Problem 1
Regress ln y against t; slope estimates k if measurement assumptions support the transform.
Faculty use
Parameter Estimation from Dynamic Data
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Patterns can reveal model bias, delays, changing variance, outliers, or omitted dynamics.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind fitting rate constants or model parameters and inspecting residual structure. Use the formulas least squares minimizes Σ(y_i−ŷ_i(θ))²; log-linearization for exponentials.
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 experimental data, engineering design, control, reliability, environmental systems, biomedical modelling, and research communication, 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.