[color=#000000]Remember: The derivative of a function [i]f[/i] at [i]x[/i] = [i]a[/i], if it even exists at [i]x[/i] = [i]a[/i], can be geometrically interpreted as the slope of the tangent line drawn to the graph of [i]f[/i] at the point ([i]a, f(a))[/i]. [br][br]Hence, the [/color][color=#ff00ff]y-coordinate (output) of the pink point = the slope of the tangent line [/color][color=#000000]drawn to the graph of [i]f[/i] at the [/color][b][color=#000000]BIG BLACK POINT[/color][/b][color=#000000]. (Note that the [/color][b][color=#ff00ff]pink point[/color][/b][color=#000000] and the [b]BLACK POINT[/b] always have the same x-coordinate.) [/color]
1) Move the point along the graph, left and right.[br]2) Answer the questions below.
What do you call the pink graph that is formed by moving the black point left and right?[br]
The pink graph is the graph of the derivative of f, also called f'.
What is the equation of the pink graph?
When the graph of f has a minimum or a maximum, explain what happens to the pink graph.
The pink graph crosses the x-axis. There is an x-intercept on the pink graph at the exact location of a min or max on the black graph.
3) Change the function in the applet to f(x)=(x^2) - 3.[br]4) Answer the questions below.
What is the equation of the new pink graph?
When the graph of f has a minimum or a maximum, explain what happens to the pink graph.
The pink graph crosses the x-axis. There is an x-intercept on the pink graph at the exact location of a min or max on the black graph.