Worksheet: Lagrange Error Bound

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In this worksheet, we will practice using the Lagrange error bound (Taylor's theorem with remainder) to find the maximum error when using Taylor polynomial approximations.

Q1:

Find the error bound 𝑅 when using the second Taylor polynomial for the function 𝑓 ( 𝑥 ) = √ 𝑥 at 𝑥 = 4 to approximate the value √ 5 . Approximate to five decimal places.

  • A | 𝑅 | ≤ 0 . 0 0 6 8 1
  • B | 𝑅 | ≤ 0 . 0 0 1 9 5
  • C | 𝑅 | ≤ 0 . 5 3
  • D | 𝑅 | ≤ 0 . 0 1 2
  • E | 𝑅 | ≤ 0 . 0 3 6

Q2:

Determine the least degree of the Maclaurin polynomials 𝑛 needed for 𝑔 ( 𝑥 ) = 𝑒 , approximating 𝑒 where 𝑅 ( − 0 . 7 5 ) < 0 . 0 0 1 .

  • A 𝑛 = 4
  • B 𝑛 = 3
  • C 𝑛 = 2
  • D 𝑛 = 5
  • E 𝑛 = 6

Q3:

Find the error bound 𝑅 when using the third Maclaurin polynomial for the function 𝑓 ( 𝑥 ) = 𝑒 at 𝑥 = 0 to approximate the value 𝑒 .

  • A | 𝑅 | ≤ 1 1 , 5 0 0
  • B | 𝑅 | ≤ 1 1 , 4 2 5
  • C | 𝑅 | ≤ 1 1 5 , 0 0 0
  • D | 𝑅 | ≤ 1 1 , 3 7 5
  • E | 𝑅 | ≤ 1 6 2 5

Q4:

Determine the least degree of the Maclaurin polynomials 𝑛 needed to approximate the value of s i n 0 . 3 with an error less than 0.001 using the Maclaurin series of 𝑓 ( 𝑥 ) = 𝑥 s i n .

  • A 𝑛 = 1
  • B 𝑛 = 2
  • C 𝑛 = 3
  • D 𝑛 = 4
  • E 𝑛 = 5

Q5:

Find the error bound 𝑅 when using the third Talyor polynomial for the function 𝑓 ( 𝑥 ) = 𝑥 s i n at 𝑥 = 9 0 ∘ to approximate the value of s i n 9 4 ∘ .

  • A | 𝑅 | ≤ 8 . 8 9 8 × 1 0
  • B | 𝑅 | ≤ 8 . 9 9 8 × 1 0
  • C | 𝑅 | ≤ 9 . 8 9 8 × 1 0
  • D | 𝑅 | ≤ 9 . 8 9 8 × 1 0
  • E | 𝑅 | ≤ 9 . 8 9 8 × 1 0

Q6:

Determine the least degree of the Maclaurin polynomials 𝑛 for ℎ ( 𝑥 ) = ( 𝑥 + 1 ) l n , approximating ℎ ( 0 . 5 ) where | 𝑅 ( 0 . 5 ) | < 0 . 0 0 0 1 .

  • A 𝑛 = 7
  • B 𝑛 = 9
  • C 𝑛 = 6
  • D 𝑛 = 8
  • E 𝑛 = 1 0

Q7:

Determine the lowest degree of the Maclaurin polynomial 𝑛 needed to approximate the function 𝑓 ( 𝑥 ) = 𝑒 in the interval [ − 1 , 1 ] with an error less than 0.001 .

Q8:

Determine the lowest degree of the Maclaurin polynomial 𝑛 needed to approximate the function 𝑓 ( 𝑥 ) = 𝑒 in the interval [ − 4 , 4 ] with an error less than 0.001.

  • A 17
  • B 19
  • C 18
  • D 15
  • E 16

Q9:

Determine the lowest degree of the Maclaurin polynomial 𝑛 needed to approximate the function 𝑓 ( 𝑥 ) = 𝑥 s i n in the interval [ − 𝜋 , 𝜋 ] with an error less than 0.001.

  • A 14
  • B 11
  • C 12
  • D 10
  • E 13

Q10:

Determine the lowest degree of the Maclaurin polynomial 𝑛 needed to approximate the function 𝑓 ( 𝑥 ) = 𝑥 c o s on the interval − 𝜋 2 , 𝜋 2 with an error less than 0.001 .

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