Home / AP Calculus AB : 7.3 Sketching Slope Fields- Exam Style questions with Answer- MCQ

AP Calculus AB : 7.3 Sketching Slope Fields- Exam Style questions with Answer- MCQ

Question
Slope Field Image
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) \( y = \frac{1}{1 + x^2} \)
▶️ Answer/Explanation
Solution
All options satisfy \( y(0) = 1 \). Compute slopes at \( (0, 1) \):
– (A) \( y = \cos x \), \( \frac{dy}{dx} = -\sin x \), slope = 0.
– (B) \( y = 1 – x^2 \), \( \frac{dy}{dx} = -2x \), slope = 0.
– (C) \( y = e^x \), \( \frac{dy}{dx} = e^x \), slope = 1.
– (D) \( y = \sqrt{1 – x^2} \), \( \frac{dy}{dx} = -\frac{x}{\sqrt{1 – x^2}} \), slope = 0.
– (E) \( y = \frac{1}{1 + x^2} \), \( \frac{dy}{dx} = -\frac{2x}{(1 + x^2)^2} \), slope = 0.
The slope field at \( (0, 1) \) is horizontal (slope = 0), narrowing to (A), (B), (D), (E).
For (E), \( \frac{dy}{dx} = -2x y^2 \), matching a field with negative slopes for \( x > 0 \), \( y > 0 \), and positive for \( x < 0 \), consistent with the image’s diagonal lines.
✅ Answer: E
Question
Slope Field Image
Shown above is a slope field for which of the following differential equations?
(A) \(\frac{dy}{dx} = xy + x\)
(B) \(\frac{dy}{dx} = xy + y\)
(C) \(\frac{dy}{dx} = y + 1\)
(D) \(\frac{dy}{dx} = (x + 1)^2\)
▶️ Answer/Explanation
Solution
The slope field shows diagonal lines with slopes depending on both \( x \) and \( y \).
(A) \(\frac{dy}{dx} = xy + x = x (y + 1)\): Slope is zero at \( x = 0 \), positive for \( x > 0 \), \( y > -1 \), negative for \( x < 0 \), \( y > -1 \).
(B) \(\frac{dy}{dx} = xy + y = y (x + 1)\): Slope is zero at \( y = 0 \), depends on \( y \).
(C) \(\frac{dy}{dx} = y + 1\): Slope is zero at \( y = -1 \), depends only on \( y \).
(D) \(\frac{dy}{dx} = (x + 1)^2\): Slope is zero at \( x = -1 \), depends only on \( x \).
The field’s horizontal slopes at \( x = 0 \) and increasing magnitude with \( |x| \) and \( |y| \) match (A).
✅ Answer: A
Question
The figure below shows a slope field for one of the differential equations given below. Identify the equation.
(A) \(\frac{dy}{dx} = y – x\)
(B) \(\frac{dy}{dx} = -xy\)
(C) \(\frac{dy}{dx} = 2x\)
(D) \(\frac{dy}{dx} = \frac{x}{y}\)
Slope Field Image
▶️ Answer/Explanation
Solution
The slope field has horizontal slopes at \( x = 0 \) and negative slopes when \( x > 0 \) and \( y > 0 \).
(A) \(\frac{dy}{dx} = y – x\): Slope at \( x = 0 \) is \( y \), not zero for all \( y \).
(B) \(\frac{dy}{dx} = -xy\): Slope is zero at \( x = 0 \), negative when \( x > 0 \), \( y > 0 \).
(C) \(\frac{dy}{dx} = 2x\): Slope is zero at \( x = 0 \), positive when \( x > 0 \).
(D) \(\frac{dy}{dx} = \frac{x}{y}\): Slope is zero at \( x = 0 \), positive when \( x > 0 \), \( y > 0 \).
Only (B) matches the constant zero slope at \( x = 0 \) and negative slopes in the first quadrant.
✅ Answer: B
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