This applet has two graphs. The left graph is of a function that can be modified with the black points. A line between two points on the curve is shown. The horizontal distance of the points from a point on the curve can be controlled with a slider and check boxes. The line can be to the right, to the left or centered on the [math]x[/math] point. The line between two points on a curve is called the secant line, which is not related the secant function.[br]Clicking the "Animate h" button will cause the 'h' value to decrease repeatedly. This will leave a path on the right graph showing how the slope varies with 'h'. Selecting "Show trace" will show all points on this path.

Set the [math]x[/math] location where you would like to get the slope of the curve.[br]How does the slope vary as the distance between points is decreased?[br]What happens when the distance between points is zero?[br]How does slope vary with different combinations of left and right check boxes?[br]Does the value near h=0 vary with different combinations of left and right check boxes?[br]What happens when one of the points on the curve is outside the domain of the function?[br][br]Mathematicians have solved the problem of an undefined slope by defining the slope at a point as the slope of the tangent line which is calculated as the limit as h -> 0 of the slope of the secant line. Checking the tangent line box shows the tangent line. Selecting show trace will show all values of the secant line slope for 0