Almost a Derivative
Plotted below is the result of the code from the previous activity. As you can see, the code created a new point whose x coordinate is 1, the same as
A, but whose y coordinate is the slope of the secant line g. I've hidden h since we don't need it in this exercise.
Click and move A along the graph of f. Notice that the new point leaves a bread crumb trail. Can you describe this breadcrumb trail in words?The breadcrumb trail is a record of the slopes of the line
g as A moves along the graph of f.Quick Check: If you slide A to (0,0) on the graph of f, what is the approximate slope of the secant line at A?
Because this secant line almost matches the graph of
f(x) at point A, you might say that the slope of this secant line is "growing" at almost the same rate that f(x) is growing at A. Furthermore, as you move A and leave a breadcrumb trail, you are keeping track of this estimate. We'll continue to think along these lines in the coming activities.