Unit 7 of AP Calculus AB: Differential Equations.
This unit covers slope fields, separation of variables and exponential models — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers slope fields, separation of variables and exponential models — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Slope Fields
A slope field is a visual representation of a differential equation dy/dx = f(x, y) where short line segments are drawn at lattice points with slopes equal to the value of dy/dx at that point. Students must be able to match a slope field to its differential equation and sketch or identify solution curves through a given point. The exam tests both reading slope fields and reasoning about behavior (e.g., where slopes are zero, positive, or undefined).
Key Points
- At any point (x, y), the slope of the segment equals the value of dy/dx evaluated at that point
- Horizontal segments (slope = 0) occur where dy/dx = 0; use this to identify isoclines
- A solution curve must be tangent to every slope segment it passes through
- If dy/dx depends only on y (not x), columns of segments are parallel; if only on x, rows are parallel
The slope field for dy/dx = x − y has horizontal segments (slope = 0) along the line y = x. A solution curve passing through (0, 1) initially slopes downward. Which of the following could be the differential equation: (A) dy/dx = x + y, (B) dy/dx = x − y, (C) dy/dx = y − x?
Check where horizontal segments appear: if dy/dx = 0, then x − y = 0, so y = x — a diagonal line, which is consistent with option (B). At the point (0, 1), dy/dx = 0 − 1 = −1, so the slope is negative, confirming the curve initially decreases. Option (C) would give dy/dx = 1 − 0 = 1 at (0,1), which is positive — contradicting the observed downward slope, so (B) is correct.
2 Separation Of Variables
Separation of variables is the primary technique for solving differential equations on the AP exam. Students must algebraically separate all y terms (including dy) to one side and all x terms (including dx) to the other, then integrate both sides. A constant of integration C must be included, and if an initial condition is given, students must solve for C to find the particular solution.
Key Points
- Rewrite dy/dx = f(x)g(y) as (1/g(y)) dy = f(x) dx, then integrate both sides
- Always include +C on one side; do not write C on both sides
- Use the initial condition (x₀, y₀) to solve for C after integrating
- Check whether the solution should remain in implicit form or be solved explicitly for y
Solve the initial value problem: dy/dx = 2xy, y(0) = 3.
Separate variables: (1/y) dy = 2x dx. Integrate both sides: ln|y| = x² + C. Exponentiate: |y| = e^(x² + C) = Ae^(x²) where A = e^C. Applying the initial condition y(0) = 3 gives A = 3, so the particular solution is y = 3e^(x²).
3 Exponential Models
Exponential growth and decay arise from the differential equation dy/dt = ky, where k is a constant. Students must know that the general solution is y = Ce^(kt), where C is the initial value y(0). If k > 0 the quantity grows; if k < 0 it decays. The AP exam frequently presents these in context (population, radioactive decay, Newton's Law of Cooling) and requires students to set up, solve, and interpret the model.
Key Points
- dy/dt = ky has general solution y = Ce^(kt); C = y(0) is the initial amount
- Doubling time satisfies e^(kt) = 2, so t = ln(2)/k; half-life satisfies t = ln(2)/|k|
- Newton's Law of Cooling is dT/dt = k(T − T_ambient), solved by substituting u = T − T_ambient to get exponential form
- Always define what C and k represent in context when writing a final answer
A bacteria population satisfies dy/dt = 0.4y. At t = 0, the population is 500. Find the population when t = 5.
The differential equation is of the form dy/dt = ky with k = 0.4, so the general solution is y = Ce^(0.4t). Applying the initial condition y(0) = 500 gives C = 500, so y = 500e^(0.4t). At t = 5: y = 500e^(2) ≈ 500 × 7.389 ≈ 3695.
Questions, answered.
What is Differential Equations?
Differential Equations is Unit 7 of AP Calculus AB, covering slope fields, separation of variables and exponential models.
How to study for AP Calculus AB Unit 7?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 30+ review questions across 5 different game modes.