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Reviewer calibration key

Review assumptions and reasoning, not just final numbers. Solve prompts yourself before looking here.

  • CAL1: domain x>5; inverse exp(y)+5; δ=ε/4 makes |4x+2−10|<ε. A hole with its value undefined or deliberately changed supplies the example.
  • CAL2: quotient 2x+h tends to 2x; exp(−3x)(1−3x); 4x/(1+2x²). Units are output units per input unit.
  • CAL3: derivative 2−32/x², critical x=4, values at 1,4,8 are 34,16,20. On [5,8], derivative is positive; minimum at 5 with value 82/5.
  • CAL4: width 2/n, right sum 8/n² times n(n+1)/2 =4(1+1/n), limit 4. Substitution u=x³ gives e−1. Parts gives antiderivative exp(2x)(x/2−1/4), definite value (e²+1)/4.
  • CAL5: c=3/8; CDF 0 below 0, t³/8 on [0,2], 1 above 2; mean 3/2; tail beyond 1 is 7/8. Point mass is zero; density has inverse-time units if t is time. Cutoff ≥ln(1000)/3 in time units consistent with rate.
  • CAL6: Taylor bound 1/3000. Trapezoid error ≤1/(6n²), so n≥19. A discontinuity such as 1/x across zero defeats the root theorem.
  • CAL7: successful code checks are necessary but insufficient. Look for retained numerical output, exact comparisons, distinct tail/quadrature budgets, explicit continuity assumptions, failure handling and independent changed-case reasoning.

Do not award a pass for a copied key. A different correct method earns the same credit; diagnose discrepancies and retest with a new case.