# Slope of a Curve

- Author:
- Dr. Doug Davis, 3D

- Topic:
- Derivative

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 point. The line between two points on a curve is called the secant line, which is not related the secant function.
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 location where you would like to get the slope of the curve.
How does the slope vary as the distance between points is decreased?
What happens when the distance between points is zero?
How does slope vary with different combinations of left and right check boxes?
Does the value near h=0 vary with different combinations of left and right check boxes?
What happens when one of the points on the curve is outside the domain of the function?
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