# Polynomial secant and tangent lines

- Author:
- paul.prue

- Topic:
- Calculus, Tangent Line or Tangent

A and , as long as . (You can think of as the "center" and as the "radius" of an .)
If , the two points determining the secant line collide! In that case, the secant line coincides with the green at . 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 .

*secant line*connects two points of the graph of a function. In this demonstration, the cyan-colored secant line connects the points*interval*-- the 1-dimensional analogue of a disk -- in the domain of*tangent line*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?

- for what value of
- 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 ?

- With