AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.8 Sketching Graphs of Functions and Their Derivatives
Notes
Read one graph to sketch another:
- Zeros of $f'$ are where $f$ has horizontal tangents.
- Where $f'$ is above the axis, $f$ rises; below, $f$ falls.
- Where $f'$ rises, $f$ is concave up; peaks and valleys of $f'$ are inflection points of $f$.
Practice
The graph of $f'$ crosses the $x$-axis from below to above at $x=1$. The graph of $f$ has:
$f'$ goes from negative to positive.
The graph of $f'$ has a local max at $x=4$. Then $f$ has:
$f''$ changes from positive to negative.
If $f$ is a parabola opening up, $f'$ is:
The derivative of a quadratic is linear with positive slope.
Ready for more? Take the Unit 5 test.