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Lesson Worksheet: Lagrange Error Bound Mathematics • Higher Education

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 ๐‘“(๐‘ฅ)=โˆš๐‘ฅ at ๐‘ฅ=4 to approximate the value โˆš5. Round to five decimal places.

  • A|๐‘…|โ‰ค0.01200๏Šจ
  • B|๐‘…|โ‰ค0.00195๏Šจ
  • C|๐‘…|โ‰ค0.03600๏Šจ
  • D|๐‘…|โ‰ค0.53000๏Šจ
  • E|๐‘…|โ‰ค0.00681๏Šจ

Q2:

Find the Lagrange error bound when using the third Maclaurin polynomial for the function ๐‘“(๐‘ฅ)=๐‘’๏— at ๐‘ฅ=0 to approximate the value ๐‘’๏Šฑ๏Šฆ๏Ž–๏Šจ.

  • A|๐‘…|โ‰ค11,500๏Šฉ
  • B|๐‘…|โ‰ค11,425๏Šฉ
  • C|๐‘…|โ‰ค1625๏Šฉ
  • D|๐‘…|โ‰ค11,375๏Šฉ
  • E|๐‘…|โ‰ค115,000๏Šฉ

Q3:

Determine the least degree of the Maclaurin polynomials ๐‘› needed to approximate the value of sin0.3 with an error less than 0.001 using the Maclaurin series of ๐‘“(๐‘ฅ)=๐‘ฅsin.

  • A๐‘›=3
  • B๐‘›=4
  • C๐‘›=2
  • D๐‘›=1
  • E๐‘›=5

Q4:

Determine the least degree of the Maclaurin polynomials ๐‘› needed for ๐‘”(๐‘ฅ)=๐‘’๏—, approximating ๐‘’๏Šฑ๏Šฆ๏Ž–๏Šญ๏Šซ where |๐‘…(โˆ’0.75)|โ‰ค0.001๏Š.

  • A๐‘›=4
  • B๐‘›=3
  • C๐‘›=2
  • D๐‘›=6
  • E๐‘›=5

Q5:

Determine the least degree of the Maclaurin polynomials ๐‘› for โ„Ž(๐‘ฅ)=(๐‘ฅ+1)ln, approximating โ„Ž(0.5) where |๐‘…(0.5)|<0.0001๏Š.

  • A๐‘›=7
  • B๐‘›=9
  • C๐‘›=6
  • D๐‘›=8
  • E๐‘›=10

Q6:

Find the Lagrange error bound when using the third Talyor polynomial for the function ๐‘“(๐‘ฅ)=๐‘ฅsin at ๐‘ฅ=90โˆ˜ to approximate the value of sin94โˆ˜.

  • A|๐‘…|โ‰ค8.998ร—10๏Šฉ๏Šฑ๏Šญ
  • B|๐‘…|โ‰ค8.898ร—10๏Šฉ๏Šฑ๏Šฌ
  • C|๐‘…|โ‰ค9.898ร—10๏Šฉ๏Šฑ๏Šฎ
  • D|๐‘…|โ‰ค9.898ร—10๏Šฉ๏Šฑ๏Šญ
  • E|๐‘…|โ‰ค9.898ร—10๏Šฉ๏Šฑ๏Šฌ

Q7:

Determine the lowest degree of the Maclaurin polynomial ๐‘› needed to approximate the function ๐‘“(๐‘ฅ)=๐‘’๏Šฑ๏— in the interval [โˆ’4,4] with an error less than 0.001.

  • A15
  • B17
  • C16
  • D19
  • E18

Q8:

Determine the lowest degree of the Maclaurin polynomial ๐‘› needed to approximate the function ๐‘“(๐‘ฅ)=๐‘ฅsin in the interval [โˆ’๐œ‹,๐œ‹] with an error less than 0.001.

  • A14
  • B10
  • C13
  • D12
  • E11

Q9:

Determine the lowest degree of the Maclaurin polynomial ๐‘› needed to approximate the function ๐‘“(๐‘ฅ)=๐‘’๏Šฑ๏Šฉ๏— in the interval [โˆ’1,1] with an error less than 0.001 .

Q10:

Determine the lowest degree of the Maclaurin polynomial ๐‘› needed to approximate the function ๐‘“(๐‘ฅ)=๐‘ฅcos on the interval ๏“โˆ’๐œ‹2,๐œ‹2๏Ÿ with an error less than 0.001 .

This lesson includes 5 additional questions and 45 additional question variations for subscribers.

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