Mathematics / Lesson 04
Change and accumulation
See derivatives and integrals as two views of how local change builds a whole.
orient
Why this idea had to exist
A speedometer reports change now; an odometer accumulates change across a trip. Calculus connects those two instruments. It gives precise language for local behavior and for the total built from many small contributions.
intuition
Build a picture you can reason with
Zoom into a smooth curve until a tiny piece looks straight. Its local tilt is the derivative. Tile the region under a rate curve with increasingly thin strips; their combined area approaches the integral.
formalize
Give the intuition a precise edge
The derivative f′(x) is the limit of [f(x+h)-f(x)]/h as h approaches zero. The definite integral is the limit of weighted sums over finer partitions. The fundamental theorem connects them: accumulating a rate and then differentiating returns the rate under appropriate conditions.
work through
Follow the decisions, not just the symbols
For position s(t)=t², the average velocity from t to t+h is ((t+h)²-t²)/h=2t+h. Let h approach zero and the local velocity becomes 2t. The cancellation is the reasoning; the power rule is its reusable shortcut.
experiment
Change one thing and watch the model answer
Move two points together on a parabola and watch the secant slope approach a tangent slope. Then accumulate rectangles under that slope curve and compare the total with the original change in position.
retrieve
Close the page and reconstruct it
Answer before opening the explanation. Retrieval is evidence only when the answer is produced without a hint.
Why does a derivative need a limiting process?
A local rate concerns an interval of zero width, so we infer it from average rates over intervals that approach zero without dividing by zero.
transfer
Move the idea into a new setting
Given a graph of electrical power over a day, explain what its slope and its area represent physically.
reflect
Leave with a diagnostic habit
Keep units attached: derivatives divide output units by input units; integrals multiply a rate by its input unit.
Source record
Follow the idea back.
- Reference
- Calculus, Volume 1
- Publisher
- OpenStax
- License
- CC BY 4.0
- Accessed
- 2026-08-12
- URL
- https://openstax.org/details/books/calculus-volume-1