GraphingCalculator 3.5;
Window 76 16 838 840;
PaneDivider 328;
Slider -4 1;
SliderSteps 99;
SliderControlValue 20;
SliderVariable h;
2D.BottomLeft -0.515625 -6.5;
Expr y=e^x;
Text "The purple graph is the function whose derivative we seek.";
Color 3;
Expr a=1;
Text "<I>a</I> is the x value where we will approximate the tangent line.

If <I>h</I> is the displacement from <I>a</I> where the secant line will be drawn, then";
Color 6;
Expr m=(e^[a+h]-e^[a])/h;
Text "where <I>m</I> is the slope of the secant line between the two points:";
Color 4;
Expr vector(a,e^[a]);
Text "and";
Color 4;
Expr vector(a+h,e^[a+h]);
Text "The secant line between the above two points is the red graph which goes through (<I>a</I>, <I>exp</I>(<I>a</I>)) with slope <I>m</I>:";
Color 2;
Expr y=e^[a]+m*[x-a];
Text "
<I>m</I> approximates the derivative of the purple function at <I>x</I>=<I>a</I> when <I>h</I> is small. ";
Color 17;
Expr m;
Text "
(For some settings of the range of the parameter <I>h</I>, <I>h</I> will take the value <I>h</I>=0. The computer cannot compute 0/0 and gives some spurious value for <I>m</I>. It then plots some random line instead of the tangent line. The settings choosen here avoid that issue. If you wish to illustrate that issue,  click on the symbol <I>h</I> in the slider at the bottom of the graph and change the number of steps to 100.)
";