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 |
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:
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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.
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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:
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:
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On the following graph you will see the graph of a function f and three straight lines of different colors:
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The red line has slope equal to
| f'(-1) | f'(1) | f(-1) | f(1) | \displaystyle \frac{f(1)-f(-1)}{2} |
The purple line has slope equal to
| f'(-1) | f'(1) | f(-1) | f(1) | \displaystyle \frac{f(1)-f(-1)}{2} |
The blue line has slope equal to
| f'(-1) | f'(1) | f(-1) | f(1) | \displaystyle \frac{f(1)-f(-1)}{2} |
The following exercise is similar to Example 2 in Section 3.5 of Applied Calculus.
Let f(x) = 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 =
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.
On the following graph you will see the graph of a function f and various tangent lines.
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.
