Courses / Mathematics / Calculus: local rates and accumulated change / Project Build and defend your work Explain and optimize a changing system Choose a geometric growth, motion or bounded design model. Connect a local rate to finite change, calculate accumulated change where appropriate, and justify an optimum only within a stated feasible domain. Use captures from at least two calculus mechanisms and include a counterexample to a tempting shortcut.
Choose an audience, investigate alternatives, copy evidence from your experiments, and produce an artifact. You can open this project at any time. Reports, numerical transfer checks and human review are separate outcomes.
Your project workspace Choose the kind of work you want to produce School investigation College investigation Workplace application Create a motion or growing-circle investigation with a predicted rate, a finite-change calculation and a plotted explanation. Distinguish signed displacement from distance or instantaneous area rate from later area.
1. Collect and compare evidence Required experiment sources:
Include every source above and at least one pair with different inputs from the same experiment. Explain confounding changes yourself; the presence check does not judge your interpretation.
Open an experiment and change an input or capture a trial to make evidence available here.
2. Write and revise your report Function, domain and units State the function and its allowed inputs. Identify which quantity is a rate, level or accumulated amount, and declare units.
Local sensitivity Derive a derivative or partial derivative and explain it in plain language. State held-fixed quantities and test a nonzero-width approximation.
Finite or accumulated change Compute an exact finite change or integral for a new case. Explain why it may differ from rate×interval, and handle sign changes explicitly.
Feasible optimum or decision Compare candidate settings, boundaries and constraints. If using a zero derivative, explain why it identifies the desired optimum rather than merely a stationary point.
Counterexample and interpretation Use a corner, missing point, domain violation or finite-step discrepancy to challenge a shortcut. State what the mathematical model does not establish about a real system.
Your artifact: code, analysis, prototype or decision brief This field stores up to 12,000 characters of inert text. Code runs only through the explicit bounded editor in the relevant lessons; project attachments never execute.
Load a text artifact Review your work against the rubric Function domains and units are correct and visible. Local derivatives are distinguished from finite average rates. Accumulation and signed quantities are interpreted correctly. Optimization checks feasible boundaries and does not rely only on a stationary condition. A concrete counterexample or edge case limits the conclusion. 14 evidence requirements remain Capture evidence from Shrink an interval without dividing by zero. Capture evidence from Accumulation can cancel without erasing motion. Capture evidence from A changing radius creates a changing area. Capture evidence from Change one input while holding another fixed. Capture evidence from Follow a gradient through a computation. Capture evidence from Use a slope to improve a choice. Capture evidence from Build the largest enclosure. Include two different input configurations from the same experiment to compare. Write your function, domain and units. Write your local sensitivity. Write your finite or accumulated change. Write your feasible optimum or decision. Write your counterexample and interpretation. Add your project artifact: code, analysis, prototype or decision brief. Record my report Download report and evidence Editing the report or evidence returns it to draft. Recording verifies field presence and input comparisons, not correctness of your reasoning or code.
3. Check transfer to unfamiliar cases These authored cases use different numbers or conditions from the lesson practice. They are public local checks, not secure certification. Each case permits three submissions. Opening, first responses and attempt counts survive imports; no reset creates fresh evidence.
Case calculus-project-limit Open case calculus-project-limit Case calculus-project-accumulation Open case calculus-project-accumulation Case calculus-project-partial Open case calculus-project-partial Case calculus-project-optimum Open case calculus-project-optimum
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