Differential equations and growth
What students will learn
Model and analyze differential equations with initial conditions, numerical Euler steps, exponential growth, and logistic limits.
Before this topic
- Derivative interpretation and rules
- Antiderivatives, substitution, and logarithms
Read and connect
Read: Modeling Situations with Differential Equations →
Translate a rate statement into a differential model with variables, signs, units, and an initial condition.
Read: Verifying Solutions for Differential Equations →
Differentiate and substitute throughout the domain, checking an initial condition separately.
Read: Sketching Slope Fields →
Compute and draw local slope-field directions from both coordinates, leaving undefined locations unfilled.
Read: Reasoning Using Slope Fields →
Reason about a solution family, equilibrium, direction, and concavity from a slope field and equation.
Read: Approximating Solutions Using Euler’s Method →
Build Euler estimates step by step from the current point and distinguish an approximation from an exact solution.
Read: Finding General Solutions Using Separation of Variables →
Separate variable factors, check lost equilibria, and verify the resulting solution family.
Read: Finding Particular Solutions Using Initial Conditions and Separation of Variables →
Use an initial condition to choose a solution, branch, and connected interval of validity.
Read: Exponential Models with Differential Equations →
Fit and interpret exponential growth or decay models, separating a continuous relative rate from interval percent change.
Read: Logistic Models with Differential Equations →
Use a logistic sign chart and initial population to distinguish capacity, equilibrium, and future growth-rate behavior.
Check understanding
Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.
Solve an initial-value problem with a pole →
Separate variables, use the initial condition, and state the largest connected solution interval containing it.
Fit and interpret an exponential model →
Determine the proportional rate from two observations, retain inverse-time units, and predict the later amount.
Track two Euler steps →
Evaluate the slope at each current estimated point and report the result as an approximation.
Compare three logistic starting states →
Distinguish a trajectory that passes through fastest growth, one whose growth rate falls from the start, and the zero equilibrium.
Standards and earlier reading lists
These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.