Aim for FiveCalculus AB

Free response without a calculator (Part B)

Medication level: slope field and separable 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

A patient receives a medication through an IV line at a constant rate of 6 milligrams per hour. The body removes the medication at a rate equal to one-fifth of the amount present. The amount of medication in the patient's body, in milligrams, is modeled by a function M of time t, in hours, that satisfies the differential equation dM/dt = 6 − M/5. At time t = 0 there is no medication in the body, so M(0) = 0. It can be shown that 0 ≤ M(t) < 30 for all t ≥ 0.

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 nine points (t, M) where t = 0, 2 or 4 and M = 10, 30 or 40. Find the slope at the points with M = 10, at the points with M = 30, and at the points with M = 40, and describe the segment you would draw at each of these points (horizontal, rising to the right, or falling to the right).

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Part (b)

3 points

Write an equation for the line tangent to the graph of M at t = 0, and use it to approximate M(1). Find d²M/dt² in terms of M. Is your approximation of M(1) an overestimate or an underestimate? Give a reason for your answer.

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Part (c)

4 points

Find M(t), the particular solution to the differential equation with the initial condition M(0) = 0.

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Part (d)

1 point

According to the model, at what time t does the amount of medication in the body reach 15 milligrams?

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