Secant Lines and the Slope of a Curve
The following applet can be used to approximate the slope of the curve y=f(x) at x=a. Simply enter the function f(x) and the values a and b. The applet automatically draws the secant line through the points (a,f(a)) and (b,f(b)). As b approaches a. the slope of the secant line approaches the slope of the line tangent to f(x) at x=a.
By selecting "h= " instead of "b= ", the applet automatically draws the secant line through the points (a,f(a)) and (a+h,f(a+h)). As h approaches 0. the slope of the secant line approaches the slope of the line tangent to f(x) at x=a. In other words,
the applet can be used to investigate the following two equivalent definitions for the derivative of f(x) at x=a.
The values a. b and/or h can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x -axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.
Here is a list of functions that can be used with this applet.