Problem 1
Because x is odd, a_n=0 and b_n=2(−1)^{n+1}/n.
Faculty use
Fourier Series and Periodic Representation
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The midpoint of the left and right limits under standard hypotheses.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind computing sine/cosine coefficients and interpreting convergence. Use the formulas f~a₀/2+Σ(a_n cos nx+b_n sin nx); coefficients are inner products.
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 heat conduction, vibrating strings, structural modes, diffusion, electrostatics, acoustics, and field simulation, 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.