Regions, trajectories and local models: mixed practice
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 110 minutes, as an unvalidated planning estimate. This collection contains 8 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.
Show setup, domain restrictions and intermediate reasoning before final values.
Static cases differ from the live widgets; use the data stated in each prompt.
These public tasks have worked guidance and are not an independently administered examination.
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0 of 8 tasks have recorded responses. No overall mastery score is assigned.
Evaluate ∫₋₁³ dx/(x²−2x+10), derive the scale and check a primitive. Explain why there are no singularities on the integration interval.
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 base lies between y=3x and y=x² on [0,3]. Find volumes for rectangular sections of height d/2 and semicircular sections of diameter d, where d is the vertical base width. Compare their ratio.
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 geometric area between y=x³ and y=4x on [−2,3]. Show every boundary switch and compare with the signed integral of x³−4x.
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 particle starts at (3,−2) with v(t)=(4,3t),0≤t≤2. Find position and endpoint displacement, set up distance, and decide when speed increases. Explain whether the displacement magnitude equals distance.
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=4−cosθ, find the point and tangent line at θ=π/2. Separately explain why a formula with denominator zero must be checked before calling the tangent vertical.
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.
In the first quadrant find the overlap area of disks bounded by r=4cosθ and r=2sinθ. Locate all common Cartesian points and the switch angle. Leave an exact inverse-tangent expression if useful.
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.
At a=−1, f,f′,f″,f‴ are 1,−2,6,−12. Construct P₃, evaluate it at −1/2, and give an error interval if independently |f⁽⁴⁾|≤24 on [−1,−1/2]. Explain why the table cannot supply that bound.
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 particle starts at (0,0) with v(t)=(1,e^(−t²)),0≤t≤1/2. Use the first three nonzero terms of e^(−t²) to approximate its final y-coordinate and bound that error. Give the exact x-coordinate and an integral for total distance; do not confuse the y approximation with distance.
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
Evaluate ∫₋₁³ dx/(x²−2x+10), derive the scale and check a primitive. Explain why there are no singularities on the integration interval.
A base lies between y=3x and y=x² on [0,3]. Find volumes for rectangular sections of height d/2 and semicircular sections of diameter d, where d is the vertical base width. Compare their ratio.
A particle starts at (3,−2) with v(t)=(4,3t),0≤t≤2. Find position and endpoint displacement, set up distance, and decide when speed increases. Explain whether the displacement magnitude equals distance.
For r=4−cosθ, find the point and tangent line at θ=π/2. Separately explain why a formula with denominator zero must be checked before calling the tangent vertical.
In the first quadrant find the overlap area of disks bounded by r=4cosθ and r=2sinθ. Locate all common Cartesian points and the switch angle. Leave an exact inverse-tangent expression if useful.
At a=−1, f,f′,f″,f‴ are 1,−2,6,−12. Construct P₃, evaluate it at −1/2, and give an error interval if independently |f⁽⁴⁾|≤24 on [−1,−1/2]. Explain why the table cannot supply that bound.
A particle starts at (0,0) with v(t)=(1,e^(−t²)),0≤t≤1/2. Use the first three nonzero terms of e^(−t²) to approximate its final y-coordinate and bound that error. Give the exact x-coordinate and an integral for total distance; do not confuse the y approximation with distance.
