|
|
The invention of the calculus by Isaac Newton and Gottfried Leibniz in the 17th century allowed for the solution of two very important mathematical problems: finding the slope of a line that is tangent to a curve at some point and finding the area of a region that is irregular (for which there is no previously known formula, such as the area of rectangles or triangles). These techniques paved the way for much of the industrial revolution, with an unprecedented increase in science and engineering.
One of the most amazing things about the calculus is the simplicity of many of its most basic ideas. The tangent line problem is an example of this simplicity. Given a curve, can the rate at which that curve is changing be found at a single point? Consider the following graph with a "tangent line" sketched in. What is the slope of that line?
Prior to this course, you have been taught to find slopes of lines (and then equations) using two separate points. Here, you are being asked to find the slope of a line based on only one point.
The great achievement of Newton and Liebniz was to approximate the slope of the line by moving only an infinitesimally small distance from the point in question and imagining a second point on the curve. If the two points were close enough together, then the slope through the two points would approximate the slope of the tangent line at the point in question.
Thus, using the familiar slope formula for the average change in position between two points:
This formula is also called the difference quotient. It finds the ratio of the change in y with respect to x. Here, we will use "function notation" to get an equivalent form. If we let the two points be
 |
and |  , |
where h is an infinitesimally small increase in x, we get the slope of the line between the two points from the following:
The variable h is the horizonatal distance between the point where we want to form a tangent line and the "second point" that is infinitesimally close to the first. Thus, the horizontal distance between the two points is almost zero. The distance is not actually zero ... only very close to zero. Therefore, to find the slope of a line tangent to a curve at a single point, we set up a difference quotient using (x, f(x)) and (x+h, f(x+h)) and take the limit as h approaches zero.
 |
Using the function depicted above and c = 1 yields the following:
 |
and |  |
 |
|
This slope of the tangent line is given a specific name in calculus. It is called the derivative of the function. The equation of the tangent line can now easily be found, using the slope and the point of tangency.
The graphing calculator can be used to approximate the derivative using the definition of the derivative at a point.
Enter into the STAT list L1 values for h that grow steadily smaller, such as 0.5, 0.25, 0.1, 0.01, and so on.
Carefully enter the difference quotient into L2. For the function above, the following was entered:
L2="(3(1+L1)–(1+L1)^2-2)/L1" Note the use of quotes to allow editing of the data (in case of typing errors). The resulting data table is shown at the right. It clearly appears that the values are approaching a limit of 1 as h approaches 0. While this does not "prove" the derivative is 1 when x = 1, it does reinforce our faith in the analytic work shown above.
The graphing calculator approximation can be used for any function. Consider the following approximation of the derivative of y = sin (x) at p/6:
Later, you will learn that the "formula" for the derivative of sin(x) is cos(x). From trigonometry, you should recall that
which is what it appears the limit is from the table.
For other visualizations of the Secant and Tangent Line concept, the following sites are suggested:
- Manipula Math Applet Collection: Secant and Tangent Lines
[http://www.ies.co.jp/math/java/calc/limsec/limsec.html]- Manipula Math Applet Collection: Average Rate of Change and Derivatives
[http://www.ies.co.jp/math/java/calc/heihen/heihen.html]- The Definition of the Derivative - S.O.S. Math
[http://www.sosmath.com/tables/derivative/derivative.html]
| <<< Back to Mr. Whitney's Mathematics Home Page |
Site Created & Maintained by |
|
Matthew C. Whitney |
