Aim for FiveCalculus BC

Free response without a calculator (Part B)

Slope field and separable differential equation

  • Units 3, 4 and 7
  • 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 differential equation dy/dx = (x + 1)(1 + y²). Let y = f(x) be the particular solution to the differential equation with initial condition f(0) = 1.

Suggested time: 15 minutes

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Part (a)

1 point

A slope field for this differential equation is to be drawn at the six points (x, y) where x = −2, −1 or 0 and y = 0 or 1. Find the slope at each of the six points, and describe the short segment you would draw at each point (horizontal, rising to the right, or falling to the right, and which segments are steeper).

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Part (b)

1 point

Use the line tangent to the graph of y = f(x) at x = 0 to approximate f(0.2).

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Part (c)

2 points

Find d²y/dx² in terms of x and y. It can be shown that f(x) ≥ 1 for 0 ≤ x ≤ 0.2. Is the approximation in part (b) an overestimate or an underestimate of f(0.2)? Give a reason for your answer.

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Part (d)

5 points

Find y = f(x), the particular solution to the differential equation with the initial condition f(0) = 1.

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