Lesson Worksheet: Taylor Series Mathematics • Higher Education

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 π‘₯=βˆ’2.

𝑓(βˆ’2)𝑓(βˆ’2)οŽ˜π‘“(βˆ’2)()𝑓(βˆ’2)()𝑓(βˆ’2)(οŠͺ)
3βˆ’158βˆ’6

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

  • A3βˆ’(π‘₯+2)+5(π‘₯+2)+8(π‘₯+2)βˆ’6(π‘₯+2)οŠͺ
  • B3βˆ’(π‘₯+2)+52(π‘₯+2)+43(π‘₯+2)βˆ’14(π‘₯+2)οŠͺ
  • C3βˆ’(π‘₯βˆ’2)+5(π‘₯βˆ’2)+8(π‘₯βˆ’2)βˆ’6(π‘₯+2)οŠͺ
  • D3βˆ’(π‘₯βˆ’2)+52(π‘₯βˆ’2)+43(π‘₯βˆ’2)βˆ’14(π‘₯βˆ’2)οŠͺ
  • E3βˆ’(π‘₯+2)+52(π‘₯+2)+83(π‘₯+2)βˆ’32(π‘₯+2)οŠͺ

Q2:

Consider the function 𝑓(π‘₯)=(2+π‘₯).

Find the Taylor series representation of the function 𝑓(π‘₯)=(2+π‘₯) at π‘₯=βˆ’1.

  • Aβˆ’1+12(π‘₯+1)βˆ’(12)(11)(π‘₯+1)2+(12)(11)(10)(π‘₯+1)6βˆ’(12)(11)(10)(9)(π‘₯+1)24+β‹―οŠ¨οŠ©οŠͺ
  • B1+(π‘₯+1)+(π‘₯+1)2+(π‘₯+1)6+(π‘₯+1)24+β‹―οŠ¨οŠ©οŠͺ
  • C1+12(2)(π‘₯+1)+(12)(11)(2)(π‘₯+1)2+(12)(11)(10)(2)(π‘₯+1)6+(12)(11)(10)(9)(2)(π‘₯+1)24+β‹―οŠ¨οŠ¨οŠ©οŠ©οŠͺοŠͺ
  • Dβˆ’1+(π‘₯+1)βˆ’(π‘₯+1)2+(π‘₯+1)6βˆ’(π‘₯+1)24+β‹―οŠ¨οŠ©οŠͺ
  • E1+12(π‘₯+1)+(12)(11)(π‘₯+1)2+(12)(11)(10)(π‘₯+1)6+(12)(11)(10)(9)(π‘₯+1)24+β‹―οŠ¨οŠ©οŠͺ

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

  • AFinite
  • BInfinite

Q3:

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

Find the Taylor series expansion of 𝑓(π‘₯)=(π‘₯)cos at π‘₯=πœ‹.

  • Aβˆ’(π‘₯βˆ’πœ‹)βˆ’(π‘₯βˆ’πœ‹)6βˆ’(π‘₯βˆ’πœ‹)120βˆ’β‹―οŠ©οŠ«
  • Bβˆ’1βˆ’(π‘₯βˆ’πœ‹)2βˆ’(π‘₯βˆ’πœ‹)24βˆ’(π‘₯βˆ’πœ‹)720βˆ’β‹―οŠ¨οŠͺ
  • Cβˆ’(π‘₯βˆ’πœ‹)+(π‘₯βˆ’πœ‹)6βˆ’(π‘₯βˆ’πœ‹)120+β‹―οŠ©οŠ«
  • Dβˆ’1+(π‘₯βˆ’πœ‹)2βˆ’(π‘₯βˆ’πœ‹)24+(π‘₯βˆ’πœ‹)720βˆ’β‹―οŠ¨οŠͺ
  • Eβˆ’1+(π‘₯βˆ’πœ‹)βˆ’(π‘₯βˆ’πœ‹)2+(π‘₯βˆ’πœ‹)6βˆ’(π‘₯βˆ’πœ‹)24+β‹―οŠ¨οŠ©οŠͺ

Write the Taylor series expansion of 𝑓(π‘₯) in sigma notation.

  • Aβˆžο‰οŠ²οŠ¦ο‰οŠ°οŠ§οŠ¨ο‰οŠ°οŠ§ο„š(βˆ’1)(π‘₯βˆ’πœ‹)(2π‘š+1)!
  • Bβˆžο‰οŠ²οŠ¦οŠ¨ο‰ο„šβˆ’(π‘₯βˆ’πœ‹)(2π‘š)!
  • Cβˆžο‰οŠ²οŠ¦ο‰οŠ°οŠ§ο‰ο„š(βˆ’1)(π‘₯βˆ’πœ‹)π‘š!
  • Dβˆžο‰οŠ²οŠ¦οŠ¨ο‰οŠ°οŠ§ο„šβˆ’(π‘₯βˆ’πœ‹)(2π‘š+1)!
  • Eβˆžο‰οŠ²οŠ¦ο‰οŠ°οŠ§οŠ¨ο‰ο„š(βˆ’1)(π‘₯βˆ’πœ‹)(2π‘š)!

Q4:

For a function 𝑓: 𝑓(βˆ’4)=6, 𝑓(βˆ’4)=βˆ’6 and 𝑓(βˆ’4)=βˆ’1𝑛𝑓(βˆ’4)()() for 𝑛β‰₯2. Find the Taylor series expansion of 𝑓 at π‘₯=βˆ’4.

  • AβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)6(𝑛)(𝑛!)(π‘₯+4)
  • BβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠο„š(βˆ’1)6(𝑛!)(π‘₯+4)
  • CβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)6(𝑛!)(π‘₯+4)
  • DβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)6(𝑛!)(π‘₯+4)
  • EβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠ¨οŠο„š(βˆ’1)6(𝑛!)(π‘₯+4)

Q5:

For the function 𝑓: 𝑓(3)=2, 𝑓(3)=7 and 𝑓(3)=βˆ’12𝑛𝑓(3)()() for 𝑛β‰₯2.

Find the first five terms of the Taylor series representation of 𝑓 at π‘₯=3.

  • A2+7(π‘₯βˆ’3)βˆ’12(π‘₯βˆ’3)βˆ’14(π‘₯βˆ’3)βˆ’112(π‘₯βˆ’3)οŠͺ
  • B2+7(π‘₯βˆ’3)βˆ’74(π‘₯βˆ’3)+724(π‘₯βˆ’3)βˆ’7192(π‘₯βˆ’3)οŠͺ
  • C2+7(π‘₯βˆ’3)βˆ’7(π‘₯βˆ’3)+212(π‘₯βˆ’3)βˆ’21(π‘₯βˆ’3)οŠͺ
  • D2+7(π‘₯βˆ’3)βˆ’72(π‘₯βˆ’3)+74(π‘₯βˆ’3)βˆ’78(π‘₯βˆ’3)οŠͺ
  • E2+7(π‘₯βˆ’3)βˆ’72(π‘₯βˆ’3)+78(π‘₯βˆ’3)βˆ’796(π‘₯βˆ’3)οŠͺ

Q6:

The Taylor series representation of the function 𝑓 is given by 𝑓(π‘₯)=2βˆ’4(π‘₯βˆ’7)+3(π‘₯βˆ’7)βˆ’(π‘₯βˆ’7)6+β‹―οŠ¨οŠͺ.

Find the value of the fourth derivative of 𝑓 at π‘₯=7.

  • A18
  • B3
  • C72
  • D0
  • E6

Find the value of the third derivative of 𝑓 at π‘₯=7.

Q7:

What is the necessary and sufficient condition for the function 𝑓 to have a Taylor series representation about π‘₯=π‘Ž?

  • A𝑓(π‘Ž)() is not equal to 𝑓(π‘Ž) for any positive integer 𝑛. (𝑓() is the 𝑛th derivative of 𝑓.)
  • B𝑓(π‘Ž) exists.
  • C𝑓(π‘Ž) is not equal to 0.
  • D𝑓(π‘Ž)() is not equal to 0 for any positive integer 𝑛. (𝑓() is the 𝑛th derivative of 𝑓.)
  • E𝑓(π‘Ž)() exists for any positive integer 𝑛. (𝑓() is the 𝑛th derivative of 𝑓.)

Q8:

What are the first four terms of the Taylor series of the function 𝑓(π‘₯)=√π‘₯ about π‘₯=4?

  • A𝑓(π‘₯)=2+12(π‘₯βˆ’4)βˆ’18(π‘₯βˆ’4)+116(π‘₯βˆ’4)βˆ’β‹―οŠ¨οŠ©
  • B𝑓(π‘₯)=2+14(π‘₯βˆ’4)βˆ’164(π‘₯βˆ’4)+1512(π‘₯βˆ’4)βˆ’β‹―οŠ¨οŠ©
  • C𝑓(π‘₯)=2+12(π‘₯βˆ’4)βˆ’196(π‘₯βˆ’4)+1192(π‘₯βˆ’4)βˆ’β‹―οŠ¨οŠ©
  • D𝑓(π‘₯)=2+12(π‘₯βˆ’4)βˆ’116(π‘₯βˆ’4)+1192(π‘₯βˆ’4)βˆ’β‹―οŠ¨οŠ©
  • E𝑓(π‘₯)=2+14(π‘₯βˆ’4)βˆ’132(π‘₯βˆ’4)+3256(π‘₯βˆ’4)βˆ’β‹―οŠ¨οŠ©

Q9:

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

Find the Taylor series representation of 𝑓 about π‘₯=1.

  • AβˆžοŠοŠ²οŠ§οŠοŠ°οŠ§οŠο„š(βˆ’1)𝑛(π‘₯βˆ’1)𝑛!
  • BβˆžοŠοŠ²οŠ§οŠοŠ°οŠ§οŠο„š(βˆ’1)(π‘›βˆ’1)(π‘₯βˆ’1)𝑛!
  • CβˆžοŠοŠ²οŠ§οŠοŠ°οŠ§οŠο„š(βˆ’1)(π‘₯βˆ’1)𝑛
  • DβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠο„š(βˆ’1)(π‘₯βˆ’1)𝑛
  • EβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠο„š(βˆ’1)(π‘›βˆ’1)(π‘₯βˆ’1)𝑛!

Find the interval of convergence of the Taylor series representation of 𝑓 about π‘₯=1.

  • A[0,2]
  • B(0,2]
  • C(βˆ’βˆž,∞)
  • D(βˆ’1,1]
  • E[βˆ’1,1]

What is the radius of convergence of the Taylor series representation of 𝑓 about π‘₯=1?

  • A2
  • B∞
  • C0
  • D12
  • E1

Q10:

Consider the function 𝑓(π‘₯)=π‘’οŠ¨ο—.

Find the Taylor series representation of 𝑓 about π‘₯=3.

  • AβˆžοŠοŠ²οŠ¦οŠ¬οŠο„šπ‘’(π‘₯βˆ’3)𝑛!
  • BβˆžοŠοŠ²οŠ¦οŠο„š(2π‘₯βˆ’3)𝑛!
  • CβˆžοŠοŠ²οŠ¦οŠοŠ¬οŠο„š2𝑒(π‘₯βˆ’3)𝑛!
  • DβˆžοŠοŠ²οŠ¦οŠ©οŠο„šπ‘’(2π‘₯βˆ’3)𝑛!
  • EβˆžοŠοŠ²οŠ¦οŠοŠο„š2(π‘₯)𝑛!

Find the interval of convergence of the Taylor series representation of 𝑓 about π‘₯=3.

  • A52,72
  • Bο€Ό52,72
  • C52,72
  • Dο€Ό52,72
  • E(βˆ’βˆž,∞)

What is the radius of convergence of the Taylor series representation of 𝑓 about π‘₯=3?

  • A∞
  • B2
  • C12
  • D0
  • E1

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