Analysis, Lesson 2
Small Changes Build a Total
A rate tells what is happening now. Accumulation remembers every small contribution.
Predict a history, then move one clock through the rate graph, shaded area, and water level.
All three flows will finish with 10 units after 10 seconds.
Choose before the tank and graph appear.
Connect rate, area, and total
Answer all five before the three flow histories are explained.
Approximate the whole
Make the pieces thinner
A curved rate does not stay constant. Treat it as nearly constant on small intervals and add the rectangle contributions.
Reason from approximation to limit
Complete all five before the error and limiting process are explained.
Design a history
Reach the same total in a different way
Set the rate for five one-second intervals. Build a total of 10 without copying the same story.
Separate the history from its final total
Answer all five before each local contribution and cumulative step is explained.
The analysis bridge
Return to rules with a new idea of change.
Algebra describes the operations; analysis shows how their effects build over time.