The Secant and Tangent Lines

In the tutorial on average rates of change we saw that the average rate of change of f gives the slope of the secant line through two points on its graph:

msec = Slope of secant line through P and Q = f(a+h) - f(a) h

As h approaches zero, we saw in the preceding tutorial that this quantity approaches the instantaneous rate of change at x = a, which we called f ' (a). Let us now see what is happening on the graph as h approaches zero: The quantity a+h (on the x axis) moves closer and closer to a, and so the point Q moves closer and closer to P. To see this process in action, press the "Make h smaller" button under the picture below.

See what happens to the secant line? It becomes more and more like the tangent line and, in the limit as h approaches 0, it becomes the tangent line.

We are led to the following conclusion, which is perhaps the most important in calculus:

Or, more simply,

Here is a summary of these concepts.
 

The following exercise is similar to Example 2 in Section 3.5 of Applied Calculus.

Let f(x) = 3x² + 4x. Use a difference quotient (formula given in the above box) with h = 0.0001 to estimate the slope of the tangent line to the graph of f at the point where x = 2.

We can also visulaize the slope of the tangent graphically as follows: Start with any smooth curve, and then zoom in closer and closer until the curve looks like a straight line. This straight line is the tangent line, and its slope is the derivative.

Here is an illustration of zooming in to a point on a graph where x = 0.75.

Notice how the curve appears to "flatten" as we zoom in; the zoomed-in curve (also shown below) appears almost straight.

Note We have zooomed in to a particular point on the curve; that is, we have always kept that point in the center of the viewing window as we zoomed. Zooming in on another point leads to the derivative at that point.

Using Technology to Zoom in to a Curve at a Specified Point (a, f(a)) Here, you are given the x-coordinate (x = a) of a point on the graph, and are asked to estimate the slope of the graph at that point in question. 1. Graph the function using a window that shows the point . 2. Set Left end-point = xMin = a - 0.001 Right end-point = xMax = a + 0.0001 3. If your grapher has a "zoomfit" feature ("ZOOM → "Zoomfit" on the TI-83) use that to graph the function with the given xMin and xMax. If you are using the On-Line Function Evaluator and Grapher or the Excel Grapher, fill in the left and right end-points with these values and leave yMin and yMax blank and hit "Graph". Otherwise,use your grapher to compute f(a - 0.0001), f(a), and f(a+0.0001) and use the maximum of these values as yMax and the minimum as yMin. You need not use all the digits; just use enough to distinguish these two numbers and to make sure that both ends of the graph are visible on the screen (that is, round up for yMax and round down for yMin).. For instance, if f(a - 0.0001) = 0.356 375 432, f(a) = 0.354 374 801, f(a + 0.0001) = 0.354 374 734, then you can use yMin = 0.354, and yMax = 0.357, 4. If the graph does not appear absolutely straight, repeat Steps 2 and 3 using smaller and smaller values of h until it does appear absolutely straight. As the graph is now indistinguishable from a straight line, the slope of the tangent will be well approximated by the slope of this straight line. Notice that the slope of this line is given by the balanced difference quotient, with the point of interest (x = a) in the center: f'(a) ≈ m = f(a + h) - f(a - h) 2h = f(a + 0.0001) - f(a - 0.0001) 0.0002

You could now try the exercises in Section 3.5 in Applied Calculus or Section 10.5 in Finite Mathematics and Applied Calculus), but you will need the material in the next tutorial to answer some questions about the derivative function and its graph.

Alternatively, press "game version" on the sidebar to go to the game version of this tutorial (it has different examples to try and is a lot of fun!).