Problem 1
Use the precise definitions, state all hypotheses, apply ∫_a^b f=F(b)-F(a), and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Fundamental Theorem of Calculus, Part II and Substitution
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Fundamental Theorem of Calculus, Part II and Substitution.
Problem 4
The argument must address evaluating integrals from antiderivatives and justified changes of variables with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in accumulation, probability, physics, engineering, and numerical quadrature, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.