Problem 1
Inverse time.
Faculty use
Units, Scaling, Assumptions and Model Validation
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Check units, initial conditions, limiting behavior/signs, parameter plausibility, and comparison with observations or an independent method.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind checking dimensional consistency, plausible limits, and the meaning of parameters. Use the formulas [rate]=[state]/[time]; nondimensional variable = quantity/reference scale.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to motion, heat flow, circuits, fluids, population change, and engineering balance laws, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.