Calculus II: convergence, approximation and geometric models
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 165 minutes, as an unvalidated planning estimate. This collection contains 11 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.
Choose a method and state its assumptions before opening answer guidance. Use sketches, tables and intermediate calculations when they support your argument.
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For aₙ=3/[n(n+3)], n≥1, derive the Nth partial sum and its limit. Find the term limit and explain why that limit alone would not prove convergence.
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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.
Classify Σₙ≥1 1/(n²+4) and give a justified upper bound for its tail after N=25. Separately classify Σₙ≥1 n/(n²+4). State which comparison direction or limit is used.
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.
Classify Σₙ≥1 (−1)^(n+1)/√n as absolute, conditional or divergent. Find a sufficient number N of initial terms to guarantee alternating error at most 1/20. Separately classify Σₙ≥1 n²/3ⁿ, showing the ratio limit.
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.
Let aₙ=2^((-1)^n)/3^n for integers n≥1. (a) Write the first four terms and calculate aₙ₊₁/aₙ separately for odd and even n. Explain why the ordinary ratio-limit test gives no conclusion. (b) Compute the nth-root limit, justify convergence using the root test, and give an independent geometric upper comparison. (c) Bound the tail R_N=Σₙ>N aₙ and find a sufficient N for R_N≤1/1000; distinguish a sufficient bound from the true smallest N. (d) Check the total by grouping consecutive odd/even pairs, explaining why the pairing is legitimate. Finally replace 3 by b>0 and classify the series for all b, including b=1. Identify what changes in the root, ratio and term tests.
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.
Find the radius and full real convergence interval of Σₙ≥1 (x+1)ⁿ/(n·2ⁿ). Differentiate the series on its open interval and identify the resulting sum. Explain why differentiating does not automatically preserve endpoint convergence.
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.
Construct the degree-three Maclaurin polynomial for e^(2x). Use |f⁽⁴⁾(t)|≤16e^(1/2)<32 on [0,1/4] to bound its error at x=1/4. Explain why matching derivatives alone does not prove an arbitrary smooth function equals its infinite Taylor series.
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.
For x=t²+1,y=t³−t, 0≤t≤2, find dy/dx and d²y/dx² where valid. At t=1 give the tangent line and speed. Set up distance traveled over the interval and diagnose the tangent at t=0 using the velocity vector.
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.
For r=2(1+cosθ), 0≤θ≤2π, find the enclosed area and total curve length. Identify the cusp and justify the absolute value in the length calculation.
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.
Rotate y=2x/3 on 0≤x≤3 about the x-axis. Find the generating segment length and the lateral surface area, and identify the omitted surfaces. Explain why a disk-volume integral cannot answer the surface question.
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.
A vertical rectangular plate is 3 m wide and extends from depth1 m to depth4 m in a synthetic fluid of weight density12 N/m³. Derive its gauge hydrostatic force. Separately find ideal pumping work for a full vertical tank of base area2 m² and height3 m containing the same fluid, pumped to an outlet1 m above its top.
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.
A learner finds ratio-test limit1 for Σ1/n² and concludes divergence. Another finds the first100 terms of Σ1/n small and concludes convergence. Repair both arguments with valid proofs, and explain what evidence would have been insufficient even if their numerical calculations were accurate.
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
For aₙ=3/[n(n+3)], n≥1, derive the Nth partial sum and its limit. Find the term limit and explain why that limit alone would not prove convergence.
Classify Σₙ≥1 1/(n²+4) and give a justified upper bound for its tail after N=25. Separately classify Σₙ≥1 n/(n²+4). State which comparison direction or limit is used.
Classify Σₙ≥1 (−1)^(n+1)/√n as absolute, conditional or divergent. Find a sufficient number N of initial terms to guarantee alternating error at most 1/20. Separately classify Σₙ≥1 n²/3ⁿ, showing the ratio limit.
Let aₙ=2^((-1)^n)/3^n for integers n≥1. (a) Write the first four terms and calculate aₙ₊₁/aₙ separately for odd and even n. Explain why the ordinary ratio-limit test gives no conclusion. (b) Compute the nth-root limit, justify convergence using the root test, and give an independent geometric upper comparison. (c) Bound the tail R_N=Σₙ>N aₙ and find a sufficient N for R_N≤1/1000; distinguish a sufficient bound from the true smallest N. (d) Check the total by grouping consecutive odd/even pairs, explaining why the pairing is legitimate. Finally replace 3 by b>0 and classify the series for all b, including b=1. Identify what changes in the root, ratio and term tests.
Find the radius and full real convergence interval of Σₙ≥1 (x+1)ⁿ/(n·2ⁿ). Differentiate the series on its open interval and identify the resulting sum. Explain why differentiating does not automatically preserve endpoint convergence.
Construct the degree-three Maclaurin polynomial for e^(2x). Use |f⁽⁴⁾(t)|≤16e^(1/2)<32 on [0,1/4] to bound its error at x=1/4. Explain why matching derivatives alone does not prove an arbitrary smooth function equals its infinite Taylor series.
For x=t²+1,y=t³−t, 0≤t≤2, find dy/dx and d²y/dx² where valid. At t=1 give the tangent line and speed. Set up distance traveled over the interval and diagnose the tangent at t=0 using the velocity vector.
Rotate y=2x/3 on 0≤x≤3 about the x-axis. Find the generating segment length and the lateral surface area, and identify the omitted surfaces. Explain why a disk-volume integral cannot answer the surface question.
A vertical rectangular plate is 3 m wide and extends from depth1 m to depth4 m in a synthetic fluid of weight density12 N/m³. Derive its gauge hydrostatic force. Separately find ideal pumping work for a full vertical tank of base area2 m² and height3 m containing the same fluid, pumped to an outlet1 m above its top.
A learner finds ratio-test limit1 for Σ1/n² and concludes divergence. Another finds the first100 terms of Σ1/n small and concludes convergence. Repair both arguments with valid proofs, and explain what evidence would have been insufficient even if their numerical calculations were accurate.
