Linked Derivative Graphs

The tangent line on $f$ is drawn in the color of $f'$, because its slope is the height of $f'$ at the cursor. The tangent line on $f'$ is drawn in the color of $f''$, because its slope is the height of $f''$. The sign strips under $f$ mark where $f$ is increasing or decreasing and where it is concave up or concave down. Dotted guide lines run through all three graphs at each local extremum and point of inflection. Show numbers gives the value of $f$, of $f'$, and of $f''$ at the cursor.

The two presets that start with $f'(x) =$ hide the graph of $f$ until $f$ is pressed. A typed function is read with R’s rules. The box accepts x^3 - 5*x and rejects 2x, because R needs * between two things being multiplied.

Under More:

• Trace. With Trace $f'$ by hand on, dragging along $f$ drops a dot on the middle graph at the height of the slope of $f$. Press $f'$ to compare the dots with the graph of $f'$.
• Extrema and points of inflection. With Show numbers on, the list gives each input in view where $f'$ or $f''$ is 0. Each entry gives the sign change, or its absence, and whether the point is a local extremum or a point of inflection.
• Derivative formulas. The formulas for $f'$ and $f''$ are written out for each preset.

Show R gives the R code that approximates $f'$ and $f''$ with step 0.001, as in the Lesson 28 Notes. The code prints both values at the cursor. While $f$ is hidden, the code starts from the given $f'$.

Keys: press the left or right arrow key to move the cursor one step. Hold Shift with an arrow key to move it five steps. Press + or − to zoom.