Concept preview

What you will be able to do

1. 01Translate between the graphs of $f$, $f'$, and $f''$: read where $f$ rises or falls from the sign of $f'$, and read the concavity of $f$ from the sign of $f''$.
2. 02Sketch the graph of $f'$ from a graph of $f$ (and vice versa), placing zeros of $f'$ at the turning points of $f$ and zeros of $f''$ at the inflection points of $f$.
3. 03Match a function to the correct graph of its derivative, and avoid the pervasive error of confusing a graph of $f$ with a graph of $f'$.

Start with the link between $f$ and $f'$. At every input $x$, the VALUE (height) of $f'$ equals the SLOPE of $f$. So where the graph of $f$ is rising, $f'$ sits above the axis (positive); where $f$ is falling, $f'$ sits below the axis (negative); and where $f$ has a horizontal tangent — a local max or min — $f'$ touches zero, crossing the $x$-axis. Steep parts of $f$ push $f'$ far from the axis; gentle parts keep $f'$ near it. Reading $f'$ off $f$ is therefore a matter of tracing the steepness of $f$ and plotting it as a height.