Introduction to the Derivative 3

3.5 The Derivative: Numerical and Graphical Viewpoints

Previous tutorial: Part A: Numerical Approach

This tutorial: Part B: Graphical Viewpoint

Next tutorial: Part C: The Derivative Function

(This topic is also in Section 3.5 in Applied Calculus or Section 10.5 in Finite Mathematics and Applied Calculus))

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:

m_sec

Slope of secant line through P and Q

f(a+h) - f(a)

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 becomess the tangent line.

This discussion leads us the following conclusion, which is perhaps the most important in calculus:

The slope of the tangent line to the graph of the function f at the point with x-coordinate x = a is given by the derivative f'(a) (the derivative of f at x = a).

Or, to put it roughly,

The derivative at x = a is the slope of the tangent at x = a

Here is a summary of these concepts.

Slope of the Secant Line and Slope of the Tangent Line

The slope of the secant line through (a, f(a)) and (a+h, f(a+h)) is the same as the average rate of change of f over the interval [a, a+h], or the difference quotient:

m_sec =
f(a+h) - f(a)

h

The slope of the tangent line through (a, f(a)) is the same as the instantaneous rate of change of f at the point a, or the derivative:

m_tan = f'(x) = lim
h0
f(a+h) - f(a)

h

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

Let f(x) = 3x² + 4x. Use a difference quotient (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 different points leads to views of different portions of the curve.

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

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 Excel Grapher or the On-Line Function Evaluator and 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 distnguish 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

yMin = 0.354, and
yMax = 0.357,

4. If the graph does not appear straight, repeat Steps 2 and 3 using smaller and smaller values of h until it does appear straight. The curve should now be indistinguishable from a straight line; therefore, the "slope" of the curve 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.

Note You can calculate the balanced difference quotient using the two end-points of the graph in the window corresponding to

x₁ = a - 0.0001	Our present xMin
x₂ = a + 0.0001	Our present xMax

To obtain the corresponding y-coordinates on a graphing calculator, use "trace" or "CALC Value" to position the cursor at the two end-points, and read off the corresponding y-coordinates y₁ and y₂ as accurately as possible. On the Excel Grapher, just position the cursor carefully over the end-points of the graph to read off their coordinates. If you are using the On-Line Function Evaluator and Grapher, then just enter the values of x₁ and x₂ under "values of x," press "Evaluate," and read off the corresponding values of y₁ and y₂. The slope is then

balanced difference quotient

y₂ - y₁

x₂ - x₁

You could now try some of 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 the questions about graphs.

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