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	Visual Calculus - Linear Approximations
		to visualize the tangent line as an approximation to the graph of the function; and
		We know that the tangent line is an approximation to f near x = a.
		The problem that we want to consider is how close to a is g a good approximation to f.
	archives.math.utk.edu /visual.calculus/2/linear_app.6/index.html (169 words)

	NationMaster - Encyclopedia: Linear approximation (Site not responding. Last check: )
		Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself.
		The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.
		Linear function approximation based on coarse coding and (8.3) was used to learn a one-dimensional square-wave function (shown at the top of Figure 8.4).
	www.nationmaster.com /encyclopedia/Linear-approximation (305 words)

	Approximation
		Each time a newer set of laws is proposed, it is required that in the limiting situations in which the older set of laws were tested against experiments, the newer laws are nearly identical to the older laws, to within the measurement uncertainties of the older measurements.
		Numerical approximations are sometimes from using a small sample of significant digits.
		Approximation theory is a branch of mathematics, a quantitative part of functional analysis.
	www.xasa.com /wiki/en/wikipedia/a/ap/approximation.html (225 words)

	Linear Approximation
		Tangent Planes and Linear Approximation in 2 Variables One of the key concepts of calculus is that of linear approximation and local linearity.
		Linear Approximation and Newton's Method Linear Approximation Suppose that is a differentiable...
		Linear interpolation is often used to approximate a value of some...
	electronic.rssg.info /Linear-approximation.php (610 words)

	Matches for: Author/Editor=(Sard_Arthur) (Site not responding. Last check: )
		Many approximations are linear, that is, conform to the principle of super-position, and may profitably be studied by means of the theory of linear spaces.
		"Linear approximation" sets forth the pertinent parts of that theory, with particular attention to the key spaces $C_n, B, K$, and Hilbert space, spaces which are powerful tools in the analysis, appraisal, and design of approximations, ranging from formulas of mechanical quadrature to approximations of operators by operators.
		Because it affords a detailed treatment of a timely and important subject, "Linear approximation" is of interest to scientists and engineers as well as to mathematicians.
	www.mathaware.org /bookstore?arg9=Arthur_Sard&fn=100&l=20&pg1=CN&r=1&s1=Sard%5FArthur (115 words)

	Linear Approximation of Functions
		Linear approximation is an example of how differentiation is used to approximate functions by linear ones close to a given point.
		This approximation may be used to linearize non algebraic functions such as sine, cosine, log, exponential and many other functions in order to make their computation easier.
		Linear approximation is one of the simplest approximations to transcendental functions that cannot be expressed algebraically.
	www.analyzemath.com /calculus/applications/linear_approximations.html (378 words)

	Springer Online Reference Works
		A whole series of exact and asymptotically-exact results is known for the approximation of function classes by concrete linear methods (in particular, by polynomial or spline methods), see [1]–[12],.
		When (1) and (3) coincide and when it is possible to construct a concrete linear method solving both problems, the cases considered are, in a well-known sense, optimal.
		Exact constants in the Jackson inequalities for trigonometric approximation in the uniform and in an integral metric are known (cf.
	eom.springer.de /a/a012970.htm (1479 words)

	8.3 Linear Methods
		In this case the approximate value function is said to be linear in the parameters, or simply linear.
		Assuming linear gradient-descent function approximation, consider the effect of the size and density of the circles.
		Linear function approximation based on coarse coding and (8.3) was used to learn a one-dimensional square-wave function (shown at the top of Figure 8.4).
	www.cs.ualberta.ca /~sutton/book/ebook/node88.html (3009 words)

	FRB: Finance and Economics Discussion Series: Screen Reader Version - Two Pitfalls of Linearization Methods*
		A related phenomenon is a wide usage of a linear quadratic (LQ) framework in analyzing optimal policy problems, since the combination of a quadratic criterion function with a linear constraint is well known to yield a linear behavior for the optimal solution.
		In this case, since the nonlinear optimality conditions are a correct representation of the original model, a linear approximation of derived optimality conditions would yield a correct linear representation of the original model.
		The reason why this approach works is that, though (10) is not any kind of approximation of (7) in general, the two expressions are equivalent up to the second order for all outcomes that are consistent with the constraint.
	www.federalreserve.gov /pubs/feds/2007/200764/index.html (2372 words)

	Calculus Online: Lab 6
		In the last two problems, we have found polynomials which approximate a given function by equating a certain number of derivatives of the function and the polynomial at some point.
		For instance, in Question 1, we formed the linear approximation by equating the value of the function and its first derivative to those same quantities in a linear function.
		Then in Question 2, we formed a quadratic approximation by equating the value of a function, its derivative and its second derivative with the corresponding quantities in a quadratic polynomial.
	www.ugrad.math.ubc.ca /coursedoc/math100/labs/lab6/lab6.html (1222 words)

	Linear Approximation and Error Estimation
		The values of the function are close to the values of the linear function whose graph is the tangent line.
		For this reason, the linear function whose graph is the tangent line to y = f(x) at a specified point (a, f(a)) is called the linear approximation of f(x) near x = a.
		Thus, the the linear approximation to f(x) near x = a is given by
	www.zweigmedia.com /ThirdEdSite/calctopic1/linearapprox.html (835 words)

	Linear least squares approximation - ALGLIB
		It should be noted that such an approximation method (linear combination of basis functions) and such an estimate function (sum of squared deviations) is not the only one which someone can choose.
		Linear combinations of the functions of this basis allow us to get an arbitrary line on a plane.
		For simplicity purposes, we assume that the approximation is built for [0,M-1] with an equidistant grid.
	www.alglib.net /interpolation/linearleastsquares.php (1451 words)

	Approximation Algorithms
		Approximation Algorithms for Bin Packing: A Survey, E. Coffman, Jr., M. Garey, and D. Johnson, Approximation Algorithms for NP-Hard Problems, D. Hochbaum (editor), PWS Publishing, Boston (1997), 46-93.
		Linear Approximation of Shortest Superstrings, by Avrim Blum, Tao Jiang, Ming Li, John Tromp, and Mihalis Yannakakis in STOC'91 and Journal of ACM, 41:4 (1994), 630-647.
		On the approximation of shortest common supersequences and longest common subsequences.
	www.csd.uwo.ca /~bma/teaching/approx/resources.html (274 words)

	linear_approximation - The Wordbook Encyclopedia
		In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
		The linear approximation is obtained by dropping the remainder:
		One can also use linear approximations for vector functions of a vector variable, in which case the derivative at a point is replaced by the Jacobian matrix.
	www.thewordbook.com /linear_approximation (282 words)

	Linear Approximation
		Linear Approximation is the simplest way of approximating the value of a function by using a bare minmum of conditions.
		The method depends upon using the value (f(a)) and rate of change (derivative) at the point x = a to approximate value of f(c), where c is in close proximity of a.
		Graphically, this method approximates the value of the function at a point by replacing the curve by a tangent to it at a nearby point.
	www.math.metrostate.edu /welcome/microworlds/linapprox.html (109 words)

	Calculus I Notes, Section 2-9
		Here is a view of the cube root function along with the linear approximation over the interval from (1,15).
		It should be noted that this approximation technique is only good "near" to the value for x which you use to calculate the slope of the tangent line (calculate the derivative).
		#2: True or False: When you calculate a value of a function using a linear approximation, the proximity of point of the function that you use and the point you are trying to estimate is extremely important.
	www.blc.edu /fac/rbuelow/calc/nt2-9.html (535 words)

	Search Results for "approximation"
		...The branch of mathematics concerned with the approximation of periodic functions by the Fourier series and with generalizations of such approximations to a wider...
		Mathematics A method of solving equations in which the errors resulting from an initial approximation are reduced by succeeding approximations until all errors...
		...two boards are of equal length, we assume that the equality is reckoned to some order of approximation determined by the context; if we did not, we would be required...
	www.bartleby.com /cgi-bin/texis/webinator/sitesearch?FILTER=col61&x=10&y=12&query=approximation (349 words)

	Astron. Astrophys. 327, 813-824 (1997)
		The reason for this discrepancy is in the linear approximation of expression for the magnetic tension.
		Then the vertical component of the magnetic tension is diminished in the nonlinear regime with respect to the linear approximation.
		This term is underestimated in the linear approximation.
	aa.springer.de /papers/7327002/2300813/sc3.htm (742 words)

	Linear Approximation Applet
		One of the key concepts of calculus is that of linear approximation and local linearity.
		One measure of the goodness of an approximation is the size of the region where the error is small enough.
		A second measure of the goodness of an approximation is the size of the region where the ratio of the error and dely is small.
	www.slu.edu /classes/maymk/banchoff/LocalLinearity.html (353 words)

	Linear Approximation Of Random Processes And Sampling Design Problems (ResearchIndex)
		Abstract: Linear approximation of random processes is considered as a three-layers problem: for a single process; for a fixed method, an optimization of a sample points design; for a class of random processes, the best approximation order.
		The close relationship between the smoothness properties of a function and the best rate of its linear approximation is one of the basic ideas of conventional (deterministic) approximation theory.
		4 Large deviations in the piecewise linear approximation of Ga..
	citeseer.ist.psu.edu /seleznjev99linear.html (647 words)

	LMS Proceedings Abstract, paper PLMS 1408 (Site not responding. Last check: )
		The aim of this paper is to provide an analysis of non-linear approximation in the $L_p$-norm $p = d / (d - 1)$ of functions of bounded variation on $\mathbb{R}^d$ with $d > 1$ by polynomials in the Haar system.
		The approximation schemes that we discuss in this paper are mostly related to Haar thresholding and $m$-term approximation.
		The main aim of this paper is to extend their results to the case $d \geq 2$.
	www.lms.ac.uk /publications/proceedings/abstracts/p1408a.html (166 words)

	Linear Approximation
		The derivative supports a quick and effective means for approximating the values of complicated functions.
		To construct the best linear approximation of sin(x) near 0, we compute f(a) and f '(a).
		Of course, the graph of the linear function L
	oregonstate.edu /instruct/mth251/cq/Stage9/Lesson/linearApprox1.html (352 words)

	Springer Online Reference Works
		In the theory of linear approximation methods, special attention is given to problems of convergence.
		Interpolation splines with a specific choice of the knots (nodes, joints) ensure the best order of approximation not only for the function itself, but also for some of its derivatives, depending on the degree of the polynomials out of which the spline is composed (see [7], [8]).
		In some particular cases, for specific linear approximation methods exact or asymptotically-exact estimates for the errors of approximation were obtained for some classes of functions.
	eom.springer.de /a/a012980.htm (1047 words)

	Linear Filters (Neural Network Toolbox) (Site not responding. Last check: )
		Single-layer linear networks can be designed directly or trained with the Widrow-Hoff rule to find a minimum error solution.
		Multiple layers in a linear network do not result in a more powerful network, so the single layer is not a limitation.
		If the relationship between inputs and targets is linear or a linear approximation is desired, then linear networks are made for the job.
	www.weizmann.ac.il /matlab/toolbox/nnet/linfil13.html (171 words)

	Linear Approximation and Error Estimation
		The values of the function are close to the values of the linear function whose graph is the tangent line.
		For this reason, the linear function whose graph is the tangent line to y = f(x) at a specified point (a, f(a)) is called the linear approximation of f(x) near x = a.
		Thus, the the linear approximation to f(x) near x = a is given by
	people.hofstra.edu /faculty/Stefan_Waner/RealWorld/calctopic1/linearapprox.html (835 words)

	Linear Approximations and L'Hopital's Rule
		Our first three examples were intended to familiarize you with the technique and to give you a feel for how good (or bad) linear approximations tend to be.
		We show a graphical interpretation of this in Figure 3.6, where the straight line is the linear approximation (in this case, the secant line joining the first two data points).
		In this case, the graph in Figure 3.8 indicates the limit to be approximately 0.5.
	euphrates.wpunj.edu /courses/maen507/Week05/section01.htm (1643 words)

	Piecewise Linear Approximation
		Such a local approximation is obtained by fitting a regression surface to the data...
		The concept is familiar from elementary calculus where a smooth function is approximated by a piecewise linear approximation.
		To draw a piecewise linear approximation, use the low frequency asymptote up to the break frequency, and the high frequency asymptote thereafter.
	electronic.rssg.info /Piecewise-linear-approximation.php (253 words)

	Linear Approximation
		Linear Approximation is the simplest way of approximating the value of a function by using a bare minmum of conditions.
		The method depends upon using the value (f(a)) and rate of change (derivative) at the point x = a to approximate value of f(c), where c is in close proximity of a.
		Graphically, this method approximates the value of the function at a point by replacing the curve by a tangent to it at a nearby point.
	www.maa.org /projectwelcome/microworlds/linapprox.html (109 words)

	1_2_3a_update
		Approximation theory is concerned with using one function to approximate another, usually to save effort in using the function.
		One characteristic of least squares approximations is that the error tends to be larger near the ends of the intervals.
		The primary spline functions in use in approximation are the linear and cubic splines.
	il.water.usgs.gov /proj/feq/fequtl98.i2h/1_2_3a_update.html (5508 words)

	Calculus I (Math 2413) - Applications of Derivatives - Linear Approximations (Site not responding. Last check: )
		Now, the approximations are nothing more than plugging these into the linear approximation.
		So, at x=8.05 this linear approximation does a very good job of approximating the actual value. However, at x=25 it doesn’t do such a good job.
		This is actually a somewhat important linear approximation. In optics this linear approximation is often used to simplify formulas. This linear approximation is also used to help describe the motion of a pendulum and vibrations in a string.
	tutorial.math.lamar.edu /AllBrowsers/2413/LinearApproximations.asp (370 words)

	World Web Math: Vector Calculus: Linear Approximations
		provides an approximate value for the change in f and is called the linear approximation formula for f at x=c.
		The definition for linear approximation for a multi-variable function will be (not surprisingly) similar to that of one variable.
		One must note that it is possible for f to have first-order partial derivatives at every point, but not to have linear approximations at a certain point P in its domain.
	web.mit.edu /wwmath/vectorc/scalar/linapprox.html (420 words)

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