# Math Insight

### Applet: The slope of a secant line

The line (in red) passing through two points (large blue and smaller cyan) on the graph of the function $f$ (in green) is a secant line. The large blue point is at the location $(x_0,f(x_0))$, which you can change by dragging the point or typing in a value for $x_0$. The smaller cyan point is at the location $(x_0+\Delta x,f(x_0+\Delta x))$, which you can change by dragging the point. You can also change $\Delta x$ by typing in a value or changing the length of the purple $\Delta x$ line segment (drag small cyan point). You cannot change $\Delta y$ directly, as it is calculated as $\Delta y = f(x_0+\Delta x)-f(x_0)$. The slope of the secant line is $\frac{\Delta y}{\Delta x}$.

As you let $\Delta x$ approach zero, the two points become closer together, and the secant line becomes closer to the tangent line of the graph of $f$. However, if you set $\Delta x=0$, then the secant line is not defined, and the slope $\frac{\Delta y}{\Delta x}=\frac{0}{0}$ is also not defined. However, if $\Delta x$ is very small, but not zero, the secant line becomes very close to the tangent line, which can be thought of as the limit of the secant line as $\Delta x$ approaches zero. If you click the “show limit for $\Delta x=0$” check box, then when you enter $\Delta x=0$, the applet instead shows the limiting tangent line. The slope of the tangent line, denoted by $\frac{dy}{dx}$ is the limit $\lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}$ and is also displayed.

You can use the buttons at the top to zoom in and out as well as pan the view.

Applet file: secant_line_slope.ggb

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#### General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.