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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