Free response without a calculator (Part B)
Analyzing f from the graph of f′
- Units 2, 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. You may want to sketch it before you start.
Graph of f′ (described in words)
The function f is defined on the closed interval −4 ≤ x ≤ 6, and f(0) = 5. The graph of f′, the derivative of f, is continuous and is made of the following four pieces, from left to right:
1. A line segment from (−4, −2) to (0, 2).
2. A line segment from (0, 2) to (1, 0).
3. The lower half of the circle of radius 2 centered at (3, 0). It runs from (1, 0) down to its lowest point (3, −2) and back up to (5, 0), and lies below the x-axis for 1 < x < 5.
4. A line segment from (5, 0) to (6, 3).
Source: Hypothetical function
Suggested time: 15 minutes
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Part (a)
2 pointsFind all values of x in the open interval −4 < x < 6 at which f has a relative minimum. Justify your answer.
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Part (b)
2 pointsFind the x-coordinate of each point of inflection of the graph of f. Give a reason for your answer.
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Part (c)
3 pointsFind the absolute minimum value of f on the closed interval −4 ≤ x ≤ 6. Justify your answer.
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Part (d)
2 pointsLet h be the function defined by h(x) = x·f(x). Write an equation for the line tangent to the graph of h at x = 0.
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