Free response without a calculator (Part B)
Table of function and derivative values
- Units 2, 3, 4 and 5
- 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
The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table gives values of the functions and their derivatives at selected values of x.
Values of f, f′, g and g′
| x | f(x) | f′(x) | g(x) | g′(x) |
|---|---|---|---|---|
| 1 | 6 | −3 | 2 | 3 |
| 2 | 4 | −1 | 3 | 2 |
| 3 | 3 | −2 | 5 | 2 |
| 4 | −3 | −8 | 8 | 4 |
Source: Hypothetical data
Suggested time: 15 minutes
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Part (a)
2 pointsLet h be the function defined by h(x) = f(g(x)). Find h′(1).
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Part (b)
2 pointsLet k be the function defined by k(x) = f(x)/g(x). Write an equation for the line tangent to the graph of k at x = 2.
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Part (c)
1 pointLet g⁻¹ be the inverse function of g. Find (g⁻¹)′(5).
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Part (d)
2 pointsFind lim (x→2) (f(x) − 4)/(g(x) − 3). Show the work that leads to your answer.
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Part (e)
2 pointsMust there be a value c, for 1 < c < 4, such that f′(c) = −3? Justify your answer.
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