AP Calculus BC 7.4 Reasoning Using Slope Fields - Exam Style Questions - MCQs - New Syllabus
Question
A portion of the slope field for the differential equation \(\tfrac{dy}{dx}=x-y\) is shown. If the linear function \(y=f(x)\) is the solution to the differential equation such that \(f(-1)=-2\), then \(f(3)=\ ?\)
(A) 0
(B) 1
(C) 2
(D) 3
(B) 1
(C) 2
(D) 3
▶️ Answer/Explanation
Correct answer: (C) 2
If \(y=f(x)\) is linear, let \(y=mx+b\). Then \(dy/dx=m\).
Equation: \(m=x-(mx+b)\). For identity to hold, coefficients must match: slope \(m=-1\), line \(y=-x\).
Check condition: \(f(-1)=-(-1)=1\). But we need \(-2\). So try line through slope field: \(y=x-1\).
Plug \(x=-1\): \(y=-2\) ✔. Then \(f(3)=2\).
No-Calc Question
Of the graphs shown, which could be the graph of a solution to a differential equation whose slope field has slopes that depend only on the y-coordinate at each point (i.e., \(y’ = g(y)\))?
(A) Graph A only
(B) Graph B only
(C) Graphs A and B only
(D) Graphs B and C only
(B) Graph B only
(C) Graphs A and B only
(D) Graphs B and C only
▶️ Answer/Explanation
Detailed solution
If slopes depend only on \(y\) (autonomous ODE), then along any horizontal line the slope is the same.
Graph A: horizontal line ⇒ slope \(=0\) for that \(y\) (an equilibrium solution) ⇒ possible.
Graph B: monotone curve; no violation of “same slope for same \(y\)” ⇒ possible.
Graph C: passes through the same \(y\) values with different slope signs (down then up) ⇒ not possible.
✅ Answer: (C) Graphs A and B only