Free response without a calculator (Part B)
Cooling soup: data table
- Units 2, 4, 5, 6 and 8
- 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
A pot of soup is taken off a stove and set on a counter. The temperature of the soup, in degrees Celsius (°C), is modeled by a twice-differentiable function H of time t, where t is measured in minutes. Selected values of H(t) are given in the table.
Temperature of the soup
| t (minutes) | H(t) (°C) |
|---|---|
| 0 | 88 |
| 3 | 76 |
| 7 | 64 |
| 12 | 52 |
| 15 | 46 |
Source: Hypothetical data
Suggested time: 15 minutes
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Part (a)
2 pointsUse the data in the table to approximate H′(9.5). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of the problem.
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Part (b)
2 pointsIs there a time t, for 3 < t < 7, at which H′(t) = −3? Justify your answer.
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Part (c)
3 pointsUse a trapezoidal sum with the four subintervals indicated by the table to approximate (1/15)∫₀¹⁵ H(t) dt. Using correct units, explain the meaning of (1/15)∫₀¹⁵ H(t) dt in the context of the problem.
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Part (d)
2 pointsIt is known that H′(15) = −1.6 and that H″(t) > 0 for 0 ≤ t ≤ 20. Use the line tangent to the graph of H at t = 15 to approximate H(17). Is this approximation an overestimate or an underestimate of H(17)? Give a reason for your answer.
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