Differential Equations
What this unit covers
The topics below follow the published Calculus AB course framework for Unit 7. This unit is worth 6–12% of the exam, so budget your time against that rather than against how long the unit takes to teach.
Lessons in this unit
- Slope Fields13 min · 3 objectivesInterpret a slope field as the geometry of a differential equation · Sketch short segments from dy/dx at sample points · Match a differential equation to its slope field and trace solution curves
- Separation of Variables15 min · 3 objectivesSeparate a differential equation so each variable is on its own side · Integrate both sides and solve for the general solution · Apply an initial condition to find the particular solution
- Exponential Growth & Decay14 min · 3 objectivesRecognize dy/dt = ky as the exponential growth/decay model · Write its solution y = y₀·e^(kt) and interpret the constants · Distinguish growth (k > 0) from decay (k < 0)
- Modeling with Differential Equations14 min · 3 objectivesTranslate a verbal description of a rate into a differential equation · Solve applied growth and decay problems, including half-life and doubling time · Interpret solutions and their long-term behavior in context
Formulas in Unit 7
Every term in Unit 7
All 12 terms we publish for Differential Equations, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.
- Separation of variables
- Rearrange to g(y)dy = f(x)dx, integrate both sides, add a single constant, then apply the initial condition to solve for it.
- Slope field
- Short segments showing dy/dx at each point. Solution curves follow the segments without crossing one another.
- Why the constant matters
- Solve for C before simplifying further. Substituting the initial condition after exponentiating is a frequent source of error.
- Exponential growth and decay
- dy/dt = ky has solution y = y₀e^(kt). Growth for positive k, decay for negative.
- Interpreting k
- The constant relative rate of change: k = 0.05 means the quantity grows by about 5% per unit time.
- Half-life and doubling time
- From y = y₀e^(kt), the half-life is ln(2)/|k| and the doubling time is ln(2)/k.
- Newton's law of cooling
- dT/dt = k(T − T_env), separable, with solution approaching ambient temperature exponentially.
- Verifying a proposed solution
- Differentiate the candidate and substitute into the differential equation; both sides must agree identically, and the initial condition must hold.
- Sketching a slope field solution
- Start at the given point and follow the segments in both directions. Solution curves never cross one another.
- Reading equilibrium solutions
- Set dy/dx = 0 and solve for y. Those constant solutions appear as horizontal rows of flat segments in the slope field.
- Domain of a particular solution
- The interval containing the initial condition on which the solution is defined and differentiable. Free-response questions ask for it explicitly.
- Sign of the constant in exponential models
- Positive k grows, negative k decays. Solve for k from a second data point rather than guessing.
What examiners penalize here
- When asked to sketch a solution curve on a given slope field, start at the initial point and keep your curve tangent to the nearby segments, never crossing them at odd angles. Follow the flow smoothly in both directions from the starting point.
- Free-response separable-equation problems award points for the separated equation, the antiderivatives of both sides, and finding C from the initial condition. Show each stage explicitly — a correct final formula with no work in between leaves points on the table.
- For half-life or doubling-time questions, set y = y₀/2 (or 2y₀) in y = y₀·e^(kt); the y₀ cancels, leaving an equation you solve with a natural log. The half-life depends only on k, never on the starting amount.
- When a problem gives a data point to find k, substitute it into y = y₀·e^(kt), isolate the exponential, and take a natural log — do not try to guess k. Carry enough decimal places through so the final rounded answer stays accurate.
Practice Calculus AB
Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.
Questions about this unit
How much of the AP Calculus AB exam is Unit 7?
Unit 7, Differential Equations, is worth 6–12% of the Calculus AB multiple-choice section according to the published course framework. Across all 8 units that makes it a substantial share — heavier than an even split would give it.
What topics are covered in Calculus AB Unit 7?
Differential Equations covers Slope fields, Separation of variables, Exponential models and Growth. We publish 12 terms with definitions for this unit, all of them on this page.
How should I study Calculus AB Unit 7?
Read the 4 lessons below first — about 55 minutes — then drill the 12 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.
All 8 units of AP Calculus AB
Unit names, topics and exam weights follow the published College Board course framework for AP Calculus AB. AP® is a trademark registered by the College Board, which does not endorse this site.