Modeling with derivatives
What students will learn
Build motion, related-rate, and optimization models with meaningful variables, feasible domains, and justified conclusions.
Before this topic
- Differentiation tools
- Derivative signs, extrema, and units
Find a resource or check in this book →
Unit 4: Contextual Applications of DifferentiationUnit 5: Analytical Applications of Differentiation
Read and connect
Read: Straight-Line Motion: Connecting Position, Velocity, and Acceleration →
Calculate and compare position, velocity, acceleration, direction, and changing speed using sign evidence.
Read: Rates of Change in Applied Contexts Other Than Motion →
Interpret marginal and other contextual rates, separating instantaneous change from a finite interval change.
Read: Introduction to Related Rates →
Distinguish fixed constraints from snapshots, then differentiate all changing quantities with respect to the shared clock.
Read: Solving Related Rates Problems →
Build a geometric relationship, differentiate before inserting a snapshot, and interpret the resulting signed rate.
Read: Introduction to Optimization Problems →
Name an objective, constraint, and feasible domain before choosing an optimization procedure.
Read: Solving Optimization Problems →
Justify an optimum over the whole feasible domain and report the requested setting, output, and units.
Connect: Interpreting the Meaning of the Derivative in Context →
Revisit the signed rate interpretation and units while defining variables for a model.
Before this connection: Derivative interpretation.
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.
Model a rising depth in a conical tank →
Use similar triangles to link changing water dimensions, differentiate before substituting, and interpret the depth rate.
Justify the least-material cylinder →
Use the volume constraint, positive-radius domain, and complete derivative signs to justify dimensions and minimum area.
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.