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Topic: Difference quotient

	Summary: Introduction to the Derivative
		The balanced difference quotient for that value of h is the slope of the (secant) line passing through the end-points.
		This slope is the difference quotient for h = 0.0001.
		(A) The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives.
	www.zweigmedia.com /ThirdEdSite/Calcsumm3.html (2022 words)

	Numerical Analysis of Applied Partial Differential Equations
		The forward difference is the difference equation that would be used on the left boundary, when the function proceeds to the right.
		The backward difference would be used to evaluate functions at the opposite boundary, where the function would have no reference outside of this boundary.
		From this point, the difference approximations may be used to supply the replacement equations, which in turn are evaluated to reveal the unknowns.
	www.geocities.com /b_ward.rm/na.html (1809 words)

	Derivative -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an (The difference in pitch between two notes) interval if it is differentiable at every x within the interval.
		This expression is (A unit of force equal to the force that imparts an acceleration of 1 m/sec/sec to a mass of 1 kilogram; equal to 100,000 dynes) Newton's difference quotient.
		Constant rule: The derivative of any (A number representing a quantity assumed to have a fixed value in a specified mathematical context) constant is (The quantity that registers a reading of zero on a scale) zero.
	www.absoluteastronomy.com /encyclopedia/d/de/derivative.htm (2895 words)

	Quotient rule - Wikipedia, the free encyclopedia
		In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.
		This proof, of course, is just another, more systematic (even if outmoded) way of proving the theorem in terms of limits, and so is equivalent to the first proof above - and even reduces to it, if you make the right substitutions in the right places.
		Another variation to this mnemonic is given when the quotient is written with the numerator as Hi the denominator as Ho: "Ho-dee-Hi minus Hi-dee-Ho over Ho-Ho."
	en.wikipedia.org /wiki/Quotient_rule (443 words)

	Derivative - FreeEncyclopedia (Site not responding. Last check: 2007-10-08)
		Derivatives are defined by taking the limit of a secant slope, as its two points of intersection (with f(x)) converge; the secant approaches a tangent.
		This is expressed by Newton's difference quotient; where h is Δx (the distance between the x-coordinates of the secant's points of intersection):
		Should the reader have an understanding of algebra and the Cartesian coordinate system, the reader should be able to independently determine that this line has a slope of 2 at every point.
	openproxy.ath.cx /de/Derivative.html (1600 words)

	Untitled (Site not responding. Last check: 2007-10-08)
		Given the function, y = f(x), The difference quotient measures the rate of change in the exogenous variable (y) to an endogenous variable (x).
		The derivative is a measure of the instantaneous rate of change, which depends on where the function is evaluated, not the average as with the difference quotient.
		To calculate the derivative using the difference quotient, plug in /\x = 0.
	core.ecu.edu /econ/whiteheadj/5360/ch6.htm (363 words)

	Quaternion Analysis (Site not responding. Last check: 2007-10-08)
		Because of the non-commutativity of quaternions, the definition of quaternion Taylor series is unclear, and the difference quotient is completely useless.
		Since the quaternions possess a multiplicative inverse, one is compelled to try out the difference quotient that works so well with functions of real and complex numbers.
		The requirement that a difference quotient should exist in the limit is too strong to be useful for the quaternions.
	www.zipcon.net /~swhite/docs/math/quaternions/quaternion_analysis.html (467 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		Suppose you are asked to find the equation of the line tangent to a curve f(x) = x to the power of three minus seven x to the power of 2 plus twelve x plus 2 at the point (3,2) without using calculus.
		Describe how you would use the difference quotient function to find this equation, and then find the equation of the tangent line.
		What he was saying is that the slope of the tangent line to f at (3,2) is -3, so you need to find the equation of the one and only straight line with slope -3 which goes through the point (x,y) = (3,2).
	mathforum.org /library/drmath/view/54446.html (433 words)

	RCMP
		Difference Between the Difference Quotient and the Continuum Derivative
		difference between numerical forward difference quotient and the continuum derivative
		Case 1b differs from 1a in that the measured angle is in degrees.
	chemistry.mtu.edu /~jdbailey/ma.2150/rcmp.1.rev.h.html (1185 words)

	secant_approximation.html
		Here's why: The difference quotient plays a major role in understanding the first derivative of a function (which is one of the most important tools in modern mathematics).
		The difference quotient is a fundamental piece of the derivative.
		What we've found is that the derivative of f, evaluated at x=a, can be approximated by the difference quotient and, as we approximate the tangent line better and better - as we take smaller and smaller values of h, we get closer and closer to the slope of the tangent line.
	www.ms.uky.edu /~cvl/General_Teaching/secant_approximation1.html (1414 words)

	Comments for Chapter 12 (Site not responding. Last check: 2007-10-08)
		It's the same old difference quotient that was presented in Calculus I with the same old limit taken as h or
		If we find that this has a limit as h =>0, we define the resulting value as "The derivative of f(x) with respect to x at the point where x = a." or some similar statement, depending on the state of the liver of whomever is doing the writing.
		The difference quotient also goes to 0 for the irrationals, so the DQ goes to 0 for all x.
	www.ccaurora.edu /mat203/comments_for_chapter_12.htm (1932 words)

	DifferenceQuotient
		The difference quotient at x for a function f is given by:
		The difference quotient gives the slope of the "secant line", the line shown in red in the
		Part a) showed that the difference quotient for this function simplifies to
	cis.stvincent.edu /carlsond/ma109/diffquot.html (141 words)

	Articles - Calculus (Site not responding. Last check: 2007-10-08)
		This is avoided, however, because the quotient Distance/Time is not used for a single instant (as in a still photograph).
		Rather a formula is developed for the quotient Distance/Time in which division by zero can be avoided, by a method called "taking the limit".
		The derivative is defined as a limit of a difference quotient.
	www.anfolk.com /articles/Calculus (1653 words)

	Labor Economics (Site not responding. Last check: 2007-10-08)
		The difference quotient – Shows how the value of the function changes with respect to a change in its argument (Dy/Dx)
		Difference quotient – The difference quotient for the function y = f(x) for an initial value x
		The difference quotient for any linear function y = a + bx is
	garnet.acns.fsu.edu /~fdv3471/math06.html (641 words)

	l2d1
		The derivative is defined as the limit as h approaches zero of the difference quotient, which in mathematical notation is,
		Since the difference quotient can be interpreted as the slope of a secant line joining the two points, (x, f(x)) and (x+h, f(x+h)), the limit as h approaches zero is the limiting value of the slope as the moving point, (x+h, f(x+h)) approaches the fixed point, (x, f(x)).
		Alternatively, if we are only interested in finding the value of the derivative at a particular point, we can substitute the x value at that point for the symbol x in the difference quotient and then find the limit.
	www.austin.cc.tx.us /rgrmth/calc1/l2d1.htm (314 words)

	Ball Grid Array Example bga-4 for the Surface Evolver (Site not responding. Last check: 2007-10-08)
		The first method is a one-way difference quotient moving only the top pad, by changing the value of "height".
		The third method is a one-way difference quotient, moving the top by changing "height" and moving the other surface vertices by linear interpolation in z.
		As a rule, the maximum accuracy of single difference is half the digits of machine precision (dz being the other half of digits), and for central difference it is two-thirds the digits (one-third being used in dz).
	www.susqu.edu /brakke/evolver/bga/bga-4.htm (1174 words)

	Guardian Unlimited \| Science \| The Essential Difference front page
		Systemising is the drive to analyse and explore a system, to extract underlying rules that govern the behaviour of a system; and the drive to construct systems.
		The empathy quotient is intended to measure how easily you pick up on other people's feelings and also how strongly you are affected by other people's feelings.
		The systemising quotient gives a score based on how interested you assess yourself to be in the operateion of systems and processes.
	www.guardian.co.uk /life/news/page/0,12983,937443,00.html (619 words)

	MATH 170 008 - Review Notes for Test #1 -- 9/9/04
		There are limits which are not obvious difference quotients (for instance 2.3: 11, 13, 15, 29 -- see assignment #6).
		In the amended 4a and 6a, many wrote down the relevant difference quotient correctly, but did not offer a graph of it.
		In problem 4, one cannot simplify the difference quotient by squaring it.
	math.boisestate.edu /~kerr/170fa04/review_suggestions_1.html (1223 words)

	Derivative of the Sine Function
		Let us first attempt to simplify the difference quotient for the sine function.
		The very first expression that we encounter in the difference quotient is the quantity sin(x+h).
		We can now write down the difference quotient and follow our instincts, gathering like terms where they appear.
	oregonstate.edu /instruct/mth251/cq/Stage6/Lesson/sine.html (358 words)

	C++ Programming : Program 2-A
		The sum, difference, product and quotient of these two values are calculated and then outputted using the 'cout' command.
		The difference of 10 & 2 is 8.
		The quotient of 10 & 2 is 5.
	www.syvum.com /cgi/online/serve.cgi/squizzes/cpp/cpp_p2a.html?0 (149 words)

	Numbers
		Feb 21st 2005 03:26 AM A program that asks the user to enter two numbers, obtains the two numbers from the user and prints the sum,product,difference, and quotient of the two numbers.
		" << quotient << endl; cout << "
		The program will then calculate the sum, the product, the difference, and the quotient of the two numbers that have been input.
	www.daniweb.com /code/snippet199.html (296 words)

	MATH 170 001 - Review Notes for Test #1 -- 2/11/05 (Site not responding. Last check: 2007-10-08)
		Graph a difference quotient and use the graph to determine m
		How a tangent-line equation relates to a difference quotient.
		There are limits which are not obvious difference quotients (for instance 2.3: 11, 13, 15, 29)
	math.boisestate.edu /~kerr/170sp05/review_suggestions_1.html (322 words)

	Bibliography for On-Line Encyclopedia of Integer Sequences (Site not responding. Last check: 2007-10-08)
		Jordan, {\em Calculus of Finite Differences}, Budapest, 1939.
		Levy and F. Lessman, {\em Finite Difference Equations}, Pitman, London, 1959.
		Th\'{e}or\^{e}t, ``Fonctions g\'{e}n\'{e}ratrices pour une classe d'\'{e}quations aux diff\'{e}rences partielles,'' {\em Ann.
	www.research.att.com /~njas/sequences/eisbib.html (7328 words)

	[No title]
		Also be able to construct a sketch in which the important values are clearly labeled.
		Be able to write the difference quotient for various functions using the form above.
		Be able to simplify the difference quotient for polynomial functions and functions such as EMBED Equation.DSMT4 and EMBED Equation.DSMT4 .
	course1.winona.edu /jquella/M160T1_Guide.doc (952 words)

	Differentiation (Site not responding. Last check: 2007-10-08)
		, is the limit of the difference quotient, just like f(x) in the reals.
		Multiply f(z) by a constant c and the difference quotient is multiplied by c.
		Those who wish to be rigorous may want to review the theorems on limits, products, and quotients.
	www.mathreference.com /cx,deriv.html (301 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		What is the difference between the forward and backward difference quotients?
		What is the error in estimating the derivative by the difference quotient?
		What is the two-sided estimate (or balanced difference quotient estimate) for the derivative?
	bradley.bradley.edu /~delgado/202/Review2f00.txt (483 words)

	2-5 (Site not responding. Last check: 2007-10-08)
		Given two points P and Q on a curve, describe the ``secant line'' through those two points; i.e., what points does the secant line go through, what is the slope of the secant line, etc.
		Explain the difference between a secant line and a tangent line.
		What is the difference between ``average'' velocity and ``instantaneous'' velocity?
	www.sonoma.edu /users/f/fordb/M161F00/ReadingOutlines/2-5/2-5.html (166 words)

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