Polynomial secant and tangent lines

A secant line connects two points of the graph of a function. In this demonstration, the cyan-colored secant line connects the points

and

, as long as

. (You can think of

as the "center" and

as the "radius" of an interval -- the 1-dimensional analogue of a disk -- in the domain of

.) If

, the two points determining the secant line collide! In that case, the secant line coincides with the green tangent line to the graph of

. 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

, it provides a linear (i.e., polynomial of degree 1) approximation of the function

in a small interval around

. Try moving the slider controls for

and

to explore various secant and tangent lines to the graph of

Click and drag the black dot on the horizontal line segment near the top of the graph, below "". The value of changes, and the tangent and secant lines move accordingly.
Drag the black dot on the line segment below "" to change the value of .

The red curve above the

-axis near

shows how large (in absolute value) the difference between

and the linear (tangent line) approximation is, for a small interval around

Move the slider for close to 0.
Watch the effect on the red curve as you move the slider for . While keeping near 0,
- for what value of is the shaded region below the red curve the smallest you can make it?
- for what value of is the shaded region the largest you can make it?
Now set about equal to 1, so the secant and tangent lines move independently as you change the value of .
- With , can you find a value of for which the secant and tangent lines approximately coincide?
- Can you find such a value of if you change to ?

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