AP Calculus BC 7.3 Sketching Slope Fields - MCQs - Exam Style Questions
Calc-Ok Question
Let \(h\) be a continuous function of \(x\). Which of the following could be a slope field for a differential equation of the form \(\displaystyle \frac{dy}{dx}=h(x)\)?
▶️ Answer/Explanation
Detailed solution
If \(\dfrac{dy}{dx}=h(x)\), the slope at \((x,y)\) depends **only on \(x\)**. Therefore, along any vertical line \(x=\text{const}\) all short line segments must have the **same** slope.
• (A) shows constant slopes along vertical lines ⇒ consistent with \(h(x)\). ✔️
• (B) and (D) have slopes varying with both \(x\) and \(y\) ⇒ not of the form \(h(x)\).
• (C) has slopes constant along **horizontal** lines (depend only on \(y\)) ⇒ corresponds to \(dy/dx=g(y)\), not \(h(x)\).
✅ Answer: (A)