Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 50 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.
Attempt all three unfamiliar cases before opening solutions. Use paper for graphs and exact calculations; the suggested time is optional.
Show intermediate coordinates, units, domain restrictions and forward checks. A numerical answer alone does not establish the requested argument.
This public mixed practice belongs to the extension unit, not the AP examination score. Written responses need review against the rubric; exposure and revisions remain visible.
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0 of 3 tasks have recorded responses. No overall mastery score is assigned.
A cart follows x(t)=t²−4t, y(t)=2t+1 for 0≤t≤3 seconds, with coordinates in meters. Construct the two component graphs and the Cartesian path with equal spatial scales, labeling t=0,1,2,3. Determine the component extrema. Eliminate t and prove both directions of the restricted Cartesian description. On [0,3], calculate both average component rates and the Cartesian secant slope. Do these rates determine average travel speed? Explain, and give the straight-line displacement lower bound for that speed.
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 linear map sends the vertices (0,0),(1,0),(0,1) to (0,0),(2,1),(−1,1). Recover its matrix M using column vectors, construct the image triangle on equal axes, and verify the image of (1,2). Let R be the quarter-turn matrix [[0,−1],[1,0]]. Compare RM and MR on (1,2), showing intermediate points. Recover the unique predecessor of (0,3) under M and verify it. If the second column of M is changed to (4,2), classify the predecessors of (6,3) and (6,4), giving the complete solution set when nonempty.
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 daily inspection, a stipulated two-state device model has P(available tomorrow | available today)=0.7 and P(available tomorrow | busy today)=0.2. Build the row-stochastic matrix and forecast twice from available proportion 0.6. Derive the interval of reachable next-day available proportions. For proposed next-day proportions 0.35 and 0.9, solve for predecessors and assess probability feasibility. If both conditional probabilities were instead 0.4, classify targets 0.4 and 0.5. Finally, explain why a correct calculation over twenty steps does not establish the model assumptions in a real device population.
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
A cart follows x(t)=t²−4t, y(t)=2t+1 for 0≤t≤3 seconds, with coordinates in meters. Construct the two component graphs and the Cartesian path with equal spatial scales, labeling t=0,1,2,3. Determine the component extrema. Eliminate t and prove both directions of the restricted Cartesian description. On [0,3], calculate both average component rates and the Cartesian secant slope. Do these rates determine average travel speed? Explain, and give the straight-line displacement lower bound for that speed.
A linear map sends the vertices (0,0),(1,0),(0,1) to (0,0),(2,1),(−1,1). Recover its matrix M using column vectors, construct the image triangle on equal axes, and verify the image of (1,2). Let R be the quarter-turn matrix [[0,−1],[1,0]]. Compare RM and MR on (1,2), showing intermediate points. Recover the unique predecessor of (0,3) under M and verify it. If the second column of M is changed to (4,2), classify the predecessors of (6,3) and (6,4), giving the complete solution set when nonempty.
At daily inspection, a stipulated two-state device model has P(available tomorrow | available today)=0.7 and P(available tomorrow | busy today)=0.2. Build the row-stochastic matrix and forecast twice from available proportion 0.6. Derive the interval of reachable next-day available proportions. For proposed next-day proportions 0.35 and 0.9, solve for predecessors and assess probability feasibility. If both conditional probabilities were instead 0.4, classify targets 0.4 and 0.5. Finally, explain why a correct calculation over twenty steps does not establish the model assumptions in a real device population.
