Secant & Tangent Lines as Average and Instantaneous Rates of Change

The graph below lets you interactively explore secant lines, tangent lines, and the graph of the derivative.

Graph controls: You can pan by clicking and dragging the graph. You can also zoom in/out using your mouse wheel, or by clicking the arrow below the wrench, or by adjusting the bounds of the graph by clicking on the wrench. Finally, click on any dots on the graph to see the coordinates of those points.

Feel free to explore on your own (you can even type in a new function in the first pane), and/or follow the guidelines below the graph.

What's Happening in This Graph?

1. Click and drag the red dot to change the second point on the secant line. As you do so, the slope of the line is displayed on the second pane on the left.

2. Now click and drag the black dot. This will change the first point on the secant line, keeping the horizontal distance h between the two points the same.

3. Try clicking the play button on either pane 3 or 4. This will animate the secant line. You can control the animation speed if you’d like.

4. Now, reset the dots so that you can see them (one way is to use the sliders in panes 3 and 4). Click on the circle in pane 5. The red line that appears is the tangent line passing through the black dot.

5. The slope of the tangent line is given in pane 6. Drag the red dot closer to the blue dot and keep an eye on panes 2 and 6. You’ll see that the slope of the secant line (given in pane 2) starts getting closer to that of the tangent line (pane 6) as you move the red dot closer to the black dot. In calculus speak, the slope of the secant line is the average rate of change between the points “black” and “red.” As we move the red dot towards the black one, this average rate of change gets closer to the instantaneous rate of change at the point where the black dot is. Remember, the instantaneous rate of change at a point (like at the black dot) is the derivative of the function at that point, and geometrically, the derivative at a point is the slope of tangent line at that point.

6. Now turn off the red dot (by clicking the circle in pane 10) and also the secant line (by clicking the circle in pane 7). Move the black dot around to see how the slope of the tangent line (the derivative) changes as we change the point of tangency.

7. Next, click the circle in pane 8. The red graph you see is the derivative function. At each x-value, the y-value of this red curve is the derivative of the blue curve at that same x-value. Drag the black dot around to see how the y-values on the red curve compare to the slopes of the tangent lines to the blue curve. If you change the function (in pane 1) you can explore other combinations that have regions where the slope is positive, negative, and zero.

8. Finally, feel free to redo these steps with a different function in pane 1 and keep exploring.