3A-1. The Derivative at a Point
Instructions:
- Use the input box for f(x) to change the function formula. Use the input box for c or click and drag the point on the graph to move it along the graph.
- Adjust the slider tool for h to adjust how far apart (horizontally) the points P and Q are on the graph. Click the "" button to bring Q closer to P.
- Use the Secant check box to show the secant line between P and Q and to display the slope of this line as a y-coordinate (in dark blue). Moving the slider tool for h or clicking the "h \to 0" button will leave a trace of this point on the graph (generating a graph of the difference quotient function).
- Use the Difference Quotient check box to show/hide the graph of the difference quotient function around the point P. Note that this function is undefined at x = c but that there is just a hole in the graph. What is the y-value of this point?
- Use the Tangent check box to show/hide the tangent line to the graph of f(x) at the point P. Observe the relationship between the hole in the difference quotient function and the slope of the tangent line.
The Derivative at a Point
The derivative of a function f at a point x=c is the instantaneous rate of change of the function at that point. We define this instantaneous rate to be the limit of the average rates of change over progressively smaller intervals around the point x=c.
The limit concept is represented in the graph above by clicking the button repeatedly. Theoretically, the point Q can be brought closer and closer to P without ever reaching it. As we bring the point Q closer to the point P we recalculate the slope of the secant line (i.e., average rate of change). As we do this we notice that the slopes of the secant lines seem to "settle down" at (or converge to) a particular value. This value is the slope of the tangent line, which we also define to be the instantaneous rate of change, aka the derivative.