Interactive Notes
A First View of Calculus
Calculus studies two complementary operations: measuring local change and measuring accumulated quantity. The figures below are not decorative; each one is a movable definition.
1. From Average Change to Instantaneous Change
Start with a function f(x) = x² / 2 + 1. If two points on the graph are separated by a visible horizontal step, the line through them gives an average rate of change. The derivative asks what happens as that step becomes arbitrarily small.
In the limit, the secant line becomes the tangent line. The number shown beside the figure is the slope being approximated.
2. From Rectangles to Accumulated Quantity
The same function can also be read vertically. If we add many thin rectangles under the graph, their total area estimates the accumulated value of the function across an interval.
Increasing the number of slices makes the rectangular approximation approach the integral. Moving the endpoint changes the interval being accumulated.
3. The Shared Idea
Both constructions depend on replacing a difficult continuous object with many simple local measurements. The derivative shrinks one interval until only local change remains. The integral adds many local pieces until a global quantity appears.