Aim for FiveCalculus AB

Free response without a calculator (Part B)

Limits, continuity and differentiability

  • Units 1 and 2
  • 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) = (x² + x − 6)/(x − 2) for x < 2, and f(x) = ax² + bx for x ≥ 2, where a and b are constants.

Suggested time: 15 minutes

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Part (a)

2 points

Suppose a = 1 and b = 1. Find lim (x→2⁻) f(x). Is f continuous at x = 2? Justify your answer using the definition of continuity.

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Part (b)

3 points

Find the values of a and b for which f is both continuous and differentiable at x = 2. Show the work that leads to your answer.

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Part (c)

2 points

Using the values of a and b from part (b), find lim (x→2⁺) (f(x) − 5)/(x − 2). Show the work that leads to your answer.

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Part (d)

2 points

Using the values of a and b from part (b), explain why there must be a value c, for 2 < c < 6, such that f(c) = 0.

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