← Calculus syllabus

United StatesCalculus

Unit 7: Differential Equations

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does Euler's method use the slope at each step to approximate a solution curve numerically?

Topic 7.5 Approximating Solutions Using Euler's Method: approximate the solution of a differential equation at a point using Euler's method with a given step size and initial condition (BC).
A focused answer to AP Calculus BC Topic 7.5, approximating the solution of a differential equation numerically with Euler's method, using the slope and step size to step forward from an initial condition, with worked examples.
10 min answer →

How does the differential equation for proportional growth give the exponential model, and how do you use it?

Topic 7.8 Exponential Models with Differential Equations: derive and apply the exponential growth and decay model from a proportional-rate differential equation.
A focused answer to AP Calculus AB Topic 7.8, deriving the exponential model from a proportional-rate differential equation and applying it to growth, decay and half-life problems, with worked examples.
9 min answer →

How do you solve a separable differential equation for its general solution?

Topic 7.6 Finding General Solutions Using Separation of Variables: solve a separable differential equation for the general solution.
A focused answer to AP Calculus AB Topic 7.6, solving separable differential equations by separating variables and integrating both sides to find the general solution, with worked examples and the constant of integration.
9 min answer →

How does an initial condition pin down one particular solution from the general family?

Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables: solve an initial value problem by separation and applying the initial condition.
A focused answer to AP Calculus AB Topic 7.7, solving initial value problems by separating variables, integrating, and using the initial condition to find the constant, with worked examples and the domain of the particular solution.
10 min answer →

How does the logistic differential equation model bounded growth toward a carrying capacity?

Topic 7.9 Logistic Models with Differential Equations: model and analyze bounded growth with the logistic differential equation, identifying the carrying capacity and the point of fastest growth (BC).
A focused answer to AP Calculus BC Topic 7.9, modelling bounded growth with the logistic differential equation, reading off the carrying capacity, finding where growth is fastest, and analyzing long-run behavior, with worked examples.
10 min answer →

How do you translate a description of a rate of change into a differential equation?

Topic 7.1 Modeling Situations with Differential Equations: write a differential equation from a verbal description of a rate of change.
A focused answer to AP Calculus AB Topic 7.1, translating verbal descriptions of rates of change into differential equations, including proportionality and combined-rate models, with worked translations.
8 min answer →

How do you use a slope field to estimate solution curves and describe long-term behavior?

Topic 7.4 Reasoning Using Slope Fields: sketch solution curves on a slope field and reason about their behavior.
A focused answer to AP Calculus AB Topic 7.4, sketching particular solution curves on a slope field through a given point and reasoning about long-term behavior and equilibria, with worked curve-tracing examples.
8 min answer →

What is a slope field, and how do you draw one from a differential equation?

Topic 7.3 Sketching Slope Fields: construct a slope field by computing the slope at grid points from a differential equation.
A focused answer to AP Calculus AB Topic 7.3, constructing a slope field by evaluating the differential equation at grid points to draw short tangent segments, with a worked grid example and the meaning of the field.
8 min answer →

How do you check whether a given function is a solution of a differential equation?

Topic 7.2 Verifying Solutions for Differential Equations: verify that a proposed function satisfies a differential equation by substitution.
A focused answer to AP Calculus AB Topic 7.2, verifying that a proposed function solves a differential equation by differentiating and substituting into both sides, with worked checks of general and particular solutions.
8 min answer →