Free response without a calculator (Part B)
BC: Maclaurin series for e^(−x²) and its integral
- 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 the function defined by f(x) = e^(−x²) for all real numbers x, and let F be the function defined by F(x) = ∫₀ˣ e^(−t²) dt.
Suggested time: 15 minutes
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Part (a)
2 pointsWrite the first four nonzero terms and the general term of the Maclaurin series for f. (You may start from the Maclaurin series for eˣ.)
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Part (b)
2 pointsFind the value of f⁽⁶⁾(0). Use the Maclaurin series for f to determine whether f has a relative maximum, a relative minimum, or neither at x = 0. Give a reason for your answer.
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Part (c)
3 pointsWrite the first four nonzero terms of the Maclaurin series for F. Use the first two nonzero terms of this series to approximate F(1/2). Show that this approximation differs from F(1/2) by less than 1/300.
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Part (d)
2 pointsUse the Maclaurin series for f to find lim (x→0) (1 − x² − f(x))/x⁴.
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