A [i]secant line[/i] connects two points of the graph of a function. In this demonstration, the cyan-colored secant line connects the points [math](a,f(a))[/math] and [math](a+h, f(a+h))[/math], as long as [math]|h|>0[/math]. (You can think of [math]a[/math] as the "center" and [math]|h|[/math] as the "radius" of an [i]interval[/i] -- the 1-dimensional analogue of a disk -- in the domain of [math]f[/math].)[br][br]If [math]h=0[/math], the two points determining the secant line collide! In that case, the secant line coincides with the green [i]tangent line[/i] to the graph of [math]f[/math] at [math]x=a[/math]. We'll see examples of functions that simply don't have tangent lines at certain points in their domains. But when the tangent line does exist at [math]x=a[/math], it provides a linear (i.e., polynomial of degree 1) approximation of the function [math]f(x)[/math] in a small interval around [math]a[/math].[br][br]Try moving the slider controls for [math]a[/math] and [math]h[/math] to explore various secant and tangent lines to the graph of [math]f[/math].[br][list][br][*]Click and drag the black dot on the horizontal line segment near the top of the graph, below "[math]a=-0.5[/math]". The value of [math]a[/math] changes, and the tangent and secant lines move accordingly.[br][*]Drag the black dot on the line segment below "[math]h=1[/math]" to change the value of [math]h[/math].[br][/list]

The red curve above the [math]x[/math]-axis near [math]x=a[/math] shows how large (in absolute value) the difference between [math]f(x)[/math] and the linear (tangent line) approximation is, for a small interval around [math]a[/math].[br][list=1][br][*]Move the slider for [math]h[/math] close to 0.[br][*]Watch the effect on the red curve as you move the slider for [math]a[/math]. While keeping [math]h[/math] near 0,[br][list][br][*]for what value of [math]a[/math] is the shaded region below the red curve the [b]smallest[/b] you can make it?[br][*]for what value of [math]a[/math] is the shaded region the [b]largest[/b] you can make it?[br][/list][br][*]Now set [math]h[/math] about equal to 1, so the secant and tangent lines move independently as you change the value of [math]a[/math].[br][list][br][*]With [math]h=1[/math], can you find a value of [math]a[/math] for which the secant and tangent lines approximately coincide?[br][*]Can you find such a value of [math]a[/math] if you change [math]h[/math] to [math]-0.75[/math]?[br][/list][br][/list]