Free response without a calculator (Part B)
Accumulation function from the graph of f
- Units 4, 5 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 and its sources
Use the description of the graph of f below. Let g be the function defined by g(x) = ∫₋₂ˣ f(t) dt for −4 ≤ x ≤ 4.
Graph of f (described in words)
The function f is defined on the closed interval −4 ≤ x ≤ 4. Its graph is continuous and is made of the following four pieces, from left to right:
1. A horizontal line segment from (−4, 2) to (−2, 2).
2. A line segment from (−2, 2) to (0, −2). It crosses the x-axis at (−1, 0).
3. The quarter of the circle x² + y² = 4 that lies in the fourth quadrant, from (0, −2) to (2, 0). It lies below the x-axis and bends upward (it is concave up).
4. A line segment from (2, 0) to (4, 3).
Source: Hypothetical function
Suggested time: 15 minutes
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Part (a)
2 pointsFind g(4) and g(−4).
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Part (b)
2 pointsOn what open interval or intervals is the graph of g both decreasing and concave up? Give a reason for your answer.
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Part (c)
3 pointsFind the absolute minimum value of g on the closed interval −4 ≤ x ≤ 4. Justify your answer.
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Part (d)
2 pointsFind lim (x→0) g(x)/(x² + 3x). Show the work that leads to your answer.
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