Concept preview

What you will be able to do

1. 01Determine the intervals where a function is increasing or decreasing from the sign of its first derivative.
2. 02Identify local maxima and minima using the first-derivative test.
3. 03Determine concavity and locate inflection points from the second derivative.
4. 04Analyse intercepts, asymptotes and symmetry to sketch a complete curve.
5. 05Diagnose a flawed curve-sketching argument, such as assuming every zero of the second derivative is an inflection point.

A graphing calculator can plot a function in an instant, but the point of curve sketching is to understand why a graph has the shape it does, reading its features directly from the function and its derivatives. The first derivative reports where the curve rises and falls and where it levels off; the second derivative reports how the curve bends and where its bending changes; and a few algebraic facts about intercepts, asymptotes and symmetry pin down the rest. Assembling these clues into a single accurate sketch is a skill that ties together everything in the derivatives unit, and it trains the habit of seeing a function and its rate of change as two views of the same object. A good hand sketch also reveals features, such as a hidden inflection or an asymptote, that a hasty machine plot can easily obscure.