Limits and Riemann sums

Idea

A definite integral cuts the interval $[a,b]$ into finer and finer rectangles and then takes the limit. A derivative, near one point, reads an instantaneous slope from a shorter and shorter span. Here $\Delta x$ is the width of the pieces of the whole interval, and $h$ is the local span looking to the right from the point $P$.

Interaction

• Compare: use the same Scale control to shrink the interval slices and the local span together, and compare the accumulated area with the slope ratio
• Global area: change the number of partitions n and the sample, and watch the rectangle sum approach the definite integral
• Local slope: drag the point P and shrink h, and watch the one-sided secant slope approach $f^{\prime }(P)$

Suggested order: in Compare, move Scale until the slices and the span get thinner, then Global area for how the error adds up, then Local slope for the secant closing in.

What to notice

• The definite integral cuts the whole interval into pieces and adds them. The derivative zooms in near one point and takes a ratio.
• As those widths shrink, the area error is what many small intervals add up to, and the slope error is the curvature left near $P$.
• The sharper the bend, the slower a finite partition or a finite $h$ settles. When the function is differentiable, the derivative is still a definite limit.

Related

• A function and its derivative
• Riemann sum
• Tangent approximation
• Area and right-endpoint height
• Equiangular spiral

Further reading

• Riemann integral (Wikipedia)
• Derivative (Wikipedia)