Mixed weights, jump values and existence boundaries
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Use the weighted Darboux/refinement convention unless a question explicitly asks for all sufficiently small meshes.
Closed cells include their endpoints; do not discard a jump value when taking extrema.
Separate spatial cell width from the integrator increment, and justify each existence claim before computing an integral.
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On [0,1] define α(0)=−1, α(x)=x²+2·1{x≥1/3} for 0<x<1, and α(1)=6. For f(x)=1+x, prove existence under both the refinement and all-small-mesh conventions and compute ∫f dα. Use integration by parts for the smooth contribution and account for both endpoints. Check the answer against bounds from the total increment. Now change only f(1/3) to 0. Is the new weighted Darboux integral merely a different value, or does it fail to exist? Prove your conclusion for every partition.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Fix c=2/5 in [0,1]. Let α be 0 before c, θ at c, and 1 after c, where 0≤θ≤1. Let f be 0 before c, v at c, and 1 after c, where 0≤v≤1. Classify all pairs (θ,v) for which the weighted Darboux/refinement integral exists, and compute it. Then classify all-small-mesh tagged convergence. Your reasoning must cover both partitions containing c and partitions omitting it.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On [0,1] let f(x)=−1 for x<1/2, f(1/2)=7, and f(x)=2 for x>1/2. First use α(x)=x³. Prove existence under both conventions and compute the integral without invoking a continuous-integrand reduction theorem on a discontinuous function. Explain why the value 7 has no effect in this case. Then replace α by β(x)=x³+(1/4)·1{x≥1/2}. Determine whether the weighted Darboux integral exists, giving a partition-independent estimate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
On [0,1] define α(0)=−1, α(x)=x²+2·1{x≥1/3} for 0<x<1, and α(1)=6. For f(x)=1+x, prove existence under both the refinement and all-small-mesh conventions and compute ∫f dα. Use integration by parts for the smooth contribution and account for both endpoints. Check the answer against bounds from the total increment. Now change only f(1/3) to 0. Is the new weighted Darboux integral merely a different value, or does it fail to exist? Prove your conclusion for every partition.
Fix c=2/5 in [0,1]. Let α be 0 before c, θ at c, and 1 after c, where 0≤θ≤1. Let f be 0 before c, v at c, and 1 after c, where 0≤v≤1. Classify all pairs (θ,v) for which the weighted Darboux/refinement integral exists, and compute it. Then classify all-small-mesh tagged convergence. Your reasoning must cover both partitions containing c and partitions omitting it.
On [0,1] let f(x)=−1 for x<1/2, f(1/2)=7, and f(x)=2 for x>1/2. First use α(x)=x³. Prove existence under both conventions and compute the integral without invoking a continuous-integrand reduction theorem on a discontinuous function. Explain why the value 7 has no effect in this case. Then replace α by β(x)=x³+(1/4)·1{x≥1/2}. Determine whether the weighted Darboux integral exists, giving a partition-independent estimate.
