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Topic: Secant line

	A Secant Line Applet
		This applet is designed to help students develop a visual intuition about the tangent line as a limit of secant lines.
		The slope of the secant line is displayed on the control panel.
		The magenta patch on the axis and graph is centered at the hot spot with a radius of delta.
	www.slu.edu /classes/maymk/banchoff/SecantLine.html (278 words)

	PlanetMath: secant line
		The secant line (or simply the secant) of a curve is a straight line intersecting the curve in at least two distinct points.
		secant, secant of the curve, secant to the curve
		This is version 11 of secant line, born on 2004-11-22, modified 2005-05-27.
	planetmath.org /encyclopedia/Secant2.html (135 words)

	Secant line - Wikipedia, the free encyclopedia
		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 secant line whose both ends lie on the curve.
		Draw a secant line 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.
	en.wikipedia.org /wiki/Secant_line (378 words)

	Sample Configurable JCM Applet: SecantTangent
		The applet draws a tangent line to a graph at a point marked by a draggable disk (which is red in the default version).
		A secant line 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 tangent slope if, for example, you want the user to estimate the slope of the tangent line from the slopes of secant lines.
	math.hws.edu /javamath/config_applets/SecantTangent.html (567 words)

	PlanetMath: tangent line
		The word is initially a participial form tangens of the Latin verb tangere `to touch'.
		Cross-references: line, intersect, property, slope, secant line, limit, word, neighborhood, point, smooth, curve
		This is version 8 of tangent line, born on 2004-11-22, modified 2006-04-28.
	planetmath.org /encyclopedia/TangencyPoint.html (171 words)

	Center for Technology and Teacher Education: Mathematics Activities
		To construct the tangent line to the curve at C', we need the coordinates of a point through which the tangent line passes and the slope of the tangent line.
		Use the limit statement from Task 9 to relate the slopes of the secant lines to f(x) through B' and C' to the slope of the tangent line to f(x) through C'.
		A tangent line 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.
	teacherlink.org /content/math/activities/skpv4-tangentlines/guide.htm (1551 words)

	Secant Lines (Site not responding. Last check: 2007-10-11)
		The line joining these two points is a secant line to the graph of f.
		If f is differentiable at a, then as h-->0, the secant lines gather at a limiting line, which is the tangent line to the graph at (a,f(a)).
		Since the slopes of the secant lines are difference quotients, the slope of the tangent line is the limit of the difference quotients as h approaches 0...the slope of the tangent line is f '(a)!!!
	oregonstate.edu /instruct/mth251/cq/Stage5/Lesson/secantLines.html (237 words)

	Secant Applet (Site not responding. Last check: 2007-10-11)
		A tangent line 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 secant line 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 secant line becomes a better approximation of the slope of the line tangent to the blue curve at P.
	csunix1.lvc.edu /~lyons/java/secant/secant.html (443 words)

	Secant Lines/TITLE> (Site not responding. Last check: 2007-10-11)
		In addition to specifying the two points on a curve for the secant line, you may specify that a number of secant lines be drawn.
		10 secant lines 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.
	www.emba.uvm.edu /~burgmeie/epic/secant1.html (205 words)

	Slope of a Curve via Secant Lines (Site not responding. Last check: 2007-10-11)
		Goal: Define the slope of a curve at a point on the curve by means of the slopes of secant lines.
		Recall the definitions of tangent to a circle and secant of a circle.
		is defined to be the slope of the line tangent to the curve at that point.
	home.xnet.com /~fidler/triton/math/review/mat131/tangent/secant.htm (289 words)

	SparkNotes: Circles: Tangent Lines and Secant Lines
		A tangent line is a line that intersects a circle at one point.
		The point at which the circle and the line intersect is the point of tangency.
		A secant line is a line that intersects a circle at two points.
	www.sparknotes.com /math/geometry1/circles/section3.rhtml (227 words)

	Tangent Line Applet (Site not responding. Last check: 2007-10-11)
		The red line in the following applet is the tangent line to the graph.
		The tangent line is the limit of secant lines that pass through two points of the function's graph.
		Notice that the secant line approaches the tangent line as you move the points closer together.
	www.unc.edu /~upshaw/Java/TanSec.html (241 words)

	secant_approximation.html
		The tangent line at (0,1) is the horizontal line y=1.
		We said a moment ago that a tangent line 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 tangent lines, and how they are found using an approximation by secant lines, you are in a good position to approach the derivative.
	www.ms.uky.edu /~cvl/General_Teaching/secant_approximation1.html (1414 words)

	The meaning of the derivative - An approach to calculus
		The slope of a tangent line to a curve
		The slope of the secant is the average rate of change between those two points.
		Therefore, the slope of the secant line PQ is
	www.themathpage.com /aCalc/derivative.htm (1459 words)

	Definition of the Derivative
		Definition of the Slope of the Secant line
		Since the derivative is the limiting slope of the secant line, 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 secant line will be -1 to the left of 0 and 1 to the right of 0.
	www.ltcconline.net /greenl/courses/115/differentiation/derdef.htm (247 words)

	Secant Line -- from Wolfram MathWorld
		A secant line, also simply called a secant, is a line passing through two points of a curve.
		In geometry, a secant line commonly refers to a line that intersects a circle at exactly two points (Rhoad et al.
		There are a number of interesting theorems related to secant lines.
	mathworld.wolfram.com /SecantLine.html (153 words)

	Secant Lines and the Slope of a Curve (Site not responding. Last check: 2007-10-11)
		Secant Lines and the Slope of a Curve
		As b approaches a, the slope of the secant line approaches the slope of the line tangent 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.
	www.math.psu.edu /dlittle/java/calculus/secantLines.html (185 words)

	John Weber - GPC - Math 2431 - Section 2.1 Notes
		The concept of tangent lines and velocity are important for the concept of the derivative (section 2.7).
		Observe the calculated values of the slopes of the secant lines.
		, the slope of the secant line approaches the slope of the tangent line!
	www.gpc.edu /~jweber/math2431/sect0201notes.xml (429 words)

	The Secant Method (Site not responding. Last check: 2007-10-11)
		, is taken to be the x-intercept of the approximating secant line.
		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.
	www.mathcs.emory.edu /ccs/ccs315/ccs315/node18.html (290 words)

	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.
	www.jco-online.com /home.asp?page=article&Year=2001&Month=02&ArticleNum=106 (1449 words)

	Tangent Parabola
		The limit of "the secant line" is "the tangent line." We recast this in the context of polynomial approximation.
		, and the tangent line as the Taylor polynomial
		This cinches the fact that the limit of the secant polynomial is the tangent polynomial.
	math.fullerton.edu /mathews/n2003/TangentParabolaMod.html (946 words)

	Secant Method (Site not responding. Last check: 2007-10-11)
		which is the intersection of the secant line 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 secant line at the current points is drawn, the next approximate solution is taken to be the intersection of the secant line with the x axis, and the process is then repeated.
	www.cse.uiuc.edu /eot/modules/nonlinear_eqns/Secant (226 words)

	Tangent lines (Site not responding. Last check: 2007-10-11)
		The derivative is defined as a limit of slopes of secant lines.
		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 tangent line.
		If you select the ceheckbox to show the slope, the slope of the line is displayed on the screen as the nodes move.
	www.calonline.msstate.edu /tools/secant-tool.html (350 words)

	1426lab1java (Site not responding. Last check: 2007-10-11)
		The applet on this page lets you investigate how the slope of a secant line 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 secant line 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 (secant line) joining them, and the slope of that line.
	www.uta.edu /math/Calculus/secant.html (405 words)

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