Ask the Sign, Never the Size

You'll learnto read where an accumulation function rises, peaks, bends and inflects straight off the graph of what it accumulates — and why the place that graph is tallest is almost never the place the accumulation is largest.

The graph on the screen is f. Every question worth asking is about g, the running total of the signed area under it. There is no new machinery here: g′ = f and g′′ = f′, so Unit 5's whole toolkit applies with the parts shifted one slot down. What the shift changes is where you look. Direction comes from the sign of f, concavity from whether f is rising, and the value from signed area. Not one of them comes from how tall f is.

8 STEPS · 6 QUICK CHECKS · A SIGN OR A TURN, NEVER A SIZE · v1

g′ = f ⇒ direction · g′′ = f′ ⇒ concavity · signed area ⇒ value

Before you start

What you're looking at

Two panels on one horizontal scale, carried over from Topic 6.4. The upper panel is the graph you are handed, f. The shaded region under it is signed area, and the lower panel plots the running total of that area against how far along the collecting has reached. The lower curve is g.

The question

Given only the upper panel, where does the lower one rise, peak, bend and end?

Watch for

Two different partitions of the same axis. The colour of the shading changes where f crosses the axis, at 4 and 8. The haloed stretches change where f turns around, at 2 and 6. Confusing the two is this topic's most expensive habit.

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