Calculus AB — differential equations, slope fields and exponential models

Unit 7 of AP Calculus AB is short, but it supplies one of the free-response questions on most exams: a differential equation is given, a slope field is sketched or read, the concavity of a solution is decided at a point, and a particular solution is found by separating variables. The multiple-choice section adds the modeling side, where a sentence about rates has to become an equation, and exponential growth and decay. This material covers the whole AB unit; logistic models belong to the BC course and are left out.

The twelve quiz questions follow the order in which the unit builds. You translate "the rate of change of P is proportional to the difference between 100 and P" into an equation, and test four functions against dy/dx = y − x by substitution, where three of them fail by a single sign or constant. You compute the slope of a solution curve at a point straight from the equation, find where the slope field of dy/dx = x² − y has horizontal segments, and match a drawn slope field to its equation by locating its horizontal segments and checking one slope off that line. One question asks what a solution through a given point is doing there, which requires the second derivative found by differentiating the equation itself. You then solve dy/dx = x/y with y(0) = 3, identify the constant solution of dy/dx = (y − 4)(x + 1), and state the interval on which a particular solution lives. The last group is the exponential model: the general solution of dy/dt = ky, the growth constant from a doubling time of five years, and the amount left after t years when the half-life is ten years.

The wrong options are the slips these questions invite: the difference written the wrong way round or in the wrong variable, a candidate solution that is off by a constant, the first derivative mistaken for the second, a constant of integration added outside the exponential, a vertical line offered as a constant solution. Each explanation shows the substitution or the separation step by step.

The flashcards collect the definitions and facts of the unit: separable form, exponential solutions, horizontal segments of a slope field, equilibrium solutions, verification by substitution, second derivatives from a differential equation, the domain of a particular solution, doubling time and half-life.

This is independent practice written by Zestly, based on the published AP Calculus AB course framework; it is not produced or endorsed by the College Board.

• Translate a verbal statement about rates into a differential equation
• Verify a proposed solution by substitution
• Compute slopes from a differential equation and locate horizontal segments of a slope field
• Match a slope field to its differential equation
• Decide the concavity of a solution at a point using the second derivative from the equation
• Solve a separable initial value problem and state the domain of the particular solution
• Find equilibrium solutions
• Use dy/dt = ky with doubling time and half-life

Practice material written by Zestly, based on the College Board AP Calculus AB course framework (Unit 7: Differential Equations, topics 7.1–7.8: modeling, verifying solutions, slope fields, separation of variables, particular solutions, exponential models), 2026–27 course and exam description.