Free response without a calculator (Part B)
A function defined by an integral, its inverse and two integrals
- Units 3 and 6
- 9 points
- About 15 minutes
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) = ∫₂ˣ √(t³ + 1) dt for x ≥ −1.
Suggested time: 15 minutes
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Part (a)
2 pointsExplain why f has an inverse function. Let g be the inverse function of f. Find g(0) and g′(0).
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Part (b)
2 pointsLet k be the function defined by k(x) = f(x² − 2) for x ≥ 1. Write an equation for the line tangent to the graph of k at x = 2.
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Part (c)
3 pointsEvaluate ∫₀² x²√(x³ + 1) dx. Show the work that leads to your answer.
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Part (d)
2 pointsEvaluate ∫₀¹ (x² + 3)/(x + 1) dx. Show the work that leads to your answer.
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