Secant Line Approximations

An interactive example

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.