Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 75 minutes, as an unvalidated planning estimate. This collection contains 6 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 the representation used, intermediate calculations and units.
These are public practice tasks. Written reasoning needs review; completion does not certify independent mastery.
For new graph cases, construct a labeled sketch from the supplied vertices before calculating. Do not substitute the live graph values.
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0 of 6 tasks have recorded responses. No overall mastery score is assigned.
A new graph follows y=2x−1 for x<1, except its point at x=0 is replaced by (0,5). For x>1 it follows y=4−x, and f(1)=2. Sketch open and closed marks. Classify the discontinuities at 0 and 1, repair any removable one, and discuss differentiability.
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.
An exact table gives x=−1,0,2; f=2,−1,0; f′=3,4,−2; g=0,2,−1; g′=−2,5,1. Find (f∘g)′(−1), (fg)′(0), and (f⁻¹)′(2) for a differentiable local inverse at f(−1)=2. State why global invertibility is not established.
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 new continuous rate p is linear through (0,−2),(2,2),(5,2),(7,−2). Let J(x)=4+∫₀ˣp. Sketch both the rate and a qualitatively correct accumulation curve. Find absolute extrema of J on [0,7], including values and locations, and its concavity intervals.
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.
Synthetic inflow rates at 0,1,4,8 minutes are 2,6,4,0 L/min. Estimate total inflow by trapezoids, and final amount from an initial 15 L with constant outflow 1 L/min. If the inflow remains between adjacent sample heights, bound the final amount. State whether the trapezoid answer is a certified measured total.
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.
Positions at times 0,2,5 seconds are 1,7,4 metres. Compute the average velocity over each interval. A learner asserts the velocity at t=2 is the average of these two numbers. Assess the claim. Under continuous differentiability, what can the mean value theorem guarantee instead?
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 p be the new rate linear through (0,−2),(2,2),(5,2),(7,−2). Define Q(x)=∫₁^(2x)p(t)dt for 0<x<7/2. Find Q′(2), Q(2), and all intervals where Q increases. Explain the difference between the input at which you read p and the multiplier from the moving endpoint.
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 new graph follows y=2x−1 for x<1, except its point at x=0 is replaced by (0,5). For x>1 it follows y=4−x, and f(1)=2. Sketch open and closed marks. Classify the discontinuities at 0 and 1, repair any removable one, and discuss differentiability.
An exact table gives x=−1,0,2; f=2,−1,0; f′=3,4,−2; g=0,2,−1; g′=−2,5,1. Find (f∘g)′(−1), (fg)′(0), and (f⁻¹)′(2) for a differentiable local inverse at f(−1)=2. State why global invertibility is not established.
A new continuous rate p is linear through (0,−2),(2,2),(5,2),(7,−2). Let J(x)=4+∫₀ˣp. Sketch both the rate and a qualitatively correct accumulation curve. Find absolute extrema of J on [0,7], including values and locations, and its concavity intervals.
Synthetic inflow rates at 0,1,4,8 minutes are 2,6,4,0 L/min. Estimate total inflow by trapezoids, and final amount from an initial 15 L with constant outflow 1 L/min. If the inflow remains between adjacent sample heights, bound the final amount. State whether the trapezoid answer is a certified measured total.
Positions at times 0,2,5 seconds are 1,7,4 metres. Compute the average velocity over each interval. A learner asserts the velocity at t=2 is the average of these two numbers. Assess the claim. Under continuous differentiability, what can the mean value theorem guarantee instead?
Let p be the new rate linear through (0,−2),(2,2),(5,2),(7,−2). Define Q(x)=∫₁^(2x)p(t)dt for 0<x<7/2. Find Q′(2), Q(2), and all intervals where Q increases. Explain the difference between the input at which you read p and the multiplier from the moving endpoint.
