Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 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.
Attempt the set before revealing criteria or worked solutions. Choose your own methods and show the intermediate reasoning on paper or in the written responses.
Check domains, hypotheses and units. An exact answer match only checks the requested value; written work needs review.
Suggested time is guidance, not an access limit. First responses and known answer guidance remain in saved history. These public tasks do not certify independent mastery or readiness for a whole course examination.
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0 of 8 tasks have recorded responses. No overall mastery score is assigned.
For radian x, let f(x)=e^(2x) sin(3x)/(x+1). Differentiate on its real domain and find f′(0). Show where the product, quotient and chain rules enter.
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 x²+xy+y²=7, find dy/dx at (1,2). Check that the point is on the curve first.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Let f(x)=x+ln x for x>0. Explain why f is strictly increasing, then find the derivative of its inverse at y=1. State the input correspondence and the nonzero-derivative condition 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.
A position model is s(t)=t³−9t²+24t metres for 0≤t≤6 seconds. Find every interior direction reversal and every open time interval on which speed increases. Explain the role of acceleration and check signs instead of relying only on zeros.
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 one instant two walkers are 6 m east and 8 m north of a crossing. The eastward distance increases at 2 m/s while the northward distance decreases at 1 m/s. Find the rate of their separation, assuming perpendicular straight paths. Define signed rates before differentiating.
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 function is continuous on [2,7], differentiable on (2,7), and has 1≤f′(x)≤4 throughout that open interval. With f(2)=−3, bound f(7) and prove that f has at most one zero in [2,7]. Do these facts guarantee a zero?
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 rectangular enclosure uses 72 m of fencing for its outside boundary and one internal divider parallel to its width w. Let l be its other dimension. Form the area objective, find the maximizing dimensions and area, and justify a global maximum including the feasible boundaries.
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 lim as x→0 of [sin(2x)−2x]/x³ for radian x. Justify each use of L’Hopital. Then explain why differentiating numerator and denominator is invalid for lim as x→0 of (2+sin x)/(3+x).
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 radian x, let f(x)=e^(2x) sin(3x)/(x+1). Differentiate on its real domain and find f′(0). Show where the product, quotient and chain rules enter.
Let f(x)=x+ln x for x>0. Explain why f is strictly increasing, then find the derivative of its inverse at y=1. State the input correspondence and the nonzero-derivative condition used.
A position model is s(t)=t³−9t²+24t metres for 0≤t≤6 seconds. Find every interior direction reversal and every open time interval on which speed increases. Explain the role of acceleration and check signs instead of relying only on zeros.
At one instant two walkers are 6 m east and 8 m north of a crossing. The eastward distance increases at 2 m/s while the northward distance decreases at 1 m/s. Find the rate of their separation, assuming perpendicular straight paths. Define signed rates before differentiating.
A function is continuous on [2,7], differentiable on (2,7), and has 1≤f′(x)≤4 throughout that open interval. With f(2)=−3, bound f(7) and prove that f has at most one zero in [2,7]. Do these facts guarantee a zero?
A rectangular enclosure uses 72 m of fencing for its outside boundary and one internal divider parallel to its width w. Let l be its other dimension. Form the area objective, find the maximizing dimensions and area, and justify a global maximum including the feasible boundaries.
Find lim as x→0 of [sin(2x)−2x]/x³ for radian x. Justify each use of L’Hopital. Then explain why differentiating numerator and denominator is invalid for lim as x→0 of (2+sin x)/(3+x).
