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AP Calculus AB 7.4 Reasoning Using Slope Fields - MCQs - Exam Style Questions

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No-Calc Question


The slope field for a certain differential equation is shown above. Which of the following could be a solution to the differential equation with the initial condition \(y(0)=1\)?

(A) \(y=\cos x\)
(B) \(y=1-x^{2}\)
(C) \(y=e^{x}\)
(D) \(y=\sqrt{1-x^{2}}\)
(E) \(y=\dfrac{1}{1+x^{2}}\)

▶️ Answer/Explanation

All choices satisfy \(y(0)=1\), so use the slope field near \((0,1)\).
The short segments along the \(y\)-axis are horizontal ⇒ the field indicates \(y'(0)=0\).
Compute \(y'(0)\) for each option:
• (A) \(y’=-\sin x\Rightarrow y'(0)=0\).
• (B) \(y’=-2x\Rightarrow y'(0)=0\).
• (C) \(y’=e^{x}\Rightarrow y'(0)=1\) ❌ (not horizontal).
• (D) \(y’=\dfrac{-x}{\sqrt{\,1-x^{2}\,}}\Rightarrow y'(0)=0\).
• (E) \(y’=\dfrac{-2x}{(1+x^{2})^{2}}\Rightarrow y'(0)=0\).
The field also shows the solution through \((0,1)\) decreases gently for \(x>0\) and stays positive, leveling toward \(y=0\).
Among the candidates, \(y=\dfrac{1}{1+x^{2}}\) matches this monotone decreasing, always-positive shape (and is symmetric about the \(y\)-axis), consistent with the field.
✅ Answer: (E)

Question
Slope Field Image
Shown above is a slope field for which of the following differential equations?
(A) \(\frac{dy}{dx} = 1 + x\)
(B) \(\frac{dy}{dx} = x^2\)
(C) \(\frac{dy}{dx} = x + y\)
(D) \(\frac{dy}{dx} = \frac{x}{y}\)
(E) \(\frac{dy}{dx} = \ln y\)
▶️ Answer/Explanation
Solution
The slope field shows slopes depending on both \( x \) and \( y \).
(A) \(\frac{dy}{dx} = 1 + x\): Depends only on \( x \), slopes constant for fixed \( x \).
(B) \(\frac{dy}{dx} = x^2\): Depends only on \( x \), slopes zero at \( x = 0 \).
(C) \(\frac{dy}{dx} = x + y\): Slope zero along \( y = -x \), positive above, negative below.
(D) \(\frac{dy}{dx} = \frac{x}{y}\): Slope zero at \( x = 0 \), positive for \( x > 0 \), \( y > 0 \).
(E) \(\frac{dy}{dx} = \ln y\): Depends only on \( y \), slopes constant for fixed \( y \).
The field’s diagonal pattern with slopes transitioning across \( y = -x \) matches (C).
✅ Answer: C
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