Aim for FiveCalculus BC

Free response without a calculator (Part B)

BC: Taylor polynomial and Lagrange error bound

  • Unit 10
  • 9 points
  • About 15 minutes
  • BC only

A multi-part problem you solve by hand, often from a graph, a table, an equation or a differential equation. You show your work, use exact values, and justify answers with calculus reasons such as a sign change in a derivative or the conditions of a theorem. On the exam: 4 questions in Part B of the free-response section (60 minutes, no calculator). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, Part B usually includes a series question and a question on BC-only topics.

The question

Let f be a function that has derivatives of all orders for all real numbers. Selected values of f and its derivatives are f(2) = 3, f′(2) = −2, f″(2) = 5 and f‴(2) = −12. The fourth derivative of f satisfies |f⁽⁴⁾(x)| ≤ 40 for all x in the interval 1.5 ≤ x ≤ 2.5.

Suggested time: 15 minutes

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Part (a)

2 points

Write the third-degree Taylor polynomial for f about x = 2.

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Part (b)

3 points

Use the polynomial from part (a) to approximate f(1.8). Use the Lagrange error bound to show that |f(1.8) − P₃(1.8)| < 0.003.

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Part (c)

2 points

Let g be the function defined by g(x) = 4 + ∫₂ˣ f(t) dt. Write the fourth-degree Taylor polynomial for g about x = 2.

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Part (d)

2 points

Can it be concluded that f(1.8) > 3.5? Explain your reasoning.

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