If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, assuming there is just one.

A chord is a segment of a secantline whose both ends lie on the curve.

Draw a secantline to the unit circle (centered at the origin) that passes through the origin and through (cos θ, sin θ), the point on the unit circle at an angle of θ.

The applet draws a tangentline to a graph at a point marked by a draggable disk (which is red in the default version).

A secantline is drawn from this point to another point, which is marked by another draggable disk (green in the default version).

The slopes of the two lines are shown at the upper left, but you can turn off the display of the tangentslope if, for example, you want the user to estimate the slope of the tangentline from the slopes of secantlines.

Use the limit statement from Task 9 to relate the slopes of the secantlines to f(x) through B' and C' to the slope of the tangentline to f(x) through C'.

A tangentline to a curve is a line meeting a curve or surface at a point and having at that point the same direction as the curve or surface.

Secant Applet(Site not responding. Last check: 2007-10-11)

A tangentline to a curve can be thought of as a close approximation to the curve in the vicinity of the point of tangency.

The magenta number "dy/dx" is the slope of the secantline through P and Q.

By moving the vertical green line closer to the red line, the point Q moves closer to the basepoint P, and the slope dy/dx of the secantline becomes a better approximation of the slope of the linetangent to the blue curve at P.

Secant Lines/TITLE>(Site not responding. Last check: 2007-10-11)

In addition to specifying the two points on a curve for the secantline, you may specify that a number of secantlines be drawn.

10 secantlines are drawn, one for each of the 10 parts.

At your discretion, the drawing will pause after each line, with the coordinates used for the last line, the point separation (h), and the slope of the last line shown at the top of the screen.

The tangentline at (0,1) is the horizontal line y=1.

We said a moment ago that a tangentline touches a graph in only one place, but there are lots of different ways to get a line to touch this graph in only one place.

Now that you understand tangentlines, and how they are found using an approximation by secantlines, you are in a good position to approach the derivative.

Since the derivative is the limitingslope of the secantline, we can think as the first point at c be fixed and then second point be a varying value x that tend towards c.

From the graph, we can see that the slope of the secantline will be -1 to the left of 0 and 1 to the right of 0.

As b approaches a, the slope of the secantline approaches the slope of the linetangent to f(x) at x=a.

The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right.

The Secant Method(Site not responding. Last check: 2007-10-11)

, is taken to be the x-intercept of the approximating secantline.

That is, the secant method may be viewed as a modification of Newton's method, where the modification takes the form of the approximation

In other words, the secant method as described below is not a bracketing method, although variations on the bracketing requirements leads to alternate bracketing methods such as the ``Regula-Falsi'' (false-position) method.

The Crown Bracket Bonding System - Journal of Clinical Orthodontics(Site not responding. Last check: 2007-10-11)

To test the accuracy and reproducibility of the estimated cemento-enamel junction locations of the secant endpoints, a second group of teeth was evaluated.

If either secant or the incisal edge is not parallel to the contours of the bracket base, it is easy to detect which crown edge deviates from the norm or has been altered--for instance, by abrasion.

Bonding the brackets parallel to the mesial secants, as in Row A, might be appropriate if the incisal edges had been ground down or damaged by trauma, or in case of a tooth-size discrepancy between the maxillary and mandibular arches.

Secant Method(Site not responding. Last check: 2007-10-11)

which is the intersection of the secantline with the x axis.

The successive steps of the secant method are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step.

At each iteration of the secant method, the approximating secantline at the current points is drawn, the next approximate solution is taken to be the intersection of the secantline with the x axis, and the process is then repeated.

As you look at the slope of the line passing through two points on the graph of a function, then let one of the points get closer and closer to the other, you should see that the slopes get closer and closer to the slope of the tangentline.

If you select the ceheckbox to show the slope, the slope of the line is displayed on the screen as the nodes move.

1426lab1java(Site not responding. Last check: 2007-10-11)

The applet on this page lets you investigate how the slope of a secantline on a curve (the straight line joining two points on the curve) varies as the points on the curve move, and, in particular, explore its behavior quantitatively as the two points move very close together.

In the latter case, the slope of the secantline approximates the value of the slope (derivative) at that point.

Click on a second point on the curve, and you should see the straight line (secantline) joining them, and the slope of that line.