • B 𝑓 ( 𝑥 ) = 2 + 1 2 ( 𝑥 4 ) 1 9 6 ( 𝑥 4 ) + 1 1 9 2 ( 𝑥 4 )
  • C 𝑓 ( 𝑥 ) = 2 + 1 2 ( 𝑥 4 ) 1 1 6 ( 𝑥 4 ) + 1 1 9 2 ( 𝑥 4 )
  • D 𝑓 ( 𝑥 ) = 2 + 1 2 ( 𝑥 4 ) 1 8 ( 𝑥 4 ) + 1 1 6 ( 𝑥 4 )
  • E 𝑓 ( 𝑥 ) = 2 + 1 4 ( 𝑥 4 ) 1 3 2 ( 𝑥 4 ) + 3 2 5 6 ( 𝑥 4 )

Q10:

Consider the function 𝑓 ( 𝑥 ) = 𝑒 .

Find the Taylor series representation of 𝑓 about 𝑥 = 3 .

  • A ( 2 𝑥 3 ) 𝑛 !
  • B 2 𝑒 ( 𝑥 3 ) 𝑛 !
  • C 𝑒 ( 𝑥 3 ) 𝑛 !
  • D 𝑒 ( 2 𝑥 3 ) 𝑛 !
  • E 2 ( 𝑥 ) 𝑛 !

Find the interval of convergence of the Taylor series representation of 𝑓 about 𝑥 = 3 .

  • A 5 2 , 7 2
  • B 5 2 , 7 2
  • C ( , )
  • D 5 2 , 7 2
  • E 5 2 , 7 2

What is the radius of convergence of the Taylor series representation of 𝑓 about 𝑥 = 3 ?

  • A 1 2
  • B
  • C1
  • D0
  • E2

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