# Secant Line Approximations

Derek Sollberger
code monkey

## The Concept

The second major concept of calculus is the notion that we can find the slope at a particular point of a function. However if we know slope as $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ then how do we find the slope at just one point?

## The Approach

For now, we are still stuck with needing to $(x,y)$ coordinates to compute a slope. Suppose that for a function $f$, we have a couple of locations $(a,f(a)), \quad (b,f(b))$ then the slope of a secant line between them would be $m = \frac{f(b) - f(a)}{b - a}$

## An Example

Let us use the radical function $f(x) = \sqrt{x}$ and center our approach at $a = 2$. If we choose $b = 8$ as our second point, then the slope of the secant line is

(sqrt(8) - sqrt(2)) / (8 - 2)

## [1] 0.2357


Moving closer to $a = 2$, if we choose $b = 4$ as our second point, then the slope of the secant line is

(sqrt(4) - sqrt(2)) / (4 - 2)

## [1] 0.2929


## Approach!

Use the application found at http://freexstate.shinyapps.io/project/ to continue to move the endpoint $b$ closer to $a$. Once they are the same (in the limiting sense), then that slope at just that one point extends outward to what we call a tangent line.