Problem 1
The local error estimate rises, so a smaller step is needed to meet tolerance.
Faculty use
Adaptive Step Size and Event Detection
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Ground impact: g(t,y)=height(t)=0, located by interpolation/root finding between accepted steps.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind controlling error estimates while locating thresholds or discontinuities. Use the formulas accept step when estimated error≤tolerance; event g(t,y)=0.
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 simulation, control, orbital motion, chemical kinetics, heat transfer, and computational engineering, 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.