Calculus foundations: limits and local change assessment
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 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.
Try the entire set before revealing worked solutions. You may use paper for calculations and diagrams. The suggested time is guidance; no timer limits access.
Show your reasoning for every task, including domain restrictions and checks in the original statement. Exact-value matches assess only the requested result.
Written arguments require review against the supplied rubric. Recording a response or marking a lesson complete does not certify mastery.
These are public practice assessments. First responses and solution exposure are retained; another attempt is practice, not a new independent examination.
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0 of 8 tasks have recorded responses. No overall mastery score is assigned.
Let f(u)=√u and g(x)=1/(x−2). Find the real domain of f∘g and explain why x=2 is excluded. Is x=1 allowed?
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≠−2 let h(x)=(x²−4)/(x+2), and define h(−2)=9. Find the limit as x approaches −2.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For x≠0 let q(x)=x²cos(1/x). Prove its limit at zero and explain why merely computing a table is insufficient.
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.
Define f(x)=kx−2 for x<3 and f(x)=2x+1 for x≥3. Find k that makes f continuous at 3.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Use the intermediate value theorem to show that p(x)=x³−3x+1 has a zero in (0,1). State every hypothesis used. Does this argument prove uniqueness?
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(x)=(2x²+1)/(x²−4), identify the limit as x→+∞ and the one-sided behavior as x→2− and x→2+. Explain the signs and distinguish a finite limit from unbounded behavior.
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 f(x)=x²−3x, derive f′(2) from the difference quotient and give the tangent-line equation at x=2. Use that line to approximate f(2.1), then calculate and interpret the approximation error.
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.
Compare f(x)=|x−1| and g(x)=(x−1)² at x=1. Establish continuity and differentiability or its failure, using one-sided difference quotients where needed. Give g′(x) for all real 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
Let f(u)=√u and g(x)=1/(x−2). Find the real domain of f∘g and explain why x=2 is excluded. Is x=1 allowed?
For r(x)=(2x²+1)/(x²−4), identify the limit as x→+∞ and the one-sided behavior as x→2− and x→2+. Explain the signs and distinguish a finite limit from unbounded behavior.
For f(x)=x²−3x, derive f′(2) from the difference quotient and give the tangent-line equation at x=2. Use that line to approximate f(2.1), then calculate and interpret the approximation error.
Compare f(x)=|x−1| and g(x)=(x−1)² at x=1. Establish continuity and differentiability or its failure, using one-sided difference quotients where needed. Give g′(x) for all real x.
