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AP Calculus BC 7.4 Reasoning Using Slope Fields - Exam Style Questions - MCQs - New Syllabus

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Question

The figure shows a slope field. For which differential equation is this a slope field?
(A) \( \displaystyle \frac{dy}{dx}=\frac{y^{2}}{(x-2)^{2}} \)
(B) \( \displaystyle \frac{dy}{dx}=\frac{1}{(x-2)^{3}} \)
(C) \( \displaystyle \frac{dy}{dx}=\frac{y}{(x+2)^{2}} \)
(D) \( \displaystyle \frac{dy}{dx}=\frac{1}{(x-2)^{2}} \)
▶️ Answer/Explanation
Detailed solution

Across each vertical line (fixed \(x\)) the short segments are parallel ⟹ slope depends only on \(x\), not on \(y\).

Eliminate any \(y\)-dependence:
• (A) and (C) depend on \(y\) ⟹ not consistent.

The field shows a vertical line of undefined slope at \(x=2\) with positive slopes on both sides.
• (B) \(1/(x-2)^{3}\) changes sign across \(x=2\) (odd power) ⟹ slopes would be negative on one side and positive on the other.
• (D) \(1/(x-2)^{2}\) is \(>0\) for \(x\ne2\) (even power) ⟹ slopes positive on both sides and blow up near \(x=2\).

✅ Answer: (D)

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