Worksheet: Lagrange Error Bound

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

  • A|𝑅|≤0.01200
  • B|𝑅|≤0.00195
  • C|𝑅|≤0.03600
  • D|𝑅|≤0.53000
  • E|𝑅|≤0.00681

Q2:

Determine the least degree of the Maclaurin polynomials 𝑛 needed for 𝑔(𝑥)=𝑒, approximating 𝑒 where |𝑅(−0.75)|≤0.001.

  • A𝑛=4
  • B𝑛=3
  • C𝑛=2
  • D𝑛=6
  • E𝑛=5

Q3:

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

  • A|𝑅|≤11,500
  • B|𝑅|≤11,425
  • C|𝑅|≤1625
  • D|𝑅|≤11,375
  • E|𝑅|≤115,000

Q4:

Determine the least degree of the Maclaurin polynomials 𝑛 needed to approximate the value of sin0.3 with an error less than 0.001 using the Maclaurin series of 𝑓(𝑥)=𝑥sin.

  • A𝑛=3
  • B𝑛=4
  • C𝑛=2
  • D𝑛=1
  • E𝑛=5

Q5:

Find the Lagrange error bound when using the third Talyor polynomial for the function 𝑓(𝑥)=𝑥sin at 𝑥=90∘ to approximate the value of sin94∘.

  • A|𝑅|≤8.998×10
  • B|𝑅|≤8.898×10
  • C|𝑅|≤9.898×10
  • D|𝑅|≤9.898×10
  • E|𝑅|≤9.898×10

Q6:

Determine the least degree of the Maclaurin polynomials 𝑛 for ℎ(𝑥)=(𝑥+1)ln, approximating ℎ(0.5) where |𝑅(0.5)|<0.0001.

  • A𝑛=7
  • B𝑛=9
  • C𝑛=6
  • D𝑛=8
  • E𝑛=10

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.

  • A15
  • B17
  • C16
  • D19
  • E18

Q9:

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

  • A14
  • B10
  • C13
  • D12
  • E11

Q10:

Determine the lowest degree of the Maclaurin polynomial 𝑛 needed to approximate the function 𝑓(𝑥)=𝑥cos on the interval −𝜋2,𝜋2 with an error less than 0.001 .

Q11:

Find the error bound when using the fourth Taylor polynomial for the function 𝑓(𝑥)=(−𝜋𝑥)sin at 𝑥=1 to approximate the value of 𝑓45. Give your answer in scientific form to three significant figures.

  • A9.79×10
  • B2.7×10
  • C6.6×10
  • D2.67×10
  • E8.16×10

Q12:

Find the error bound when using the second Taylor polynomial for the function 𝑓(𝑥)=𝑥 at 𝑥=8 to approximate the value of 𝑓(8.5). Give your answer in scientific form to three significant figures.

  • A|𝑅|≤7.16×10
  • B|𝑅|≤7.02×10
  • C|𝑅|≤3.58×10
  • D|𝑅|≤3.01×10
  • E|𝑅|≤2.148×10

Q13:

Find the error bound when using the third Maclaurin polynomial for the function 𝑓(𝑥)=𝑒 to approximate the value of 𝑓(0.2). Give your answer to five decimal places.

  • A|𝑅|≤0.00107
  • B|𝑅|≤0.00006
  • C|𝑅|≤0.003
  • D|𝑅|≤0.03816
  • E|𝑅|≤0.00159

Q14:

Find the error bound when using the third Taylor polynomial for the function 𝑓(𝑥)=3𝑥ln at 𝑥=13 to approximate the value of 𝑓12.Give your answer in scientific form to three significant figures.

  • A|𝑅|≤3.75×10
  • B|𝑅|≤3.09×10
  • C|𝑅|≤1.56×10
  • D|𝑅|≤9.38×10
  • E|𝑅|≤4.17×10

Q15:

Find the error bound when using the third Maclaurin polynomial for the function 𝑓(𝑥)=3𝑥𝑒 to approximate the value of 𝑓(−0.1). Give your answer in scientific form to three significant figures.

  • A|𝑅|≤6.17×10
  • B|𝑅|≤4.00×10
  • C|𝑅|≤6.75×10
  • D|𝑅|≤4.17×10
  • E|𝑅|≤1.48×10

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