Free response without a calculator (Part B)
BC: Power series and convergence
- 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
The function f is defined by the power series f(x) = Σ (n = 1 to ∞) (x − 3)ⁿ/(n·2ⁿ) = (x − 3)/2 + (x − 3)²/8 + (x − 3)³/24 + ⋯ + (x − 3)ⁿ/(n·2ⁿ) + ⋯ for all x for which the series converges.
Suggested time: 15 minutes
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Part (a)
3 pointsFind the interval of convergence of the power series for f. Show the analysis that leads to your answer.
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Part (b)
3 pointsWrite the first three nonzero terms and the general term of the power series for f′(x). Find the value of f′(4).
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Part (c)
3 pointsUse the first two nonzero terms of the series for f(2) to approximate f(2). Show that this approximation differs from f(2) by less than 1/20.
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