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Calculus II practice

Taylor Series Approximations and Error worksheets

Represent standard functions, approximate nearby values, and apply alternating or Lagrange error bounds.

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Represent standard functions, approximate nearby values, and apply alternating or Lagrange error bounds.

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Conclusions are tied to visible reasoning

Generated keys distinguish sequence limits, partial sums, infinite-series convergence, endpoint behavior, and approximation error instead of collapsing them into a single answer.

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What this activity builds

Separate the exact function value, polynomial approximation, actual error, and proven bound.

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The governing limit or test, its hypotheses, the calculation, and a precise conclusion.

The worked key preserves the reasoning that distinguishes similar-looking sequence and series tasks.

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  • Have students name the mathematical object before choosing a method.
  • Require separate endpoint checks for power series.
  • Keep exact values and labeled approximations side by side.

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Math SheetNameScoreTeacherDateTaylor Series Approximations and ErrorUse standard Taylor or Maclaurin series to represent a function, approximate a nearby value, and distinguish the exact target from the polynomial approximation and its error bound.1.For f(x)=1/(1-x) centered at a=0, Write the order-4 Taylor polynomial and the infinite-series representation, state its convergence interval, andverify that x=1/10 is inside it. Use the lagrange method for the error statement.2.For f(x)=cos(x) centered at a=0, Write the order-4 Taylor polynomial and the infinite-series representation, state its convergence interval, andverify that x=4/5 is inside it. Use the lagrange method for the error statement.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateTaylor Series Approximations and ErrorUse standard Taylor or Maclaurin series to represent a function, approximate a nearby value, and distinguish the exact target from the polynomial approximation and its error bound.3.For f(x)=ln(1+x) centered at a=0, Use the lagrange remainder method to bound the error at the exact point x=2/5. Use the lagrange method for theerror statement.4.For f(x)=e^x centered at a=0, Write the order-4 Taylor polynomial and the infinite-series representation, state its convergence interval, and verifythat x=9/20 is inside it. Use the lagrange method for the error statement.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateTaylor Series Approximations and ErrorUse standard Taylor or Maclaurin series to represent a function, approximate a nearby value, and distinguish the exact target from the polynomial approximation and its error bound.5.For f(x)=cos(x) centered at a=0, Use the order-4 polynomial to approximate the exact target f(2/5)=cos(2/5). Use the alternating method for theerror statement.6.For f(x)=sin(x) centered at a=0, Use the lagrange remainder method to bound the error at the exact point x=2/5. Use the lagrange method for theerror statement.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 3 of 3

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