Problem 1
∫₀ᵗe^{−(t−τ)}dτ=1−e^{−t}.
Faculty use
Convolution and Impulse Response
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The output superposes impulse responses generated by each past input value.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind expressing the output of a linear time-invariant system as accumulated shifted response. Use the formulas (f*g)(t)=∫₀ᵗf(τ)g(t−τ)dτ; L{f*g}=F(s)G(s).
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 switching circuits, control systems, impacts, signal processing, and transient engineering response, 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.