Problem 1
Predict 1.1; average slopes (1+1.1)/2=1.05; y₁=1.105.
Faculty use
Improved Euler and Predictor–Corrector Reasoning
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
It samples slope information across the interval rather than only at the left endpoint.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind averaging beginning and predicted ending slopes. Use the formulas predictor y*=y_n+h f_n; corrector y_{n+1}=y_n+h(f_n+f*)/2.
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.