Say Which Curve It Is Before You Read Anything Off It

You'll learnto translate any feature between the graphs of f, f′ and f″ — a shape becomes a position and a position becomes a crossing — and to build a sketch from the two sign charts and the anchor points, knowing which questions a graph of f′ cannot answer at all.

Everything in this unit so far produces the same object from a different angle: a sign chart. This topic puts them together into one picture, and introduces the error that makes an otherwise correct picture give a wrong answer. Every feature of f has three descriptions — one on f itself, one on the graph of f′, one on the graph of f″ — and moving one column to the right turns a shape into a position, and a position into a crossing. A peak is a peak on whichever curve was drawn, and which curve was drawn is written on the axis label.

8 STEPS · 6 QUICK CHECKS · A TURNING POINT ON ONE CURVE IS A CROSSING ON THE NEXT · v1

f turns ⇒ f′ crosses · f′ turns ⇒ f″ crosses · a shape becomes a position becomes a crossing

Before you start

What you're looking at

A translation table first, then one function drawn three times on a shared x axis — itself on top, its first derivative in the middle, its second derivative underneath. A vertical guide cuts all three at the same input, so a feature and its translation line up.

The question

Given a feature on one of the three curves, what is the corresponding feature on the other two? And which questions about f does a picture of f′ simply not answer?

Watch for

Turning points migrating into crossings as you move down the stack, and the step where three different functions are drawn above one derivative panel that fits all of them.

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