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Worksheet: Maclaurin Series

In this worksheet, we will practice representing exponential and trigonometric functions as power series, finding the expansion around zero, and finding the interval of the series convergence.

Q1:

Consider the function 𝑓 ( π‘₯ ) = 𝑒  .

Find 𝑓 ( π‘₯ )  .

  • A 𝑒 π‘₯  l n
  • B 𝑒   
  • C 𝑒 π‘₯    l n
  • D 𝑒 
  • E l n π‘₯

Find 𝑓 ( π‘₯ )    , where 𝑓    represents the 𝑛 th derivative of 𝑓 with respect to π‘₯ .

  • A 𝑒 
  • B 𝑒 π‘₯ + 𝑒 ( βˆ’ 1 ) ( 𝑛 βˆ’ 2 ) ! π‘₯             l n for 𝑛 > 1
  • C ( βˆ’ 1 ) ( 𝑛 βˆ’ 2 ) ! π‘₯       for 𝑛 > 1
  • D 𝑒   
  • E 𝑒 π‘₯ + 𝑒 ( βˆ’ 1 ) ( 𝑛 βˆ’ 2 ) ! π‘₯         l n for 𝑛 > 1

Hence, derive the Maclaurin series for 𝑒  .

  • A 𝑒 = ο„š 𝑓 ( π‘Ž ) ( π‘₯ βˆ’ π‘Ž ) 𝑛 !  ∞       
  • B 𝑒 = ο„š π‘₯ 𝑛 !  ∞    
  • C 𝑒 = ο„š π‘₯ 𝑛 !  ∞    
  • D 𝑒 = ο„š 𝑒 𝑛 !  ∞    
  • E 𝑒 = ο„š 𝑓 ( π‘Ž ) ( π‘₯ βˆ’ π‘Ž ) 𝑛 !  ∞       

What is the radius of convergence 𝑅 of the Maclaurin series for 𝑒  ?

  • A 𝑅 = + ∞
  • B 𝑅 = 1 0 0
  • C 𝑅 = 1
  • D 𝑅 = 𝑒
  • EIt does not converge.

Q2:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ s i n .

What are the first four derivatives of 𝑓 with respect to π‘₯ ?

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ s i n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  s i n
  • B 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , 𝑓 β€² β€² β€² ( π‘₯ ) = π‘₯ c o s , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  s i n
  • C 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , and 𝑓 ( π‘₯ ) = βˆ’ π‘₯  οŠͺ  s i n
  • D 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  s i n
  • E 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ s i n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  s i n

Write the general form for the 𝑛 th derivative of 𝑓 with respect to π‘₯ .

  • A 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2     s i n
  • B 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2     s i n
  • C 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2     c o s
  • D 𝑓 ( π‘₯ ) = ( π‘₯ + 𝑛 πœ‹ )    s i n
  • E 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2     c o s

Hence, derive the Maclaurin series for s i n π‘₯ .

  • A ∞       ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !
  • B ∞       ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 ) !
  • C ∞         ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 + 1 ) !
  • D ∞            ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 + 1 ) !
  • E ∞      ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !

What is the radius 𝑅 of convergence of the Maclaurin series for s i n π‘₯ ?

  • A 𝑅 = + ∞
  • B 𝑅 = 2 πœ‹
  • C 𝑅 = πœ‹
  • D 𝑅 = 1
  • E 𝑅 = πœ‹ 2

Q3:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ c o s .

What are the first four derivatives of 𝑓 with respect to π‘₯ ?

  • A 𝑓 β€² ( π‘₯ ) = π‘₯ s i n , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = π‘₯ s i n , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  c o s
  • B 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  s i n
  • C 𝑓 β€² ( π‘₯ ) = π‘₯ s i n , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , and 𝑓 ( π‘₯ ) = βˆ’ π‘₯  οŠͺ  c o s
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = π‘₯ s i n , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  c o s
  • E 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s i n , and 𝑓 ( π‘₯ ) = π‘₯  οŠͺ  c o s

Write the general form for the 𝑛 th derivative of 𝑓 with respect to π‘₯ .

  • A 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2     c o s
  • B 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2     s i n
  • C 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2     s i n
  • D 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2     c o s
  • E 𝑓 ( π‘₯ ) = ( π‘₯ + 𝑛 πœ‹ )    c o s

Hence, derive the Maclaurin series for c o s π‘₯ .

  • A ∞       ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !
  • B ∞         ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 + 1 ) !
  • C ∞       ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 ) !
  • D ∞        ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 ) !
  • E ∞      ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !

What is the radius 𝑅 of convergence of the Maclaurin series for c o s π‘₯ ?

  • A 𝑅 = + ∞
  • B 𝑅 = 2 πœ‹
  • C 𝑅 = πœ‹
  • D 𝑅 = 1
  • E 𝑅 = πœ‹ 2

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