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AP Calculus BC Practice Quiz: Lagrange Error Bound

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: June 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

What is the primary purpose of using the Lagrange error bound in the context of a Taylor polynomial approximation?

All Questions (9)

What is the primary purpose of using the Lagrange error bound in the context of a Taylor polynomial approximation?

A) To find the exact value of the function being approximated.

B) To determine a maximum possible value for the error of the approximation.

C) To calculate the next term in the Taylor series.

D) To prove that the Taylor series converges to the function.

Correct Answer: B

According to the provided content, the Lagrange error bound is used to 'determine a maximum interval for the error of a Taylor polynomial approximation to a function.' This means it establishes an upper bound on how large the error can be. [cite: 3225]

A student calculates the Lagrange error bound for a third-degree Taylor polynomial approximation, P₃(x), of a function f(x). This bound represents a value that...

A) is the exact difference between f(x) and P₃(x).

B) the actual error of the approximation will not exceed.

C) is only valid if the Taylor series is alternating.

D) is equal to the fourth term of the Taylor series.

Correct Answer: B

The Lagrange error bound provides a 'maximum interval for the error,' meaning it is an upper bound. The actual error of the approximation must be less than or equal to this calculated value. [cite: 3225]

Under which of the following conditions might the alternating series error bound be a suitable tool for analyzing the error of a Taylor polynomial approximation?

A) When the function being approximated is always increasing.

B) When the Taylor polynomial has only even-powered terms.

C) When the terms of the corresponding Taylor series alternate in sign and decrease in magnitude.

D) When the Lagrange error bound is too difficult to calculate.

Correct Answer: C

The provided content states that 'the alternating series error bound can be used to bound the error... in some situations.' The key situation for using this specific bound is when the Taylor series is an alternating series, which is characterized by terms that alternate in sign and decrease in absolute value towards zero. [cite: 3226]

If the Lagrange error bound for a Taylor polynomial approximation of f(x) on a given interval is calculated to be 0.01, what can be definitively concluded about the actual error of the approximation within that interval?

A) The actual error is exactly 0.01.

B) The actual error is greater than or equal to 0.01.

C) The actual error is less than or equal to 0.01.

D) The actual error is exactly 0.01 at the endpoints of the interval.

Correct Answer: C

The Lagrange error bound is used to 'determine a maximum interval for the error.' Therefore, the calculated bound of 0.01 is the maximum possible error, and the actual error must be less than or equal to this value. [cite: 3225]

The skill of determining the error bound for a Taylor polynomial approximation using methods like the Lagrange error bound is specified as a topic for which AP Calculus course?

A) AP Calculus AB

B) AP Calculus BC

C) Both AP Calculus AB and AP Calculus BC

D) AP Precalculus

Correct Answer: B

The provided content explicitly marks the topics of Lagrange error bound and alternating series error bound as '(BC ONLY)'. [cite: 3225, 3226]

Which statement best describes the relationship between the Lagrange error bound and the alternating series error bound for Taylor polynomials?

A) They are identical methods that always produce the same result.

B) The Lagrange error bound is a general method, while the alternating series error bound applies to the specific case of alternating series.

C) The alternating series error bound is more general and can be used for any Taylor polynomial.

D) The Lagrange error bound is used for functions, while the alternating series error bound is used for sequences.

Correct Answer: B

The content describes the Lagrange error bound as a tool to 'determine the error bound' generally [cite: 3224], while noting the alternating series error bound can be used 'in some situations' [cite: 3226]. This implies the Lagrange error bound is the more general tool, and the alternating series error bound is for a specific type of series.

The value obtained from the Lagrange error bound formula provides...

A) the exact error of the Taylor polynomial approximation.

B) a lower limit for the error of the approximation.

C) an upper limit for the error of the approximation.

D) the rate at which the error is changing.

Correct Answer: C

The purpose of the Lagrange error bound is to 'determine a maximum interval for the error.' This 'maximum' value serves as an upper limit, or upper bound, for the true error of the approximation. [cite: 3225]

If the Lagrange error bound for a Taylor polynomial Pₙ(x) approximating f(x) on the interval I is M, which of the following statements must be true?

A) For every x in the interval I, the inequality |f(x) - Pₙ(x)| ≤ M holds.

B) The value of the (n+1)th derivative of f(x) is equal to M somewhere in the interval I.

C) The Taylor series for f(x) must be an alternating series for the bound M to be valid.

D) The error of the approximation is exactly M at some point in the interval I.

Correct Answer: A

The Lagrange error bound M establishes the 'maximum interval for the error' over the entire interval I. This means that for any point x within that interval, the absolute difference between the true function value f(x) and the approximation Pₙ(x) is guaranteed to be no greater than M. [cite: 3225]

When approximating a function with a Taylor polynomial, a student notes that the corresponding Taylor series is alternating. What is the most accurate conclusion based on the provided content?

A) The Lagrange error bound is no longer a valid method for determining the error bound.

B) The alternating series error bound can potentially be used as a method to bound the error.

C) The error of the approximation is guaranteed to be smaller than if the series were not alternating.

D) The Taylor polynomial approximation will be exact.

Correct Answer: B

The content states that 'in some situations, the alternating series error bound can be used to bound the error.' The primary condition for this situation is that the series is alternating. This opens up the possibility of using this specific error bound, in addition to the more general Lagrange error bound. [cite: 3226]