Aim for FiveCalculus BC

Free response without a calculator (Part B)

Implicitly defined curve

  • Units 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

Consider the curve given by the equation x²y + y³ = 10.

Suggested time: 15 minutes

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Part (a)

2 points

Show that dy/dx = −2xy/(x² + 3y²).

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Part (b)

2 points

Write an equation for the line tangent to the curve at the point (1, 2). Use this line to approximate the y-coordinate of the point on the curve near (1, 2) where x = 1.3.

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Part (c)

2 points

Find the coordinates of every point on the curve at which the line tangent to the curve is horizontal.

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Part (d)

3 points

Near the point found in part (c), the curve is the graph of a twice-differentiable function y = f(x). Does f have a relative minimum, a relative maximum, or neither at that point? Justify your answer.

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